File "markSceme-HL-paper1.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AA/Topic 5/markSceme-HL-paper1html
File size: 863.78 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-02ef852527079acf252dc4c9b2922c93db8fde2b6bff7c3c7f657634ae024ff1.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-9717ccaf4d6f9e8b66ebc0e8784b3061d3f70414d8c920e3eeab2c58fdb8b7c9.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../../../../../../index.html">Home</a>
</li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li class="dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Help
<b class="caret"></b>
</a>
<ul class="dropdown-menu">
<li><a href="https://questionbank.ibo.org/video_tour?locale=en">Video tour</a></li>
<li><a href="https://questionbank.ibo.org/instructions?locale=en">Detailed instructions</a></li>
<li><a target="_blank" href="https://ibanswers.ibo.org/">IB Answers</a></li>
</ul>
</li>
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
<div class="pull-right screen_only"><a class="btn btn-small btn-info" href="https://questionbank.ibo.org/updates?locale=en">Updates to Questionbank</a></div>
<p class="muted language_chooser">
User interface language:
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=en">English</a>
|
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=es">Español</a>
</p>
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="https://mirror.ibdocs.top/qb.png" alt="Ib qb 46 logo">
</div>
</div>
</div>
<h2>HL Paper 1</h2><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove by mathematical induction that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo>+</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the Maclaurin series of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math></p>
<p>LHS: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>+</mo><mn>2</mn><mi>x</mi><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><mo>=</mo><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>RHS: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mn>1</mn></mfenced><mi>x</mi><mo>+</mo><mn>1</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><mo>=</mo><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math></p>
<p>now assume true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>; i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>k</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>k</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Do not award <em><strong>M1</strong></em> for statements such as "let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>". Subsequent marks can still be awarded.</p>
<p><br>attempt to differentiate the RHS <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mo>d</mo><msup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>k</mi></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>+</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>x</mi><mo>+</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>A1</strong></em></p>
<p>so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> implies true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> true and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> true <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> true</p>
<p>therefore, true for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>R1</strong></em> only if three of the previous four marks have been awarded</p>
<p> </p>
<p><em><strong>[7</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo>+</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><msub><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><menclose notation="left"><mi>x</mi><mo>=</mo><mn>0</mn></menclose></msub><mo>=</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> may be seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mo>'''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>6</mn><mo>,</mo><mo> </mo><mo> </mo><msup><mi>f</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>12</mn></math></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mi>f</mi><mo>'''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><msup><mi>f</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≈</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>'<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>×</mo></math> Maclaurin series of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo> </mo></math>' <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≈</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>≈</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac><mo>≈</mo><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>9</mn></msup><mfenced><mrow><mo>+</mo><mi>higher</mi><mo> </mo><mi>order</mi><mo> </mo><mi>terms</mi></mrow></mfenced></mrow><msup><mi>x</mi><mn>9</mn></msup></mfrac></math></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow></mfenced><mn>3</mn></msup></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>+</mo><mo> </mo><mi>higher</mi><mo> </mo><mi>order</mi><mo> </mo><mi>terms</mi></mrow></mfenced></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></mfenced><mn>3</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mfenced><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow><mi>x</mi></mfrac></mfenced><mn>3</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>attempt to use L'Hôpital's rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mfenced><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><mn>0</mn></mrow><mn>1</mn></mfrac></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced open="[" close="]"><mrow><munder><mi>lim</mi><mrow><mi mathvariant="normal">x</mi><mo>→</mo><mn>0</mn></mrow></munder><mo> </mo><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mrow></mfenced><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>4</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mn>2</mn>
<mi>n</mi>
</msup>
</mrow>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></span> is an odd or even function, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using mathematical induction, prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mfrac> <mrow> <mi>m</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{Z}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find an expression for the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 1"> <mi>n</mi> <mo>></mo> <mn>1</mn> </math></span>, the equation of the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>even function <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos kx = \cos ( - kx)"> <mi>cos</mi> <mo></mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong>and</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is a product of even functions <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>even function <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\cos 2x)(\cos 4x) \ldots = \left( {\cos ( - 2x)} \right)\left( {\cos ( - 4x)} \right) \ldots "> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>…</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>…</mo> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>A0R1</em></strong>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider the case <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin 4x}}{{2\sin 2x}} = \frac{{2\sin 2x\cos 2x}}{{2\sin 2x}} = \cos 2x"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p>hence true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>R1</em></strong></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>, <em>ie</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(\cos 2x)(\cos 4x) \ldots (\cos {2^k}x) = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>M1 </em></strong>for “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>” or “assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span>” or equivalent.</p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span>:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{k + 1}}(x) = {f_k}(x)(\cos {2^{k + 1}}x)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>k</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sin {2^{k + 1}}x}}{{{2^k}\sin 2x}}\cos {2^{k + 1}}x"> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>k</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2\sin {2^{k + 1}}x\cos {2^{k + 1}}x}}{{{2^{k + 1}}\sin 2x}}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sin {2^{k + 2}}x}}{{{2^{k + 1}}\sin 2x}}"> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> true and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span> true <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow n = k + 1"> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span> true. Hence true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> To obtain the final <strong><em>R1</em></strong>, all the previous <strong><em>M </em></strong>marks must have been awarded.</p>
<p> </p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’ = \frac{{vu' - uv'}}{{{v^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mi>v</mi> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>−</mo> <mi>u</mi> <msup> <mi>v</mi> <mo>′</mo> </msup> </mrow> <mrow> <mrow> <msup> <mi>v</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> (or correct product rule) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}(x) = \frac{{({2^n}\sin 2x)({2^{n + 1}}\cos {2^{n + 1}}x) - (\sin {2^{n + 1}}x)({2^{n + 1}}\cos 2x)}}{{{{({2^n}\sin 2x)}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct numerator and <strong><em>A1 </em></strong>for correct denominator.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{\left( {{2^n}\sin \frac{\pi }{2}} \right)\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right) - \left( {\sin {2^{n + 1}}\frac{\pi }{4}} \right)\left( {{2^{n + 1}}\cos \frac{\pi }{2}} \right)}}{{{{\left( {{2^n}\sin \frac{\pi }{2}} \right)}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = \frac{{({2^n})\left( {{2^{n + 1}}\cos {2^{n + 1}}\frac{\pi }{4}} \right)}}{{{{({2^n})}^2}}}"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\cos {2^{n + 1}}\frac{\pi }{4}{\text{ }}( = 2\cos {2^{n - 1}}\pi )"> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>π</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f’_n}\left( {\frac{\pi }{4}} \right) = 2"> <mrow> <msubsup> <mi>f</mi> <mi>n</mi> <mo>′</mo> </msubsup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}\left( {\frac{\pi }{4}} \right) = 0"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> This <strong><em>A </em></strong>mark is independent from the previous marks.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\left( {x - \frac{\pi }{4}} \right)"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^x}\,{\text{cos}}{\,^2}x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<msup>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</math></span>, where 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5. The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is shown on the following graph which has local maximum points at A and C and touches the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at B and D.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use integration by parts to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x{\text{d}}x = } \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x + c"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <msup> <mspace width="thinmathspace"></mspace> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinates of A and of C , giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{arctan}}\,b"> <mi>a</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{R}"> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area enclosed by the curve and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis between B and D, as shaded on the diagram.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>attempt at integration by parts with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {{\text{e}}^x}"> <mi>u</mi> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = {\text{cos}}\,2x"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x = } \frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x\,{\text{d}}x - \int {\frac{{{{\text{e}}^x}}}{2}} {\text{sin}}\,2x\,{\text{d}}x"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x - \frac{1}{2}\left( { - \frac{{{{\text{e}}^x}}}{2}{\text{cos}}\,2x + \int {\frac{{{{\text{e}}^x}}}{2}} {\text{cos}}\,2x} \right)"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{4}{\text{cos}}\,2x - \frac{1}{4}\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} "> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore \frac{5}{4}\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = \frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x\, + \frac{{{{\text{e}}^x}}}{4}{\text{cos}}\,2x"> <mo>∴</mo> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x\left( { + c} \right)"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt at integration by parts with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{cos}}\,2x"> <mi>u</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = {{\text{e}}^x}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x = } {{\text{e}}^x}\,{\text{cos}}\,2x + 2\int {{{\text{e}}^x}\,{\text{sin}}\,2x\,{\text{d}}x} "> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^x}\,{\text{cos}}\,2x + 2\left( {{{\text{e}}^x}\,{\text{sin}}\,2x - 2\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} } \right)"> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x - 4\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} "> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore 5\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x"> <mo>∴</mo> <mn>5</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x\left( { + c} \right)"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>attempt at use of table <em><strong>M1</strong></em></p>
<p><em>eg</em></p>
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong> </em></p>
<p><strong>Note:</strong> <em><strong>A1</strong> </em>for first 2 lines correct, <em><strong>A1</strong> </em>for third line correct.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x = \,} {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x - 4\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} "> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mspace width="thinmathspace"></mspace> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore 5\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x"> <mo>∴</mo> <mn>5</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x\left( { + c} \right)"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{co}}{{\text{s}}^2}\,x{\text{d}}x = } \int {\frac{{{{\text{e}}^x}}}{2}} \left( {{\text{cos}}\,2x + 1} \right){\text{d}}x"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {\frac{{{\text{2}}{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x} \right) + \frac{{{{\text{e}}^x}}}{2}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>2</mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2}\left( { + c} \right)"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Do not accept solutions where the RHS is differentiated.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {{\text{e}}^x}\,{\text{co}}{{\text{s}}^{\text{2}}}\,x - 2{{\text{e}}^x}\,{\text{sin}}\,x\,{\text{cos}}\,x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mtext>2</mtext> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for an attempt at both the product rule and the chain rule.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,{\text{cos}}\,x\left( {{\text{cos}}\,x - 2\,{\text{sin}}\,x} \right) = 0"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to factorise <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{cos}}\,x}"> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </math></span> or divide by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x\left( {{\text{cos}}\,x \ne 0} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>discount <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = 0"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> (as this would also be a zero of the function)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{cos}}\,x - 2\,{\text{sin}}\,x = 0"> <mo stretchy="false">⇒</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{tan}}\,x = \frac{1}{2}"> <mo stretchy="false">⇒</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = {\text{arctan}}\left( {\frac{1}{2}} \right)"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (at A) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi + {\text{arctan}}\left( {\frac{1}{2}} \right)"> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (at C) <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct answer. If extra values are seen award <em><strong>A1A0</strong></em>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = 0 \Rightarrow x = \frac{\pi }{2}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\pi }}{2}"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> The <em><strong>A1</strong></em>may be awarded for work seen in part (c).</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {\left( {{{\text{e}}^x}\,{\text{co}}{{\text{s}}^{\text{2}}}\,x} \right)} \,{\text{d}}x = \left[ {\frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2}} \right]_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}}"> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mtext>2</mtext> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( { - \frac{{{{\text{e}}^{\frac{{3\pi }}{2}}}}}{{10}} + \frac{{{{\text{e}}^{\frac{{3\pi }}{2}}}}}{2}} \right) - \left( { - \frac{{{{\text{e}}^{\frac{\pi }{2}}}}}{{10}} + \frac{{{{\text{e}}^{\frac{\pi }{2}}}}}{2}} \right)\left( { = \frac{{{\text{2}}{{\text{e}}^{\frac{{3\pi }}{2}}}}}{5} - \frac{{{\text{2}}{{\text{e}}^{\frac{\pi }{2}}}}}{5}} \right)"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>2</mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <mtext>2</mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1(A1</strong><strong>)</strong><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for substitution of the end points and subtracting, <em><strong>(A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,3\pi = {\text{sin}}\,\pi = 0"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mi>π</mi> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,3\pi = {\text{cos}}\,\pi = - 1"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mi>π</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>π</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> and <em><strong>A1</strong></em> for a completely correct answer.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = 1 + {\text{sin}}\,2x"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sec}}\,2x + {\text{tan}}\,2x = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}"> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{\pi }{6}} {\left( {{\text{sec}}\,2x + {\text{tan}}\,2x} \right)} {\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {a + \sqrt b } \right)"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <msqrt> <mi>b</mi> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{Z}"> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = {\text{si}}{{\text{n}}^2}\,x + 2{\text{sin}}\,x\,{\text{cos}}\,x + {\text{co}}{{\text{s}}^2}\,x"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Do not award the <em><strong>M1</strong></em> for just <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^2}\,x + {\text{co}}{{\text{s}}^2}\,x"> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<p><strong><span style="background-color: #ffffff;">Note: </span></strong><span style="background-color: #ffffff;">Do not award <em><strong>A1</strong> </em>if correct expression is followed by incorrect working.</span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 + {\text{sin}}\,2x"> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sec}}\,2x + {\text{tan}}\,2x = \frac{1}{{{\text{cos}}\,2x}} + \frac{{{\text{sin}}\,2x}}{{{\text{cos}}\,2x}}"> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong></em> is for an attempt to change both terms into sine and cosine forms (with the same argument) or both terms into functions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,x"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{1}} + {\text{sin}}\,2x}}{{{\text{cos}}\,2x}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>1</mtext> </mrow> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)}^2}}}{{{\text{co}}{{\text{s}}^2}\,x - {\text{si}}{{\text{n}}^2}\,x}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for numerator, <em><strong>A1</strong></em> for denominator.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)}^2}}}{{\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Apply MS in reverse if candidates have worked from RHS to LHS.</p>
<p><strong>Note:</strong> Alternative method using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2x"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sec}}\,2x"> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,x"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{\pi }{6}} {\left( {\frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}} \right)} {\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct expression with or without limits.</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - {\text{ln}}\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)} \right]_0^{\frac{\pi }{6}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{ln}}\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)} \right]_{\frac{\pi }{6}}^0"> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mn>0</mn> </msubsup> </math></span> <em><strong>(M1)</strong><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for integration by inspection or substitution, <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{ln}}\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)}"> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math></span>, <em><strong>A1</strong></em> for completely correct expression including limits.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ln}}\left( {{\text{cos}}\,\frac{\pi }{6} - {\text{sin}}\,\frac{\pi }{6}} \right) + {\text{ln}}\left( {{\text{cos}}\,0 - {\text{sin}}\,0} \right)"> <mo>=</mo> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for substitution of limits into their integral and subtraction.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ln}}\left( {\frac{{\sqrt 3 }}{2} - \frac{1}{2}} \right)"> <mo>=</mo> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{cos}}\,x - {\text{sin}}\,x"> <mi>u</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = - {\text{sin}}\,x - {\text{cos}}\,x = - \left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \int_1^{\frac{{\sqrt 3 }}{2} - \frac{1}{2}} {\left( {\frac{1}{u}} \right)} {\text{d}}u"> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>1</mn> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct limits even if seen later, <em><strong>A1</strong></em> for integral.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - {\text{ln}}\,u} \right]_1^{\frac{{\sqrt 3 }}{2} - \frac{1}{2}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <mn>1</mn> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{ln}}\,u} \right]_{\frac{{\sqrt 3 }}{2} - \frac{1}{2}}^1"> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mn>1</mn> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ln}}\left( {\frac{{\sqrt 3 }}{2} - \frac{1}{2}} \right)\left( {{\text{ + ln}}\,1} \right)"> <mo>=</mo> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext> + ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\left( {\frac{2}{{\sqrt 3 - 1}}} \right)"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for both putting the expression over a common denominator and for correct use of law of logarithms.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\left( {1 + \sqrt 3 } \right)"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\frac{1}{2}{\text{ln}}\left( {{\text{tan}}\,2x + {\text{sec}}\,2x} \right) - \frac{1}{2}{\text{ln}}\left( {{\text{cos}}\,2x} \right)} \right]_0^{\frac{\pi }{6}}"> <msubsup> <mrow> <mo>[</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{ln}}\left( {\sqrt 3 + 2} \right) - \frac{1}{2}{\text{ln}}\left( {\frac{1}{2}} \right) - 0"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em><em><strong>A1(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{ln}}\left( {4 + 2\sqrt 3 } \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>+</mo> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{ln}}\left( {{{\left( {1 + \sqrt 3 } \right)}^2}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\left( {1 + \sqrt 3 } \right)"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is a positive constant.</p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = x\sqrt {f(x)} ">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<msqrt>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| x \right| > a">
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>></mo>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{f(x)}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left| {\frac{1}{{f(x)}}} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f(x)\cos x{\text{d}}x} "> <mo>∫</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is an increasing function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_08.15.01.png" alt="M17/5/MATHL/HP1/ENG/TZ2/09.a.i/M"></p>
<p><strong><em>A1 </em></strong>for correct shape</p>
<p><strong><em>A1 </em></strong>for correct <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercepts and minimum point</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_08.17.28.png" alt="M17/5/MATHL/HP1/ENG/TZ2/09.a.ii/M"></p>
<p><strong><em>A1 </em></strong>for correct shape</p>
<p><strong><em>A1 </em></strong>for correct vertical asymptotes</p>
<p><strong><em>A1 </em></strong>for correct implied horizontal asymptote</p>
<p><strong><em>A1 </em></strong>for correct maximum point</p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_08.20.22.png" alt="M17/5/MATHL/HP1/ENG/TZ2/09.a.iii/M"></p>
<p><strong><em>A1 </em></strong>for reflecting negative branch from (ii) in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis</p>
<p><strong><em>A1 </em></strong>for correctly labelled minimum point</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt at integration by parts <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {({x^2} - {a^2})\cos x{\text{d}}x = ({x^2} - {a^2})\sin x - \int {2x\sin x{\text{d}}x} } "> <mo>∫</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ({x^2} - {a^2})\sin x - 2\left[ { - x\cos x + \int {\cos x{\text{d}}x} } \right]"> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>]</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ({x^2} - {a^2})\sin x + 2x\cos - 2\sin x + c"> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> </math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {({x^2} - {a^2})\cos x{\text{d}}x = \int {{x^2}\cos x{\text{d}}x - \int {{a^2}\cos x{\text{d}}x} } } "> <mo>∫</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </mrow> </math></span></p>
<p>attempt at integration by parts <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{x^2}\cos x{\text{d}}x = {x^2}\sin x - \int {2x\sin x{\text{d}}x} } "> <mo>∫</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^2}\sin x - 2\left[ { - x\cos x + \int {\cos x{\text{d}}x} } \right]"> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>]</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^2}\sin x + 2x\cos x - 2\sin x"> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \int {{a^2}\cos x{\text{d}}x = - {a^2}\sin x} "> <mo>−</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {({x^2} - {a^2})\cos x{\text{d}}x = ({x^2} - {a^2})\sin x + 2x\cos x - 2\sin x + c} "> <mo>∫</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = x{({x^2} - {a^2})^{\frac{1}{2}}}"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = {({x^2} - {a^2})^{\frac{1}{2}}} + \frac{1}{2}x{({x^2} - {a^2})^{ - \frac{1}{2}}}(2x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Method mark is for differentiating the product. Award <strong><em>A1 </em></strong>for each correct term.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = {({x^2} - {a^2})^{\frac{1}{2}}} + {x^2}{({x^2} - {a^2})^{ - \frac{1}{2}}}"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </math></span></p>
<p>both parts of the expression are positive hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is positive <strong><em>R1</em></strong></p>
<p>and therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is an increasing function (for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| x \right| > a"> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> <mo>></mo> <mi>a</mi> </math></span>) <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> defined on the domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < 2\pi ">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
</math></span> by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 3\,{\text{cos}}\,2x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 4 - 11\,{\text{cos}}\,x">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
<p>The following diagram shows the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span></p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinates of the points of intersection of the two graphs.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact area of the shaded region, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\pi + q\sqrt 3 "> <mi>p</mi> <mi>π</mi> <mo>+</mo> <mi>q</mi> <msqrt> <mn>3</mn> </msqrt> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}"> <mi>q</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the points A and B on the diagram, the gradients of the two graphs are equal.</p>
<p>Determine the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-coordinate of A on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{cos}}\,2x = 4 - 11\,{\text{cos}}\,x"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span></p>
<p>attempt to form a quadratic in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {2\,{\text{co}}{{\text{s}}^2}\,x - 1} \right) = 4 - 11\,{\text{cos}}\,x"> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {6\,{\text{co}}{{\text{s}}^2}\,x + 11\,{\text{cos}}\,x - 7 = 0} \right)"> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>7</mn> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>valid attempt to solve their quadratic <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3\,{\text{cos}}\,x + 7} \right)\left( {2\,{\text{cos}}\,x - 1} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>7</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = \frac{1}{2}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{3}{\text{,}}\,\,\frac{{5\pi }}{3}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Ignore any “extra” solutions.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider (±) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_{\frac{\pi }{3}}^{\frac{{5\pi }}{3}} {\left( {4 - 11\,{\text{cos}}\,x - 3\,{\text{cos}}\,2x} \right)} \,{\text{d}}x"> <munderover> <mo>∫</mo> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \pm \right)\left[ {4x - 11\,{\text{sin}}\,x - \frac{3}{2}{\text{sin}}\,2x} \right]_{\frac{\pi }{3}}^{\frac{{5\pi }}{3}}"> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mo>)</mo> </mrow> <msubsup> <mrow> <mo>[</mo> <mrow> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Ignore lack of or incorrect limits at this stage.</p>
<p>attempt to substitute their limits into their integral <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{20\pi }}{3} - 11\,{\text{sin}}\frac{{5\pi }}{3} - \frac{3}{2}{\text{sin}}\frac{{10\pi }}{3} - \left( {\frac{{4\pi }}{3} - 11\,{\text{sin}}\frac{\pi }{3} - \frac{3}{2}{\text{sin}}\frac{{2\pi }}{3}} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>10</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{16\pi }}{3} + \frac{{11\sqrt 3 }}{2} + \frac{{3\sqrt 3 }}{4} + \frac{{11\sqrt 3 }}{2} + \frac{{3\sqrt 3 }}{4}"> <mo>=</mo> <mfrac> <mrow> <mn>16</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>11</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>11</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{16\pi }}{3} + \frac{{25\sqrt 3 }}{2}"> <mo>=</mo> <mfrac> <mrow> <mn>16</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>25</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate both functions and equate <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6\,{\text{sin}}\,2x = 11\,{\text{sin}}\,x"> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11\,{\text{sin}}\,x + 12\,{\text{sin}}\,x\,{\text{cos}}\,x = 0"> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>12</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x\left( {11 + 12\,{\text{cos}}\,x} \right) = 0"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mo>+</mo> <mn>12</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = - \frac{{11}}{{12}}\,\,\left( {{\text{or}}\,\,{\text{sin}}\,x = 0\,} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>or</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace"></mspace> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow y = 4 - 11\left( { - \frac{{11}}{{12}}} \right)"> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>11</mn> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{169}}{{12}}\,\left( { = 14\frac{1}{{12}}} \right)"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mn>169</mn> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>14</mn> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>c</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math>.</p>
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≥</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></math>, find an expression for the median of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the median</p>
<p><strong><br>EITHER</strong></p>
<p>attempts to find the area of the required triangle <em><strong>M1</strong></em></p>
<p>base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>and height is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p>area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>×</mo><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msup><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to integrate the correct function <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>a</mi><mi>m</mi></munderover><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfenced><mi>a</mi><mi>m</mi></msubsup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><msubsup><mfenced open="[" close="]"><mrow><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mi>a</mi><mi>x</mi></mrow></mfenced><mi>a</mi><mi>m</mi></msubsup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct integration and <em><strong>A1</strong> </em>for correct limits.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>sets up (their) <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>a</mi><mi>m</mi></munderover><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi></math> or area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0A0A0M1A0A0</strong></em> if candidates conclude that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mi>c</mi></math> and set up their area or sum of integrals <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>±</mo><msqrt><mfrac><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></msqrt></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mi>a</mi></math>, rejects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>-</mo><msqrt><mfrac><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></msqrt></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>a</mi><mo>+</mo><msqrt><mfrac><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></msqrt></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {{\text{e}}^x}\sin x,{\text{ }}0 \leqslant x \leqslant \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>The curvature at any point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x,{\text{ }}y)">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo stretchy="false">)</mo>
</math></span> on a graph is defined as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}">
<mi>κ<!-- κ --></mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> has a local maximum value when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the point of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</p>
<p>The curvature at any point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x,{\text{ }}y)"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> </math></span> on a graph is defined as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}"> <mi>κ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the curvature of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> at the local maximum point.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa "> <mi>κ</mi> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> and comment on its meaning with respect to the shape of the graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = {{\text{e}}^{\frac{{3\pi }}{4}}}\left( {\sin \frac{{3\pi }}{4} + \cos \frac{{3\pi }}{4}} \right) = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>cos</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^{\frac{{3\pi }}{4}}}\cos \frac{{3\pi }}{4} < 0"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo><</mo> <mn>0</mn> </math></span> <strong><em>R1</em></strong></p>
<p>hence maximum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 0 \Rightarrow 2{{\text{e}}^x}\cos x = 0"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \frac{\pi }{2}"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A0 </em></strong>if extra zeros are seen.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-28_om_14.29.02.png" alt="N16/5/MATHL/HP1/ENG/TZ0/11.e/M"></p>
<p>correct shape and correct domain <strong><em>A1</em></strong></p>
<p>max at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>, point of inflexion at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>zeros at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi "> <mi>x</mi> <mo>=</mo> <mi>π</mi> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Penalize incorrect domain with first <strong><em>A </em></strong>mark; allow <strong><em>FT </em></strong>from (d) on extra points of inflexion.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^x {{{\text{e}}^x}\sin x{\text{d}}x = [{{\text{e}}^x}\sin x]_0^\pi - \int_0^\pi {{{\text{e}}^x}\cos x{\text{d}}x} } "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = [{{\text{e}}^x}\sin x]_0^\pi - \left( {[{{\text{e}}^x}\cos x]_0^x + \int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x} } \right)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mo>+</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = [ - {{\text{e}}^x}\cos x]_0^\pi + \int_0^\pi {{{\text{e}}^x}\cos x{\text{d}}x} } "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>+</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = [ - {{\text{e}}^x}\cos x]} _0^\pi + \left( {[{{\text{e}}^x}\sin x]_0^\pi - \int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x} } \right)"> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mn>0</mn> <mi>π</mi> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = \frac{1}{2}\left( {[{{\text{e}}^x}\sin x]_0^x - [{{\text{e}}^x}\cos x]_0^x} \right)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mo>−</mo> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>x</mi> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = \frac{1}{2}({{\text{e}}^x} + 1)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{d^2}y}}{{d{x^2}}} = 2{e^{\frac{{3\pi }}{4}}}\cos \frac{{3\pi }}{4} = - \sqrt 2 {e^{\frac{{3\pi }}{4}}}"> <mfrac> <mrow> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>e</mi> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <msup> <mi>e</mi> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| { - \sqrt 2 {{\text{e}}^{\frac{{3\pi }}{4}}}} \right|}}{1} = \sqrt 2 {{\text{e}}^{\frac{{3\pi }}{4}}}"> <mi>κ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mn>1</mn> </mfrac> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = 0"> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p>the graph is approximated by a straight line <strong><em>R1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> has a local maximum at A. Find the coordinates of A.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there is exactly one point of inflexion, B, on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The coordinates of B can be expressed in the form B<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{2^a},\,b \times {2^{ - 3a}}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>a</mi> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> where <em>a</em>, <em>b</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Q}"> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>. Find the value of <em>a</em> and the value of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> showing clearly the position of the points A and B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = - 3{x^{ - 4}} - 3x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for using quotient or product rule award <em><strong>A1</strong> </em>if correct derivative seen even in unsimplified form, for example <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{ - 15{x^4} \times 2{x^3} - 6{x^2}\left( {2 - 3{x^5}} \right)}}{{{{\left( {2{x^3}} \right)}^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>15</mn> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>×</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{{{x^4}}} - 3x = 0"> <mo>−</mo> <mfrac> <mn>3</mn> <mrow> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </mrow> </mfrac> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {x^5} = - 1 \Rightarrow x = - 1"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( { - 1,\, - \frac{5}{2}} \right)"> <mrow> <mtext>A</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 12{x^{ - 5}} - 3\left( { = 0} \right)"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>12</mn> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct derivative seen even if not simplified.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mroot> <mn>4</mn> <mn>5</mn> </mroot> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>hence (at most) one point of inflexion <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> This mark is independent of the two <em><strong>A1</strong> </em>marks above. If they have shown or stated their equation has only one solution this mark can be awarded.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right)"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> changes sign at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)"> <mi>x</mi> <mo>=</mo> <mroot> <mn>4</mn> <mn>5</mn> </mroot> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>R1</strong></em></p>
<p>so exactly one point of inflexion</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt[5]{4} = {2^{\frac{2}{5}}}\left( { \Rightarrow a = \frac{2}{5}} \right)"> <mi>x</mi> <mo>=</mo> <mroot> <mn>4</mn> <mn>5</mn> </mroot> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {{2^{\frac{2}{5}}}} \right) = \frac{{2 - 3 \times {2^2}}}{{2 \times {2^{\frac{6}{5}}}}} = - 5 \times {2^{ - \frac{6}{5}}}\left( { \Rightarrow b = - 5} \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>−</mo> <mn>3</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>5</mn> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mfrac> <mn>6</mn> <mn>5</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for the substitution of their value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"><em><strong>A1A1A1A1</strong></em></p>
<p><em><strong>A1</strong></em> for shape for <em>x</em> < 0<br><em><strong>A1 </strong></em>for shape for <em>x</em> > 0<br><em><strong>A1 </strong></em>for maximum at A<br><em><strong>A1 </strong></em>for POI at B.</p>
<p><strong>Note:</strong> Only award last two <em><strong>A1</strong></em>s if A and B are placed in the correct quadrants, allowing for follow through.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>By using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi></math> or otherwise, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></munderover><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is a non-zero real number.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>attempts to express the integral in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><msup><mi>u</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>d</mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><msubsup><mfenced open="[" close="]"><msup><mi>u</mi><mi>n</mi></msup></mfenced><mn>1</mn><mn>2</mn></msubsup><mo> </mo><mo> </mo><mo>(</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><msubsup><mfenced open="[" close="]"><mrow><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></msubsup><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone the absence of or incorrect limits up to this point.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><msup><mn>1</mn><mi>n</mi></msup></mrow><mi>n</mi></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correct substitution of <span style="text-decoration:underline;"><strong>their</strong></span> limits for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> into their antiderivative for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> (or given limits for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> into their antiderivative for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><msup><mtext>sec</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mi>x</mi><mo> </mo><mtext>sec</mtext><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>applies integration by inspection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><msubsup><mfenced open="[" close="]"><mrow><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></msubsup></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A2</strong></em> if the limits are not stated.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mfenced><mrow><msup><mtext>sec</mtext><mi>n</mi></msup><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>-</mo><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correct substitution into their antiderivative.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has probability density function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnspacing="1.4ex" columnalign="left"><mtr><mtd><mfrac><mi>k</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>,</mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext><mo>.</mo></mtd></mtr></mtable></mfenced></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mo>(</mo><mi>X</mi><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>k</mi><mfenced open="⌊" close="⌋"><mrow><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mtext>arcsin</mtext><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mi>x</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arcsin</mtext><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mi>x</mi></mrow></mfenced></math>.<br>Condone absence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> up to this stage.</p>
<p> </p>
<p>equating their integrand to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mtext>arcsin</mtext><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mi>x</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone absence of limits if seen at a later stage.</p>
<p><br><strong>EITHER</strong></p>
<p>attempt to integrate by inspection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac><mo>×</mo><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo>∫</mo><mo>-</mo><mn>6</mn><mi>x</mi><msup><mfenced><mrow><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mi mathvariant="normal">π</mi></mfrac><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> up to this stage.</p>
<p><br><strong>OR</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>6</mn><mi>x</mi></math></p>
<p><br><strong>Note:</strong> Other substitutions may be used. For example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msubsup><mo>∫</mo><mn>4</mn><mn>1</mn></msubsup><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Condone absence of limits up to this stage.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><msubsup><mfenced open="[" close="]"><mrow><mn>2</mn><msqrt><mi>u</mi></msqrt></mrow></mfenced><mn>4</mn><mn>1</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Condone the use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> up to this stage.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A0M1A1A0</strong></em> for their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mfenced open="[" close="]"><mrow><mo>-</mo><mn>2</mn><msqrt><mi>u</mi></msqrt></mrow></mfenced></math> for working with incorrect or no limits.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Most candidates who attempted part (a) knew that the integrand must be equated to 1 and only a small proportion of these managed to recognize the standard integral involved here. The effect of 3 in 3<em>x<sup>2</sup></em> was missed by many resulting in very few completely correct answers for this part. Part (b) proved to be challenging for vast majority of the candidates and was poorly done in general. Stronger candidates who made good progress in part (a) were often successful in part (b) as well. Most candidates used a substitution, however many struggled to make progress using this approach. Often when using a substitution, the limits were unchanged. If the function was re-written in terms of <em>x</em>, this did not result in an error in the final answer.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is shown below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an odd function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The range of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mi>b</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempts to replace <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></msqrt><mfenced><mrow><mo>=</mo><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for an attempt to calculate both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced></math> independently, showing that they are equal.<br><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for a graphical approach including evidence that <strong>either</strong> the graph is invariant after rotation by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo></math> about the origin <strong>or</strong> the graph is invariant after a reflection in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and then in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis (or vice versa).</p>
<p> </p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an odd function <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts both product rule and chain rule differentiation to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mo>×</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>×</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac></math></p>
<p>sets their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempts to find at least one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for an attempt to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at least at one of their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> roots.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>≤</mo><mi>y</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi><mo> </mo><mo>\</mo><mo> </mo><mfenced open="{" close="}"><mi>k</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>≠</mo><mn>5</mn></math>. </p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the vertical asymptote on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the horizontal asymptote on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an algebraic method to determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is a self-inverse function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region bounded by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>7</mn></math> is rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. Find the volume of the solid generated, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>)</mo><mo> </mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></mstyle></mfenced><mo>-</mo><mn>5</mn></mrow><mrow><mfenced><mstyle displaystyle="true"><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></mstyle></mfenced><mo>-</mo><mi>k</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>k</mi><mfenced><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>-</mo><mn>5</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><mn>5</mn><mi>k</mi><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>k</mi></mrow><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn><mo>-</mo><mi>k</mi><mi>x</mi><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><mn>5</mn><mi>x</mi></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfenced></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>f</mi><mo>∘</mo><mi>f</mi></mrow></mfenced><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi></math> , (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is self-inverse) <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> The statement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>x</mi></math> could be seen anywhere in the candidate’s working to award <em><strong>R1</strong></em>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> can be done at any stage.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>-</mo><mi>x</mi><mi>k</mi><mo>=</mo><mi>k</mi><mi>y</mi><mo>-</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>-</mo><mi>k</mi><mi>y</mi><mo>=</mo><mi>x</mi><mi>k</mi><mo>-</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is self-inverse) <em><strong>R1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img style="display: block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>attempt to draw both branches of a rectangular hyperbola <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mn>5</mn><mn>3</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>5</mn><mn>7</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong>EITHER</strong></p>
<p>attempt to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mo>+</mo><mfrac><mn>4</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup></math> and divide out <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>+</mo><mfrac><mrow><mn>24</mn><mi>x</mi><mo>-</mo><mn>56</mn></mrow><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo>+</mo><mfrac><mn>24</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>5</mn><mn>7</mn></munderover><mfenced><mrow><mn>9</mn><mo>+</mo><mfrac><mn>24</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>16</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced><mo> </mo><mtext>d</mtext><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfrac><mn>16</mn><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mrow></mfenced><mn>5</mn><mn>7</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced open="⌊" close="⌋"><mrow><mfenced><mrow><mn>63</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>45</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>-</mo><mn>8</mn></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>22</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>5</mn><mn>7</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></mfenced><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>⇒</mo><mfrac><mrow><mtext>d</mtext><mi>u</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>5</mn><mo>=</mo><mn>3</mn><mfenced><mrow><mi>u</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mn>5</mn><mo>=</mo><mn>3</mn><mi>u</mi><mo>+</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>2</mn><mn>4</mn></msubsup><msup><mfenced><mfrac><mrow><mn>3</mn><mi>u</mi><mo>+</mo><mn>4</mn></mrow><mi>u</mi></mfrac></mfenced><mn>2</mn></msup><mtext>d</mtext><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>2</mn><mn>4</mn></msubsup><mn>9</mn><mo>+</mo><mfrac><mn>16</mn><msup><mi>u</mi><mn mathvariant="italic">2</mn></msup></mfrac><mo>+</mo><mfrac><mn>24</mn><mi>u</mi></mfrac><mo> </mo><mtext>d</mtext><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>9</mn><mi>u</mi><mo>-</mo><mfrac><mn>16</mn><mi>u</mi></mfrac><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>u</mi></mrow></mfenced><mn>2</mn><mn>4</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Ignore absence of or incorrect limits seen up to this point.</p>
<p><em><strong><br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>22</mn><mo>+</mo><mn>24</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math><em><strong> A1<br></strong></em></p>
<p><em><strong><br></strong></em><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> is given by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x\,{\text{tan}}\left( {\frac{{\pi xy}}{4}} \right)">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the point (1, 1) , show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2 + \pi }}{{2 - \pi }}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mo>−</mo> <mi>π</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the equation of the normal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C"> <mi>C</mi> </math></span> at the point (1, 1).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt to differentiate implicitly <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = x\,{\text{se}}{{\text{c}}^2}\left( {\frac{{\pi xy}}{4}} \right)\left[ {\left( {\frac{\pi }{4}x\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{\pi }{4}y} \right)} \right] + {\text{tan}}\left( {\frac{{\pi xy}}{4}} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mi>x</mi> <mi>y</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>π</mi> <mi>x</mi> <mi>y</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each term.</p>
<p>attempt to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1"> <mi>y</mi> <mo>=</mo> <mn>1</mn> </math></span> into their equation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{\pi }{2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{\pi }{2} + 1"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}\left( {1 - \frac{\pi }{2}} \right) = \frac{\pi }{2} + 1"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2 + \pi }}{{2 - \pi }}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mo>−</mo> <mi>π</mi> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use gradient of normal <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 1}}{{\frac{{{\text{d}}y}}{{{\text{d}}x}}}}"> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi - 2}}{{\pi + 2}}"> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </math></span></p>
<p>so equation of normal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 1 = \frac{{\pi - 2}}{{\pi + 2}}\left( {x - 1} \right)"> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{\pi - 2}}{{\pi + 2}}x + \frac{4}{{\pi + 2}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mi>x</mi> <mo>+</mo> <mfrac> <mn>4</mn> <mrow> <mi>π</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x + h)^2} + k"> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>k</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>, indicating on it the equations of the asymptotes, the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept and the local maximum.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {f(x){\text{d}}x = \ln (p)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area of the region enclosed between the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and the lines with equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2 = {\left( {x + \frac{3}{2}} \right)^2} - \frac{1}{4}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2 = (x + 2)(x + 1)"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_13.58.40.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B11.b/M"></p>
<p><strong><em>A1</em></strong> for the shape</p>
<p><strong><em>A1</em></strong> for the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><strong><em>A1</em></strong> for asymptotes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 2"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span></p>
<p><strong><em>A1</em></strong> for coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{3}{2},{\text{ }} - 4} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong><em>A1</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0,{\text{ }}\frac{1}{2}} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{x + 1}} - \frac{1}{{x + 2}}{\text{d}}x} "> <munderover> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </munderover> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\ln (x + 1) - \ln (x + 2)} \right]_0^1"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mn>1</mn> </msubsup> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \ln 2 - \ln 3 - \ln 1 + \ln 2"> <mo>=</mo> <mi>ln</mi> <mo></mo> <mn>2</mn> <mo>−</mo> <mi>ln</mi> <mo></mo> <mn>3</mn> <mo>−</mo> <mi>ln</mi> <mo></mo> <mn>1</mn> <mo>+</mo> <mi>ln</mi> <mo></mo> <mn>2</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \ln \left( {\frac{4}{3}} \right)"> <mo>=</mo> <mi>ln</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore p = \frac{4}{3}"> <mo>∴</mo> <mi>p</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_14.20.03.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B11.e/M"></p>
<p>symmetry about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis <strong><em>M1</em></strong></p>
<p>correct shape <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow <strong><em>FT </em></strong>from part (b).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\int_0^1 {f(x){\text{d}}x} "> <mn>2</mn> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\ln \left( {\frac{4}{3}} \right)"> <mo>=</mo> <mn>2</mn> <mi>ln</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>FT </em></strong>from part (e).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{arccos}}\left( {\frac{x}{2}} \right)">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{arccos}}\left( {\frac{x}{2}} \right) \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{1}{{2\sqrt {1 - {{\left( {\frac{x}{2}} \right)}^2}} }}\left( { = - \frac{1}{{\sqrt {4 - {x^2}} }}} \right)">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>4</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><strong>Note:</strong> <strong><em>M1</em></strong> is for use of the chain rule.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at integration by parts <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{arccos}}\left( {\frac{x}{2}} \right) \Rightarrow \frac{{{\text{d}}u}}{{{\text{d}}x}} = - \frac{1}{{\sqrt {4 - {x^2}} }}">
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>4</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = 1 \Rightarrow v = x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>v</mi>
<mo>=</mo>
<mi>x</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x} = \left[ {x\,\,{\text{arccos}}\left( {\frac{x}{2}} \right)} \right]_0^1 + \int_0^1 {\frac{1}{{\sqrt {4 - {x^2}} }}} dx">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mo>+</mo>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>4</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mi>d</mi>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>using integration by substitution or inspection <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {x\,\,{\text{arccos}}\left( {\frac{x}{2}} \right)} \right]_0^1 + \left[ { - {{\left( {4 - {x^2}} \right)}^{\frac{1}{2}}}} \right]_0^1">
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mo>+</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>1</mn>
</msubsup>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{ - {{\left( {4 - {x^2}} \right)}^{\frac{1}{2}}}}">
<mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</math></span> or equivalent.</p>
<p><strong>Note:</strong> Condone lack of limits to this point.</p>
<p>attempt to substitute limits into their integral <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3} - \sqrt 3 + 2">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>n</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>n</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>,</mo><mo> </mo><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>×</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>It is given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> term in the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use the chain rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></mfrac></math> or equivalent seen</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>''</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac><mo>=</mo></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mn>1</mn></msup><mfrac><mrow><mn>1</mn><mo>!</mo></mrow><mrow><mn>0</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mn>2</mn></mrow></msup></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R0</strong></em> for not starting at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>. Award subsequent marks as appropriate.</p>
<p> </p>
<p>assume true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, (so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>k</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></msup></math>) <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>M1</strong></em> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>” or “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> is true”. Subsequent marks can still be awarded.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><msup><mi>f</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>d</mi><mfenced><mrow><msup><mi>f</mi><mfenced><mi>k</mi></mfenced></msup><mfenced><mi>x</mi></mfenced></mrow></mfenced></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></mfenced><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext><mo>=</mo><msup><mi>f</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><strong>Note:</strong> The following <em><strong>A</strong></em> marks can be awarded in any order.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow><mn>2</mn></mfrac></mfenced><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for isolating <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> correctly.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for multiplying top and bottom by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></msup><mo>=</mo><mtext>RHS</mtext></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>since true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>, and true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> if true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, the statement is true for all, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></math> by mathematical induction <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>To obtain the final <em><strong>R1</strong></em>, at least four of the previous marks must have been awarded.</p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math></p>
<p>using product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><mfenced><mrow><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>h</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math></p>
<p>equating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to apply binomial theorem for rational exponents <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mfenced><mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math><em><strong> A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempt to set equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> and solve <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </strong></em>or <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>h</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math></p>
<p>equating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p>using product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mfenced><mi>x</mi></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>2</mn><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mfenced><mi>x</mi></mfenced></math><em><strong> A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mi>g</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mn>2</mn><mi>g</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>g</mi><mfenced><mn>0</mn></mfenced><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>×</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>m</mi><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </strong></em>or <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> A1</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>, </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>For the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>show that there are no <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph, showing clearly any asymptotic behaviour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}"> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The area enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4"> <mi>y</mi> <mo>=</mo> <mn>4</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,v"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>v</mi> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v"> <mi>v</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule (or equivalent) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{\left( {{x^2} - 1} \right)\left( 2 \right) - \left( {2x - 4} \right)\left( {2x} \right)}}{{{{\left( {{x^2} - 1} \right)}^2}}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2{x^2} + 8x - 2}}{{{{\left( {{x^2} - 1} \right)}^2}}}"> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>simplifying numerator (may be seen in part (i)) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {x^2} - 4x + 1 = 0"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </math></span> or equivalent quadratic equation <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>use of quadratic formula</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \frac{{4 \pm \sqrt {12} }}{2}"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mo>±</mo> <msqrt> <mn>12</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>use of completing the square</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 2} \right)^2} = 3"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> (since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + \sqrt 3 "> <mn>2</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </math></span> is outside the domain) <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not condone verification that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 \Rightarrow f'\left( x \right) = 0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> <mo stretchy="false">⇒</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<p>Do not award the final <em><strong>A1</strong></em> as follow through from part (i).</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(0, 4) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - 4 = 0 \Rightarrow x = 2"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A1</strong></em></p>
<p>outside the domain <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>award <em><strong>A1</strong></em> for concave up curve over correct domain with one minimum point in the first quadrant<br>award <em><strong>A1</strong></em> for approaching <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pm 1"> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math></span> asymptotically</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to combine fractions (using common denominator) <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\left( {x - 1} \right) - \left( {x + 1} \right)}}{{\left( {x + 1} \right)\left( {x - 1} \right)}}"> <mfrac> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3x - 3 - x - 1}}{{{x^2} - 1}}"> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2x - 4}}{{{x^2} - 1}}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4 \Rightarrow 2x - 4 = 4{x^2} - 4"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> or) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p>area under the curve is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int_0^{\frac{1}{2}} {\frac{3}{{x + 1}} - \frac{1}{{x - 1}}{\text{d}}x} "> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><strong>Note:</strong> Ignore absence of, or incorrect limits up to this point.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {3\,{\text{ln}}\,\left| {x + 1} \right| - {\text{ln}}\,\left| {x - 1} \right|} \right]_0^{\frac{1}{2}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>|</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\,{\text{ln}}\frac{3}{2} - {\text{ln}}\frac{1}{2}\left( { - 0} \right)"> <mo>=</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\frac{{27}}{4}"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>27</mn> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>area is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <mn>2</mn> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{1}{2}} {4\,{\text{d}}x} - \int_0^{\frac{1}{2}} {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2 - {\text{ln}}\frac{{27}}{4}"> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>27</mn> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\frac{{4\,{{\text{e}}^2}}}{{27}}"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { \Rightarrow v = \frac{{4\,{{\text{e}}^2}}}{{27}}} \right)"> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^2 {f\left( x \right){\text{d}}x = 10} ">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^2 {f\left( x \right){\text{d}}x = 12} ">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>12</mn>
</mrow>
</math></span>, find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext> + 2</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext> + 2</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {f\left( x \right){\text{d}}x = 10} - 12 = - 2">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
<mo>−</mo>
<mn>12</mn>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {2{\text{d}}x = \left[ {2x} \right]} _{ - 2}^0 = 4">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<msubsup>
<mrow>
<mn>2</mn>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mo>=</mo>
<mn>4</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x} = 2">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext> + 2</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x} = \int_0^2 {f\left( x \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext> + 2</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 12 <em><strong>A</strong><strong>1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{sin}}\,x">
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{co}}{{\text{s}}^3}x\,{\text{d}}x}}{{\sqrt {{\text{sin}}\,x} }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{sin}}\,x \Rightarrow {\text{d}}u = {\text{cos}}\,x{\text{d}}x">
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p>valid attempt to write integral in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u">
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{co}}{{\text{s}}^3}x\,{\text{d}}x}}{{\sqrt {{\text{sin}}\,x} }}} = \int {\frac{{\left( {1 - {u^2}} \right){\text{d}}u}}{{\sqrt u }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<msqrt>
<mi>u</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int {\left( {{u^{ - \frac{1}{2}}} - {u^{\frac{3}{2}}}} \right)} \,{\text{d}}u">
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2{u^{\frac{1}{2}}} - \frac{{2{u^{\frac{5}{2}}}}}{5}\left( { + c} \right)">
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sqrt {{\text{sin}}\,x} - \frac{{2{{\left( {\sqrt {{\text{sin}}\,x} } \right)}^5}}}{5}\left( { + c} \right)">
<mo>=</mo>
<mn>2</mn>
<msqrt>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</msqrt>
<mo>−</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or equivalent <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> up to and including the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> term.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mtext>e</mtext><msup><mi>x</mi><mn>2</mn></msup></msup><mo> </mo><mi>sin</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>d</mo><mi>x</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo>(</mo><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>g</mi><mo>'''</mo><mfenced><mi>x</mi></mfenced><mo>-</mo><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></mrow></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the result from part (c), find the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> up to and including the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> term.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">recognition of both known series <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>x</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mi>x</mi><mrow><mn>1</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>5</mn></msup><mrow><mn>5</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></math></p>
<p style="text-align:left;">attempt to multiply the two series up to and including <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> term <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mi>x</mi><mrow><mn>1</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>5</mn></msup><mrow><mn>5</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> or its derivatives to obtain Maclaurin series <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>+</mo><mfrac><mi>x</mi><mrow><mn>1</mn><mo>!</mo></mrow></mfrac><mo>×</mo><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>×</mo><mn>2</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mo>×</mo><mn>2</mn><mo>+</mo><mo>…</mo></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><msup><mi>x</mi><mn>2</mn></msup></msup><mo> </mo><mi>sin</mi><mo> </mo><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">substituting their expression and attempt to integrate <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mtext>e</mtext><msup><mi>x</mi><mn>2</mn></msup></msup><mo> </mo><mi>sin</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>d</mo><mi>x</mi><mo>≈</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>6</mn></msup></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Condone absence of limits up to this stage.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>5</mn></msup><mn>5</mn></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>7</mn></msup><mn>21</mn></mfrac></mrow></mfenced><mn>0</mn><mn>1</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>61</mn><mn>105</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempt to use product rule at least once <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mfenced><mrow><mo>=</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mfenced><mrow><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Accept working with each side separately to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math>.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>'</mo><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mfenced><mrow><mi>g</mi><mo>''</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>g</mi><mo>'''</mo><mfenced><mi>x</mi></mfenced><mo>-</mo><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><strong>Note:</strong> Accept working with each side separately to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></math>.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into a derivative <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>g</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>g</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><msup><mi>g</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mo>-</mo><mn>4</mn></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">attempt to substitute into Maclaurin formula <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>-</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>4</mn><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Do not award any marks for approaches that do not use the part (c) result.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mo>-</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>x</mi><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Condone the omission of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mo>…</mo></math> in their working.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></math> indeterminate form, attempt to apply l'Hôpital's rule <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mfenced><mrow><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></math>, using l'Hôpital's rule again</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>6</mn><mi>x</mi></mrow></mfrac><mfenced><mrow><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></mrow><mrow><mn>6</mn><mi>x</mi></mrow></mfrac></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></math>, using l'Hôpital's rule again</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>g</mi><mo>'''</mo><mfenced><mi>x</mi></mfenced></mrow><mn>6</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was well answered using both methods. A number failed to see the connection between parts (a) and (b). Far too often, a candidate could not add three fractions together in part (b). There were many good responses to part (c) with candidates showing results on both sides are equal. A number of candidates failed to use the result from (c) in part (d). There were some good responses to part (e), with candidates working successfully with the series from (d) or applying l'Hôpital's rule. In particular, some responses were missing the appropriate limit notation and candidates following method 2 did not always show that the initial expression was of an indeterminate form before applying l'Hôpital's rule. Many candidates did not attempt parts (d) and (e).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5 is shown in the following diagram. The curve intercepts the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at (1, 0) and (4, 0) and has a local minimum at (3, −1).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The shaded area enclosed by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis is 0.5. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = 3">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>,</p>
</div>
<div class="specification">
<p>The area enclosed by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4">
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> is 2.5 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>3 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use definite integral of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {f'\left( x \right){\text{d}}x} = 0.5"> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mn>0.5</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) - f\left( 0 \right) = 0.5"> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>f</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.5</mn> </math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0.5 + 3"> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.5</mn> <mo>+</mo> <mn>3</mn> </math></span></p>
<p>= 3.5 <em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_1^4 {f'\left( x \right){\text{d}}x} = - 2.5"> <msubsup> <mo>∫</mo> <mn>1</mn> <mn>4</mn> </msubsup> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mo>−</mo> <mn>2.5</mn> </math></span> <em><strong> (A1)</strong></em></p>
<p><strong>Note:</strong> <em><strong>(A1)</strong></em> is for −2.5.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right) - f\left( 1 \right) = - 2.5"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>−</mo> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>2.5</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right) = 3.5 - 2.5"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>3.5</mn> <mo>−</mo> <mn>2.5</mn> </math></span></p>
<p>= 1 <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong> A1A1A1</strong></em></p>
<p><em><strong>A1</strong></em> for correct shape over approximately the correct domain<br><em><strong>A1</strong></em> for maximum and minimum (coordinates or horizontal lines from 3.5 and 1 are required),<br><em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept at 3</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The region <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is bounded by the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>6</mn></msqrt></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be the area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> divides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> into two regions of equal area.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the gradient of a tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, clearly indicating any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>32</mn></mfrac><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<p>a curve symmetrical about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis with correct concavity that has a local maximum point on the positive <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis <strong>A1</strong></p>
<p>a curve clearly showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>→</mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p>horizontal asymptote <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mrow><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mi>arctan</mi><mfrac><mi>x</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>M1A0</strong> for obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mi>k</mi><mo> </mo><mi>arctan</mi><mfrac><mi>x</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>≠</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac></math>.</p>
<p style="text-align:left;"><strong>Note:</strong> Condone the absence of or use of incorrect limits to this stage.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mfenced><mrow><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mo> </mo><mn>0</mn></mrow></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>×</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>k</mi></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle></mrow><mn>4</mn></mfrac></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math> <strong>(M1)</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi>k</mi><msqrt><mn>6</mn></msqrt></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle></mrow><mn>4</mn></mfrac></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mfenced><mrow><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>k</mi></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi><mo>=</mo><munderover><mo>∫</mo><mi>k</mi><msqrt><mn>6</mn></msqrt></munderover><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mn>3</mn><msqrt><mn>2</mn></msqrt></mfrac><mfenced><mrow><mi>arctan</mi><mo> </mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>arctan</mi><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><msqrt><mn>2</mn></msqrt></mfrac><mo>=</mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mn>3</mn></mfenced><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> <strong>A1</strong></p>
<p style="text-align:left;">so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts product rule or quotient rule differentiation <strong>M1</strong></p>
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mrow><mo>-</mo><mn>6</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mrow><mo>-</mo><mn>3</mn></mrow></msup><mo>+</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>6</mn><mi>x</mi></mrow></mfenced><mfenced><mn>2</mn></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>4</mn></msup></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>A0</strong> if the denominator is incorrect. Subsequent marks can be awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>THEN</strong></p>
<p style="text-align:left;">attempts to express their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> as a rational fraction with a factorized numerator <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>4</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>6</mn><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>3</mn></msup></mfrac></mrow></mfenced></math></p>
<p style="text-align:left;">attempts to solve their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>m</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;">from the curve, the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math> <strong>R1</strong></p>
<p style="text-align:left;">(the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math>)</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>R1</strong> for any equivalent valid reasoning.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>6</mn><mfenced><mrow><mo>-</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced></mrow><mstyle displaystyle="true"><msup><mfenced><mrow><mstyle displaystyle="true"><msup><mfenced><mrow><mo>-</mo><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></mrow></mfenced><mn>2</mn></msup></mstyle><mstyle displaystyle="true"><mo>+</mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mrow></mfenced><mn>2</mn></msup></mstyle></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;">leading to a maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>32</mn></mfrac><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[7 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r,\,x \in {\mathbb{R}^ + }">
<mi>r</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{{{\text{lo}}{{\text{g}}_r}\,{r^2}}}\left( { = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{{{\text{2}}\,{\text{lo}}{{\text{g}}_r}\,r}}} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{{{\text{lo}}{{\text{g}}_x}\,{r^2}}}">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>x</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{2\,{\text{lo}}{{\text{g}}_x}\,r}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>x</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The folium of Descartes is a curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} + {y^3} - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, shown in the following diagram.</p>
<p><img src="images/Schermafbeelding_2018-02-07_om_18.23.15.png" alt="N17/5/MATHL/HP1/ENG/TZ0/07"></p>
<p>Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} + {y^3} - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^2} + 3{y^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3x\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3y = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Differentiation wrt <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> is also acceptable.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{3y - 3{x^2}}}{{3{y^2} - 3x}}{\text{ }}\left( { = \frac{{y - {x^2}}}{{{y^2} - x}}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>y</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>All following marks may be awarded if the denominator is correct, but the numerator incorrect.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^2} - x = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {y^2}">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^6} + {y^3} - 3{y^3} = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>6</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^6} - 2{y^3} = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>6</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3}({y^3} - 2) = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(y \ne 0)\therefore y = \sqrt[3]{2}">
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>∴</mo>
<mi>y</mi>
<mo>=</mo>
<mroot>
<mn>2</mn>
<mn>3</mn>
</mroot>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\left( {\sqrt[3]{2}} \right)^2}{\text{ }}\left( { = \sqrt[3]{4}} \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mroot>
<mn>2</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} + xy - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x({x^2} - 2y) = 0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0 \Rightarrow y = \frac{{{x^2}}}{2}">
<mi>x</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^2} = \frac{{{x^4}}}{4}">
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{{x^4}}}{4}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x({x^3} - 4) = 0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x \ne 0)\therefore x = \sqrt[3]{4}">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>∴</mo>
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>3</mn>
</mroot>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{{{\left( {\sqrt[3]{4}} \right)}^2}}}{2} = \sqrt[3]{2}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mroot>
<mn>4</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mroot>
<mn>2</mn>
<mn>3</mn>
</mroot>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A particle moves along a straight line. Its displacement, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> metres, at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = t + \cos 2t,{\text{ }}t \geqslant 0">
<mi>s</mi>
<mo>=</mo>
<mi>t</mi>
<mo>+</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>t</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span>. The first two times when the particle is at rest are denoted by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1} < {t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
<mo><</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the displacement of the particle when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}">
<mi>t</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = t + \cos 2t">
<mi>s</mi>
<mo>=</mo>
<mi>t</mi>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}s}}{{{\text{d}}t}} = 1 - 2\sin 2t">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>s</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>t</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0">
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2t = \frac{1}{2}">
<mo stretchy="false">⇒</mo>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1} = \frac{\pi }{{12}}(s),{\text{ }}{t_2} = \frac{{5\pi }}{{12}}(s)">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A0A0 </em></strong>if answers are given in degrees.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \frac{\pi }{{12}} + \cos \frac{\pi }{6}\,\,\,\left( {s = \frac{\pi }{{12}} + \frac{{\sqrt 3 }}{2}(m)} \right)">
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mo> </mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>using l’Hôpital’s rule,</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mo> </mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>-</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>4</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mn>6</mn><mi>x</mi></mrow></mfrac></math> <em><strong>(M1)A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>8</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><mn>6</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - {x^2}"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\left( {x - h} \right)^2} + k"> <mi>a</mi> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>k</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{, }}h{\text{, }}k \in \mathbb{R}"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>h</mi> <mrow> <mtext>, </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{\frac{1}{2}}^{\frac{3}{2}} {\frac{1}{{\sqrt {2x - {x^2}} }}} {\text{d}}x"> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to complete the square or multiplication and equating coefficients <em><strong>(</strong><strong>M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - {x^2} = - {\left( {x - 1} \right)^2} + 1"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 1"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 1"> <mi>h</mi> <mo>=</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 1"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of their identity from part (a) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\int_{\frac{1}{2}}^{\frac{3}{2}} {\frac{1}{{\sqrt {1 - {{\left( {x - 1} \right)}^2}} }}} {\text{d}}x} \right)"> <mrow> <mo>(</mo> <mrow> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(</strong><strong>M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {{\text{arc}}\,{\text{sin}}\left( {x - 1} \right)} \right]_{\frac{1}{2}}^{\frac{3}{2}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <mtext>arc</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{arc}}\,{\text{sin}}\left( u \right)} \right]_{ - \frac{1}{2}}^{\frac{1}{2}}"> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <mtext>arc</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone lack of, or incorrect limits up to this point.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{arc}}\,{\text{sin}}\left( {\frac{1}{2}} \right) - {\text{arc}}\,{\text{sin}}\left( { - \frac{1}{2}} \right)"> <mo>=</mo> <mrow> <mtext>arc</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>arc</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(</strong><strong>M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{6} - \left( { - \frac{\pi }{6}} \right)"> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A</strong><strong>1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3}"> <mo>=</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>0</mn></math>.</p>
<p>The binomial expansion of this expression, in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, as far as the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the restriction which must be placed on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for this expansion to be valid.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to expand binomial with negative fractional power <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mrow><mi>a</mi><mi>x</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>8</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>8</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt><mo>=</mo><mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfenced><mfrac><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mn>8</mn></mfrac></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math></p>
<p>attempt to equate coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>:</mo><mo> </mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mn>4</mn><mi>b</mi><mo>;</mo><mo> </mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>:</mo><mo> </mo><mfrac><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mi>b</mi></math></p>
<p>attempt to solve simultaneously <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>x</mi></mfenced><mo><</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1</strong></em><em><strong> mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \tan \theta ">
<mi>x</mi>
<mo>=</mo>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</math></span> show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x = } \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta } ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
</mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \tan \theta ">
<mi>x</mi>
<mo>=</mo>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{\text{d}}x}}{{{\text{d}}\theta }} = {\sec ^2}\theta ">
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>sec</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{{{({x^2} + 1)}^2}}}{\text{d}}x = \int {\frac{{{{\sec }^2}\theta }}{{{{({{\tan }^2}\theta + 1)}^2}}}{\text{d}}\theta } } ">
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sec</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mi>tan</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> The method mark is for an attempt to substitute for both <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}x">
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int {\frac{1}{{{{\sec }^2}\theta }}{\text{d}}\theta } ">
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sec</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</math></span> (or equivalent) <strong><em>A1</em></strong></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0,{\text{ }}\theta = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>θ</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1,{\text{ }}\theta = \frac{\pi }{4}">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta } ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x} = \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta } } \right) = \frac{1}{2}\int\limits_0^{\frac{\pi }{4}} {\left( {1 + \cos 2\theta } \right){\text{d}}\theta } ">
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left[ {\theta + \frac{{\sin 2\theta }}{2}} \right]_0^{\frac{\pi }{4}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mi>θ</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msubsup>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{8} + \frac{1}{4}">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>8</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A curve has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 2{y^2}{{\text{e}}^{x - 1}} = 2">
<mn>3</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equations of the tangents to this curve at the points where the curve intersects the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt to differentiate implicitly <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - \left( {4y\frac{{{\text{d}}y}}{{{\text{d}}x}} + 2{y^2}} \right){{\text{e}}^{x - 1}} = 0"> <mn>3</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>y</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correctly differentiating each term.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{3 \bullet {{\text{e}}^{1 - x}} - 2{y^2}}}{{4y}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mo>∙</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <mi>y</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>This final answer may be expressed in a number of different ways.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - 2{y^2} = 2 \Rightarrow {y^2} = \frac{1}{2} \Rightarrow y = \pm \sqrt {\frac{1}{2}} "> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>2</mn> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <mo>±</mo> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{3 - 2 \bullet \frac{1}{2}}}{{ \pm 4\sqrt {\frac{1}{2}} }} = \pm \frac{{\sqrt 2 }}{2}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo>∙</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>±</mo> <mn>4</mn> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> </mfrac> <mo>=</mo> <mo>±</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1,{\text{ }}\sqrt {\frac{1}{2}} } \right)"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> the tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - \sqrt {\frac{1}{2}} = \frac{{\sqrt 2 }}{2}(x - 1)"> <mi>y</mi> <mo>−</mo> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> and <strong><em>A1</em></strong></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1,{\text{ }} - \sqrt {\frac{1}{2}} } \right)"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> the tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + \sqrt {\frac{1}{2}} = - \frac{{\sqrt 2 }}{2}(x - 1)"> <mi>y</mi> <mo>+</mo> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </msqrt> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>These equations simplify to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \pm \frac{{\sqrt 2 }}{2}x"> <mi>y</mi> <mo>=</mo> <mo>±</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mi>x</mi> </math></span>.</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A0M1A1A0 </em></strong>if just the positive value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> is considered and just one tangent is found.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>9</mn></msubsup><mfenced><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></mfenced><mo>d</mo><mi>x</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>10</mn><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p>substituting limits into their integrated function and subtracting <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mn>9</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>9</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>1</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>-</mo><mn>10</mn><mo>×</mo><mn>3</mn><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>10</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>A mixed response was noted for this question. Candidates who simplified the algebraic fraction before integrating were far more successful in gaining full marks in this question. Many candidates used other valid approaches such as integration by substitution and integration by parts with varying degrees of success. A small number of candidates substituted the limits without integrating.</p>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The inverse of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfrac><mi>x</mi><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mn>2</mn></mfrac><msqrt><mi>r</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><img src="data:image/png;base64,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"></strong></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept an indication of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p><br>vertical asymptotes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p>horizontal asymptote <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>uses a valid method to find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the local maximum point <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> For example, uses the axis of symmetry or attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<p><br>local maximum point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)A0</strong></em> for a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> and coordinates not given.</p>
<p><br>three correct branches with correct asymptotic behaviour and the key features in approximately correct relative positions to each other <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> (this can be done at a later stage).</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempts to complete the square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mfenced><mrow><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>1</mn><mo>=</mo><mo>±</mo><msqrt><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mo>±</mo><msqrt><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac></msqrt></mrow></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msqrt></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>even if <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo></math> (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo></math>) is missing</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>±</mo><msqrt><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></msqrt></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>±</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>3</mn></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math> is rejected <em><strong>R1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for concluding that the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> must have the ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo></math>’ sign.<br>The <em><strong>R1</strong> </em>may be awarded earlier for using the condition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>3</mn></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow><mn>2</mn></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p>attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mn>2</mn><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mfrac><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mn>2</mn></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mn>2</mn></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award all available marks to this stage if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mo>-</mo><mn>7</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to solve their quadratic equation <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mfenced><mn>2</mn></mfenced><mfenced><mn>7</mn></mfenced></msqrt></mrow><mn>4</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mo>±</mo><msqrt><mn>72</mn></msqrt></mrow><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award all available marks to this stage if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msqrt><mn>2</mn></msqrt></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>3</mn></math>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mn>18</mn></msqrt></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>18</mn></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was generally well done. It was pleasing to see how often candidates presented complete sketches here. Several decided to sketch using the reciprocal function. Occasionally, candidates omitted the upper branches or forgot to calculate the <em>y</em>-coordinate of the maximum.</p>
<p>Part (b): The majority of candidates knew how to start finding the inverse, and those who attempted completing the square or using the quadratic formula to solve for y made good progress (both methods equally seen). Otherwise, they got lost in the algebra. Very few explicitly justified the rejection of the negative root.</p>
<p>Part (c) was well done in general, with some algebraic errors seen in occasions.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^{\frac{1}{2}}}">
<mi>u</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\int\limits_1^9 {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mo>∫</mo>
<mn>1</mn>
<mn>9</mn>
</munderover>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>, expressing your answer in the form arctan <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}">
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \frac{1}{2}{x^{ - \frac{1}{2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{d}}u = \frac{1}{2}{x^{ - \frac{1}{2}}}{\text{d}}x">
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> or equivalent) <em><strong>A1</strong></em></p>
<p>substitution, leading to an integrand in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{2u{\text{d}}u}}{{{u^3} + u}}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>u</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>u</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>u</mi>
</mrow>
</mfrac>
</mrow>
</math></span> or equivalent <em><strong>A1</strong></em></p>
<p>= 2 arctan <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\sqrt x } \right)\left( { + c} \right)">
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\int\limits_1^9 {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mo>∫</mo>
<mn>1</mn>
<mn>9</mn>
</munderover>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span> = arctan 3 − arctan 1 <em><strong>A1</strong></em></p>
<p>tan(arctan 3 − arctan 1) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3 - 1}}{{1 + 3 \times 1}}">
<mfrac>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p>tan(arctan 3 − arctan 1) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span></p>
<p>arctan 3 − arctan 1 = arctan <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">A car travels along the road at a speed of 24 ms<sup>−1</sup>. Let the position of the car be X and let OĈX = <em>θ</em>.</p>
<p style="text-align: left;">Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span>, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>let OX = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span></p>
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 24"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>24</mn> </math></span> (or −24) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}x}}{{{\text{d}}t}} \times \frac{{{\text{d}}\theta }}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{tan}}\,\theta = x"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{se}}{{\text{c}}^2}\,\theta = \frac{{{\text{d}}x}}{{{\text{d}}\theta }}"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{{3\,{\text{se}}{{\text{c}}^2}\,\theta }}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>24</mn> </mrow> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> </mfrac> </math></span></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </math></span> into their differential equation<span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"> </span><em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><strong>M1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = {\text{arctan}}\left( {\frac{x}{3}} \right)"> <mi>θ</mi> <mo>=</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>x</mi> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{1}{3} \times \frac{1}{{1 + \frac{{{x^2}}}{9}}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>×</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mn>9</mn> </mfrac> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 24 \times \frac{1}{{3\left( {1 + \frac{{{x^2}}}{9}} \right)}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>24</mn> <mo>×</mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mn>9</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> into their differential equation <em><strong>M1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{3} = 8"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>24</mn> </mrow> <mn>3</mn> </mfrac> <mo>=</mo> <mn>8</mn> </math></span> (rad s<sup>−1</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept −8 rad s<sup>−1</sup>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 24"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>24</mn> </math></span> (or −24) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{tan}}\,\theta = x"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to differentiate implicitly with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{se}}{{\text{c}}^2}\,\theta \times \frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}x}}{{{\text{d}}t}}"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>×</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{{3\,{\text{se}}{{\text{c}}^2}\,\theta }}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>24</mn> </mrow> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> </mfrac> </math></span></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </math></span> into their differential equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{3} = 8"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>24</mn> </mrow> <mn>3</mn> </mfrac> <mo>=</mo> <mn>8</mn> </math></span> (rad s<sup>−1</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept −8 rad s<sup>−1</sup>.</p>
<p><strong>Note:</strong> Can be done by consideration of CX, use of Pythagoras.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>let the position of the car be at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d - 24t"> <mi>d</mi> <mo>−</mo> <mn>24</mn> <mi>t</mi> </math></span> from O <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = \frac{{d - 24t}}{3}\left( { = \frac{d}{3} - 8t} \right)"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mo>−</mo> <mn>24</mn> <mi>t</mi> </mrow> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mi>d</mi> <mn>3</mn> </mfrac> <mo>−</mo> <mn>8</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> For <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = \frac{{24t}}{3}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>24</mn> <mi>t</mi> </mrow> <mn>3</mn> </mfrac> </math></span> award <em><strong>A0M1</strong></em> and follow through.</p>
<p><strong>EITHER</strong></p>
<p>attempt to differentiate implicitly with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{se}}{{\text{c}}^2}\,\theta \frac{{{\text{d}}\theta }}{{{\text{d}}t}} = - 8"> <mrow> <mtext>se</mtext> </mrow> <mrow> <msup> <mrow> <mtext>c</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 0"> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </math></span> into their differential equation <em><strong>M1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = {\text{arctan}}\left( {\frac{d}{3} - 8t} \right)"> <mi>θ</mi> <mo>=</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>d</mi> <mn>3</mn> </mfrac> <mo>−</mo> <mn>8</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{8}{{1 + {{\left( {\frac{d}{3} - 8t} \right)}^2}}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>8</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>d</mi> <mn>3</mn> </mfrac> <mo>−</mo> <mn>8</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>at O, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{d}{{24}}"> <mi>t</mi> <mo>=</mo> <mfrac> <mi>d</mi> <mrow> <mn>24</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = - 8"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><mi>x</mi><mo> </mo><mi>cos</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced></mrow><mrow><mn>5</mn><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi></mrow></mfrac></mfenced></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p> attempts to apply l’Hôpital’s rule on <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><mi>x</mi><mo> </mo><mi>cos</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced></mrow><mrow><mn>5</mn><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi></mrow></mfrac></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><mo> </mo><mi>cos</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>-</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced></mrow><mrow><mn>5</mn><mo> </mo><msup><mi>sec</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfrac></mfenced></math> <strong>M1A1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to use product and chain rule differentiation on the numerator, <strong>A1</strong> for a correct numerator and <strong>A1</strong> for a correct denominator. The awarding of <strong>A1</strong> for the denominator is independent of the <strong>M1</strong>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mn>5</mn></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[5 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The acceleration, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>, of a particle moving in a horizontal line at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>v</mi><mo>)</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is the particle’s velocity and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the particle is at a fixed origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and has initial velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>0</mn></msub><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>Initially at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the particle moves in the positive direction until it reaches its maximum displacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>. The particle then returns to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> metres represent the particle’s displacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math> its maximum displacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>-</mo><mi>k</mi><mo>)</mo></math> represent the particle’s velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> seconds before it reaches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math>, where</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>-</mo><mi>k</mi><mo>)</mo><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mo>(</mo><mi>T</mi><mo>-</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Similarly, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>+</mo><mi>k</mi><mo>)</mo></math> represent the particle’s velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> seconds after it reaches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving an appropriate differential equation, show that the particle’s velocity at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub><mo>)</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> taken for the particle to reach <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math> satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>T</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving an appropriate differential equation and using the result from part (b) (i), find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>0</mn></msub></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result to part (b) (i), show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>k</mi></msup><mo>-</mo><mn>1</mn></math>.</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>Deduce a similar expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>+</mo><mi>k</mi><mo>)</mo></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>+</mo><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>≥</mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mn>1</mn><mo> </mo><mo>d</mo><mi>t</mi><mo>=</mo><mo>∫</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfrac><mo>d</mo><mi>v</mi></math> (or equivalent / use of integrating factor) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> with initial conditions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfrac></mfenced><mo>⇒</mo><msup><mtext>e</mtext><mi>t</mi></msup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>t</mi></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mi>v</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>-</mo><mi>C</mi><mo>=</mo><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced><mo>⇒</mo><msup><mtext>e</mtext><mrow><mi>t</mi><mo>-</mo><mi>C</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced></mfrac></math></p>
<p>Attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> with initial conditions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>C</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced></mfrac><mo>⇒</mo><mi>C</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mo>=</mo><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced><mo>⇒</mo><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfrac></mfenced><mo>⇒</mo><msup><mtext>e</mtext><mi>t</mi></msup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>t</mi></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mi>v</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>-</mo><mi>C</mi><mo>=</mo><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>v</mi></mrow></mfenced><mo>⇒</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi><mo>+</mo><mi>C</mi></mrow></msup><mo>=</mo><mn>1</mn><mo>+</mo><mi>v</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn><mo>=</mo><mi>v</mi></math></p>
<p>Attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with initial conditions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mi>v</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> condone use of modulus within the ln function(s)</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>T</mi><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>T</mi></mrow></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>T</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>T</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>0</mn></msub><mo>=</mo><msup><mtext>e</mtext><mi>T</mi></msup><mo>-</mo><mn>1</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> and showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>0</mn></math>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>∫</mo><mi>v</mi><mfenced><mi>t</mi></mfenced><mo>d</mo><mi>t</mi><mo> </mo><mfenced><mrow><mo>=</mo><mo>∫</mo><mfenced><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>t</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mi>t</mi><mfenced><mrow><mo>+</mo><mi>D</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>s</mi><mo>=</mo><mn>0</mn></math> so) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math></p>
<p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub><mo>,</mo><mo> </mo><msup><mtext>e</mtext><mi>T</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub><mo>⇒</mo><mi>T</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced></math></p>
<p>Substituting into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mfenced><mrow><mo>=</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub><mo>=</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfrac></mfenced><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>s</mi><mtext>max</mtext></msub><mo>=</mo><msub><mi>v</mi><mn>0</mn></msub><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>T</mi></mrow></msup><msup><mtext>e</mtext><mi>k</mi></msup><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><mfenced><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfrac></mfenced><msup><mtext>e</mtext><mi>k</mi></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mi>k</mi></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>-</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>e</mi><mi>T</mi></msup><msup><mtext>e</mtext><mrow><mo>-</mo><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>e</mi><mrow><mi>T</mi><mo>-</mo><mi>T</mi><mo>+</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mi>k</mi></msup><mo>-</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>T</mi></mrow></msup><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>(A1)</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mi>T</mi></msup><msup><mtext>e</mtext><mrow><mo>-</mo><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>-</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mi>T</mi><mo>-</mo><mi>T</mi><mo>-</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>+</mo><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to express as a square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mfrac><mi>k</mi><mn>2</mn></mfrac></msup><mi>-</mi><msup><mtext>e</mtext><mrow><mi>-</mi><mfrac><mi>k</mi><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo> </mo><mfenced><mrow><mo>≥</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>+</mo><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>≥</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>+</mo><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>-</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>Attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>k</mi></mrow></mfrac><mfenced><mrow><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>k</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>, (when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn></math>), hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>≥</mo><mn>2</mn></math> <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>+</mo><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>≥</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mtext>arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>tan</mi><mo> </mo><mn>3</mn><mi>x</mi></mrow></mfrac></mfenced></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to differentiate numerator and denominator <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mtext>arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>tan</mi><mo> </mo><mn>3</mn><mi>x</mi></mrow></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mfenced><mfrac><mn>2</mn><mrow><mn>1</mn><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mrow><mn>3</mn><mo> </mo><msup><mtext>sec</mtext><mn>2</mn></msup><mo> </mo><mn>3</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: <em>A1</em> </strong>for numerator and<em><strong> A1</strong></em> for denominator. Do not condone absence of limits.</p>
<p> </p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A1A0M1A1</strong></em> for absence of limits.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}">
<mrow>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}">
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> defined as follows</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}\,{\text{:}}\,xy = 4">
<mrow>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}\,{\text{:}}\,{y^2} - {x^2} = 2">
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using implicit differentiation, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> for each curve in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let P(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>) be the unique point where the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}"> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}"> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </math></span> intersect.</p>
<p>Show that the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}"> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </math></span> at P is perpendicular to the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}"> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </math></span> at P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}\,{\text{:}}\,y + x\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0"> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>:</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>is for use of both product rule and implicit differentiation.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{y}{x}"> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{4}{{{x^2}}}"> <mo>−</mo> <mfrac> <mn>4</mn> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}\,{\text{:}}\,2y\frac{{{\text{d}}y}}{{{\text{d}}x}} - 2x = 0"> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>:</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>y</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{x}{y}"> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>x</mi> <mi>y</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \pm \frac{x}{{\sqrt {2 + {x^2}} }}"> <mo>±</mo> <mfrac> <mi>x</mi> <mrow> <msqrt> <mn>2</mn> <mo>+</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> <em><strong>M1</strong></em></p>
<p>product of gradients at P is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{b}{a}} \right)\left( {\frac{a}{b}} \right) = - 1"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> or equivalent reasoning <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> The <em><strong>R1</strong> </em>is dependent on the previous <em><strong>M1</strong></em>. </p>
<p> </p>
<p>so tangents are perpendicular <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>ln</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>x</mi></mstyle></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mi>y</mi></mrow><mi>x</mi></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math>, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>y</mi></mrow><mi>x</mi></mfrac><mo>=</mo><mfrac><mrow><mi>ln</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>x</mi></mstyle></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> <em><strong>(M1)</strong></em></p>
<p>attempt to find integrating factor <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi mathvariant="normal">e</mi><mrow><mo>∫</mo><mfrac><mn>2</mn><mi>x</mi></mfrac><mo>d</mo><mi>x</mi></mrow></msup><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></msup></mrow></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>2</mn><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow></mfenced><mo>=</mo><mi>ln</mi><mo> </mo><mn>2</mn><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>=</mo><mo>∫</mo><mi>ln</mi><mo> </mo><mn>2</mn><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math></p>
<p>attempt to use integration by parts <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>x</mi><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><mi>x</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>c</mi><msup><mi>x</mi><mn>2</mn></msup></mfrac></math></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>4</mn></math> into an integrated equation involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>=</mo><mn>0</mn><mo>-</mo><mn>2</mn><mo>+</mo><mn>4</mn><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>c</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><mi>x</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[7</strong></em><em><strong> marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>A particle moves in a straight line such that at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(t \geqslant 0)">
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>⩾</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span>, its velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>, in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{m}}{{\text{s}}^{ - 1}}">
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 10t{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mo>=</mo>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span>. Find the exact distance travelled by the particle in the first half-second.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \int\limits_0^{\frac{1}{2}} {10t{{\text{e}}^{ - 2t}}{\text{d}}t} ">
<mi>s</mi>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span></p>
<p>attempt at integration by parts <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - 5t{{\text{e}}^{ - 2t}}} \right]_0^{\frac{1}{2}} - \int\limits_0^{\frac{1}{2}} { - 5{{\text{e}}^{ - 2t}}{\text{d}}t} ">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - 5t{{\text{e}}^{ - 2t}} - \frac{5}{2}{{\text{e}}^{ - 2t}}} \right]_0^{\frac{1}{2}}">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone absence of limits (or incorrect limits) and missing factor of 10 up to this point.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \int\limits_0^{\frac{1}{2}} {10t{{\text{e}}^{ - 2t}}{\text{d}}t} ">
<mi>s</mi>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 5{{\text{e}}^{ - 1}} + \frac{5}{2}{\text{ }}\left( { = \frac{{ - 5}}{{\text{e}}} + \frac{5}{2}} \right){\text{ }}\left( { = \frac{{5{\text{e}} - 10}}{{2{\text{e}}}}} \right)">
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q\left( x \right) = {x^5} - 10{x^2} + 15x - 6,{\text{ }}x \in \mathbb{R}">
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = q(x)"> <mi>y</mi> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is concave up for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 1"> <mi>x</mi> <mo>></mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = q(x)"> <mi>y</mi> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> showing clearly any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 20{x^3} - 20"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>20</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>20</mn> </math></span> <strong><em>M1A1</em></strong></p>
<p>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 1,{\text{ }}20{x^3} - 20 > 0 \Rightarrow "> <mi>x</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>20</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>20</mn> <mo>></mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </math></span> concave up <strong><em>R1AG</em></strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_16.48.38.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B12.e.ii/M"></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercept at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}0)"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }} - 6)"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p>stationary point of inflexion at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}0)"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> with correct curvature either side <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo> </mo><mo>,</mo><mo> </mo><mi>y</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>y</mi><mo>-</mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that, when <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo> </mo><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the coordinates of all points on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt at implicit differentiation <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mfenced open="⌊" close="⌋"><mrow><mi>x</mi><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>M1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for LHS, <em><strong>M1</strong></em> for attempt at chain rule, <em><strong>A1</strong></em> for RHS.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo> </mo><mi>x</mi><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>-</mo><mo> </mo><mi>x</mi><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>=</mo><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mfenced><mrow><mn>2</mn><mi>y</mi><mo>-</mo><mi>x</mi><mo> </mo><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for collecting derivatives and factorising.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>y</mi><mo>-</mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>≠</mo><mn>0</mn></mrow></mfenced><mo>⇒</mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mfenced><mrow><mo>=</mo><mo>±</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></msqrt><mo>=</mo><mo>±</mo><msqrt><mn>1</mn><mo>-</mo><mn>0</mn></msqrt></mrow></mfenced><mo>=</mo><mo>±</mo><mn>1</mn></math> <em><strong>OR</strong></em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></mrow></mfenced></math> <em><strong>OR</strong></em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> If they offer values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi></math>, award <em><strong>A1</strong></em> for at least two correct values in two different ‘quadrants’ and no incorrect values.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn><mo>⇒</mo><mn>1</mn><mo>=</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mo>±</mo><mi>x</mi></mrow></mfenced><mo>⇒</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>±</mo><mn>1</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn><mo>⇒</mo><mn>0</mn><mo>=</mo><mi>cos</mi><mo> </mo><mfenced><mrow><mo>±</mo><mi>x</mi></mrow></mfenced><mo>⇒</mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>⇒</mo></mrow></mfenced><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>⇒</mo></mrow></mfenced><mfenced><mrow><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced><mo>,</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Allow ‘coordinates’ expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn></math> for example.<br><strong>Note:</strong> Each of the <em><strong>A</strong></em> marks may be awarded independently and are not dependent on <em><strong>(M1)</strong></em> being awarded.</p>
<p><strong>Note:</strong> Mark only the candidate’s first two attempts for each case of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi></math>.<br><br></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Find the coordinates of the points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} + 3x{y^2} - {x^3} = 27">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>27</mn>
</math></span> at which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt at implicit differentiation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{y^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + 3{y^2} + 6xy\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3{x^2} = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for the second & third terms, <em><strong>A1</strong></em> for the first term, fourth term & RHS equal to zero.</p>
<p>substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{y^2} - 3{x^2} = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow y = \pm x">
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>±</mo>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>substitute either variable into original equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x \Rightarrow {x^3} = 9 \Rightarrow x = \sqrt[3]{9}">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} = 9 \Rightarrow y = \sqrt[3]{9}">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</math></span>) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x \Rightarrow {x^3} = 27 \Rightarrow x = 3">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>27</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} = - 27 \Rightarrow y = - 3">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>27</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span>) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\sqrt[3]{9}{\text{,}}\,\,\sqrt[3]{9}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , (3, −3) <em><strong>A1</strong></em></p>
<p><em><strong>[9 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\arcsin x\,{\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt at integration by parts with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \arcsin x">
<mi>u</mi>
<mo>=</mo>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v' = 1">
<msup>
<mi>v</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\arcsin x\,{\text{d}}x} = x\arcsin x - \int {\frac{x}{{\sqrt {1 - {x^2}} }}{\text{d}}x} {\text{ }}">
<mo>∫</mo>
<mrow>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\arcsin x">
<mi>x</mi>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
</math></span> and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \int {\frac{x}{{\sqrt {1 - {x^2}} }}{\text{d}}x} ">
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<p> </p>
<p>solving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{x}{{\sqrt {1 - {x^2}} }}{\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span> by substitution with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 1 - {x^2}">
<mi>u</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> or inspection <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\arcsin x{\text{d}}x} = x\arcsin x + \sqrt {1 - {x^2}} + c">
<mo>∫</mo>
<mrow>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>+</mo>
<mi>c</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{{\text{sin}}\,x}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first two derivatives of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and hence find the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> up to and including the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> term.</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>Show that the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> in the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is zero.</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>Using the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{3x}} - 1">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, find the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right)">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> up to and including the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> term.</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>Hence, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{f\left( x \right) - 1}}{{{\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right)}}">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempting to use the chain rule to find the first derivative <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \left( {{\text{cos}}\,x} \right){{\text{e}}^{{\text{sin}}\,x}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to use the product rule to find the second derivative <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = {{\text{e}}^{{\text{sin}}\,x}}\left( {{\text{co}}{{\text{s}}^2}\,x - {\text{sin}}\,x} \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>attempting to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 0 \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( 0 \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
</math></span> <strong>M1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( 0 \right) = \left( {{\text{cos}}\,0} \right){{\text{e}}^{{\text{sin}}\,0}} = 1">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>; <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( 0 \right) = {{\text{e}}^{{\text{sin}}\,0}}\left( {{\text{co}}{{\text{s}}^2}\,0 - {\text{sin}}\,0} \right) = 1">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo>−</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>substitution into the Maclaurin formula <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = f\left( 0 \right) + xf'\left( 0 \right) + \frac{{{x^2}}}{{2{\text{!}}}}f''\left( 0 \right) + \ldots ">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mo>…</mo>
</math></span> <strong>M1</strong></p>
<p>so the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> up to and including the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> term is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + x + \frac{{{x^2}}}{2}">
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>attempting to differentiate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'''\left( x \right) = \left( {{\text{cos}}\,x} \right){{\text{e}}^{{\text{sin}}\,x}}\left( {{\text{co}}{{\text{s}}^2}\,x - {\text{sin}}\,x} \right) - \left( {{\text{cos}}\,x} \right){{\text{e}}^{{\text{sin}}\,x}}\left( {2\,{\text{sin}}\,x + 1} \right)">
<msup>
<mi>f</mi>
<mo>‴</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (or equivalent) <em><strong>A2</strong></em></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> into <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'''\left( x \right)">
<msup>
<mi>f</mi>
<mo>‴</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'''\left( 0 \right) = 1\left( {1 - 0} \right) - 1\left( {0 + 1} \right) = 0">
<msup>
<mi>f</mi>
<mo>‴</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>so the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> in the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is zero <em><strong>AG</strong></em></p>
<p> </p>
<p><strong><em>METHOD 2</em></strong></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{sin}}\,x}">
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</math></span> into the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{{\text{sin}}\,x}} = 1 + {\text{sin}}\,x + \frac{{{\text{si}}{{\text{n}}^2}\,x}}{{2{\text{!}}}} + \frac{{{\text{si}}{{\text{n}}^3}\,x}}{{3{\text{!}}}} + \ldots ">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</math></span></p>
<p>substituting Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{sin}}\,x}">
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{{\text{sin}}\,x}} = 1 + \left( {x - \frac{{{x^3}}}{{3{\text{!}}}} + \ldots } \right) + \frac{{{{\left( {x - \frac{{{x^3}}}{{3{\text{!}}}} + \ldots } \right)}^2}}}{{2{\text{!}}}} + \frac{{{{\left( {x - \frac{{{x^3}}}{{3{\text{!}}}} + \ldots } \right)}^3}}}{{3{\text{!}}}} + \ldots ">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>A1</strong></em></p>
<p>coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{3{\text{!}}}} + \frac{1}{{3{\text{!}}}} = 0">
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>so the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> in the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is zero <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x">
<mn>3</mn>
<mi>x</mi>
</math></span> into the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{3x}} = 1 + 3x + \frac{{{{\left( {3x} \right)}^2}}}{{2{\text{!}}}} + \frac{{{{\left( {3x} \right)}^3}}}{{3{\text{!}}}} + \ldots ">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>A1</strong></em></p>
<p>substituting <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{{\text{e}}^{3x}} - 1} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></span> into the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right) = \left( {{{\text{e}}^{3x}} - 1} \right) - \frac{{{{\left( {{{\text{e}}^{3x}} - 1} \right)}^3}}}{3} + \frac{{{{\left( {{{\text{e}}^{3x}} - 1} \right)}^5}}}{5} - \ldots ">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mn>5</mn>
</mfrac>
<mo>−</mo>
<mo>…</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {3x + \frac{{{{\left( {3x} \right)}^2}}}{{2{\text{!}}}} + \frac{{{{\left( {3x} \right)}^3}}}{{3{\text{!}}}} + \ldots } \right) - \frac{{{{\left( {3x + \frac{{{{\left( {3x} \right)}^2}}}{{2{\text{!}}}} + \frac{{{{\left( {3x} \right)}^3}}}{{3{\text{!}}}} + \ldots } \right)}^3}}}{3} + \ldots ">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</math></span> <em><strong>A1</strong></em></p>
<p>selecting correct terms from above <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {3x + \frac{{{{\left( {3x} \right)}^2}}}{{2{\text{!}}}} + \frac{{{{\left( {3x} \right)}^3}}}{{3{\text{!}}}}} \right) - \frac{{{{\left( {3x} \right)}^3}}}{3}">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3x + \frac{{9{x^2}}}{2} - \frac{{9{x^3}}}{2}">
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>substitution of <strong>their</strong> series <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{x + \frac{{{x^2}}}{2} + \ldots }}{{3x + \frac{{9{x^2}}}{2} + \ldots }}">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 + \frac{x}{2} + \ldots }}{{3 + \frac{{9x}}{2} + \ldots }}">
<mo>=</mo>
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>…</mo>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>use of l’Hôpital’s rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\left( {{\text{cos}}\,x} \right){{\text{e}}^{{\text{sin}}\,x}}}}{{\frac{{3{{\text{e}}^{3x}}}}{{1 + {{\left( {{{\text{e}}^{3x}} - 1} \right)}^2}}}}}">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find the equation of the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>–</mo><mn>3</mn><mi>x</mi></math> at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>0</mn><mo>⇒</mo></mrow></mfenced><mi>y</mi><mo>=</mo><mn>1</mn></math> <em><strong>(A1)</strong></em></p>
<p>appreciate the need to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo></mrow></mfenced><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>-</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>0</mn><mo>⇒</mo></mrow></mfenced><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>0</mn></mrow></mfrac><mo>=</mo><mo>-</mo><mn>1</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{{\text{ln}}\,k} {{{\text{e}}^{2x}}} {\text{d}}x = 12"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>k</mi> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>12</mn> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{{\text{e}}^{2x}}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> seen <em><strong>(A1)</strong></em></p>
<p>attempt at using limits in an integrated expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\left[ {\frac{1}{2}{{\text{e}}^{2x}}} \right]_0^{{\text{ln}}\,k} = \frac{1}{2}{{\text{e}}^{2\,{\text{ln}}\,k}} - \frac{1}{2}{{\text{e}}^0}} \right)"> <mrow> <mo>(</mo> <mrow> <msubsup> <mrow> <mo>[</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>k</mi> </mrow> </msubsup> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>k</mi> </mrow> </msup> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>0</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{{\text{e}}^{{\text{ln}}\,{k^2}}} - \frac{1}{2}{{\text{e}}^0}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </mrow> </msup> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>0</mn> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>Setting their equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 12"> <mo>=</mo> <mn>12</mn> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> their equation must be an integrated expression with limits substituted.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{k^2} - \frac{1}{2} = 12"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>=</mo> <mn>12</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{k^2} = 25 \Rightarrow } \right)k = 5"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>25</mn> <mo stretchy="false">⇒</mo> </mrow> <mo>)</mo> </mrow> <mi>k</mi> <mo>=</mo> <mn>5</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award final <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \pm 5"> <mi>k</mi> <mo>=</mo> <mo>±</mo> <mn>5</mn> </math></span>.</p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> have the following vector equations where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>,</mo><mo> </mo><mi>μ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub><mo>:</mo><msub><mi mathvariant="bold-italic">r</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub><mo>:</mo><msub><mi mathvariant="bold-italic">r</mi><mn>2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
</div>
<div class="question">
<p>By using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>d</mo><mi>x</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><mi>cos</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>∫</mo><mfrac><mi>u</mi><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>-</mo><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>d</mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p>attempt to use partial fractions <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mi>u</mi><mrow><mfenced><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>⇒</mo><mi>u</mi><mo>≡</mo><mi>A</mi><mfenced><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mi>B</mi><mfenced><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math></p>
<p>Valid attempt to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>u</mi><mrow><mfenced><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mfenced><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mfenced><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><mfenced><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mfenced><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>u</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>u</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>u</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone the absence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>C</mi></math> or lack of moduli here but not in the final answer.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mi>C</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone further simplification of the correct answer.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A right circular cone of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> is inscribed in a sphere with centre O and radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> as shown in the following diagram. The perpendicular height of the cone is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>, X denotes the centre of its base and B a point where the cone touches the sphere.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the cone may be expressed by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{32\pi {R^3}}}{{81}}">
<mfrac>
<mrow>
<mn>32</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to use Pythagoras in triangle OXB <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {r^2} = {R^2} - {\left( {h - R} \right)^2}">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>h</mi>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>substitution of their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2}">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> into formula for volume of cone <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{{\pi {r^2}h}}{3}">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi h}}{3}\left( {{R^2} - {{\left( {h - R} \right)}^2}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>h</mi>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi h}}{3}\left( {{R^2} - \left( {{h^2} + {R^2} - 2hR} \right)} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>h</mi>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> This <strong>A</strong> mark is independent and may be seen anywhere for the correct expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{{\left( {h - R} \right)}^2}}">
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>h</mi>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi h}}{3}\left( {2hR - {h^2}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>h</mi>
<mi>R</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at max, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}h}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>h</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}h}} = \frac{\pi }{3}\left( {4Rh - 3{h^2}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>h</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>R</mi>
<mi>h</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 4Rh = 3{h^2}">
<mo stretchy="false">⇒</mo>
<mn>4</mn>
<mi>R</mi>
<mi>h</mi>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow h = \frac{{4R}}{3}">
<mo stretchy="false">⇒</mo>
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</math></span> (since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h \ne 0">
<mi>h</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span>) <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{V_{{\text{max}}}} = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mrow>
<msub>
<mi>V</mi>
<mrow>
<mrow>
<mtext>max</mtext>
</mrow>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> from part (a)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3}\left( {2R{{\left( {\frac{{4R}}{3}} \right)}^2} - {{\left( {\frac{{4R}}{3}} \right)}^3}} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3}\left( {2R\frac{{16{R^2}}}{9} - \left( {\frac{{64{R^3}}}{{27}}} \right)} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mfrac>
<mrow>
<mn>16</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>64</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>27</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} = {R^2} - {\left( {\frac{{4R}}{3} - R} \right)^2}">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} = {R^2} - \frac{{{R^2}}}{9} = \frac{{8{R^2}}}{9}">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>8</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {V_{{\text{max}}}} = \frac{{\pi {r^2}}}{3}\left( {\frac{{4R}}{3}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>V</mi>
<mrow>
<mrow>
<mtext>max</mtext>
</mrow>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{4\pi R}}{9}\left( {\frac{{8{R^2}}}{9}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>π</mi>
<mi>R</mi>
</mrow>
<mn>9</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>8</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{32\pi {R^3}}}{{81}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>32</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f,\,\,g,">
<mi>f</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>g</mi>
<mo>,</mo>
</math></span> defined for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question">
<p>Hence, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mi>π</mi>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>Attempt to add <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) + g'\left( x \right) = - 2{{\text{e}}^{ - x}}\,{\text{sin}}\,x">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} = \left[ { - \frac{{{{\text{e}}^{ - x}}}}{2}\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)} \right]_0^\pi ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mi>π</mi>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mi>π</mi>
</msubsup>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: Condone absence of limits.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {1 + {{\text{e}}^{ - \pi }}} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>π</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I = \int {{{\text{e}}^{ - x}}} \,{\text{sin}}\,x\,{\text{d}}x">
<mi>I</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - \int {{{\text{e}}^{ - x}}} \,{\text{cos}}\,x\,{\text{d}}x">
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <strong>OR </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x + \int {{{\text{e}}^{ - x}}} \,{\text{cos}}\,x\,{\text{d}}x">
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - \int {{{\text{e}}^{ - x}}} \,{\text{sin}}\,x\,{\text{d}}x">
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I = \frac{1}{2}{{\text{e}}^{ - x}}\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)">
<mi>I</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x = \frac{1}{2}\left( {1 + {{\text{e}}^{ - \pi }}} \right)} ">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>π</mi>
</msubsup>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>π</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>