File "markSceme-HL-paper2.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AI/Topic 3/markSceme-HL-paper2html
File size: 683.83 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-02ef852527079acf252dc4c9b2922c93db8fde2b6bff7c3c7f657634ae024ff1.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-9717ccaf4d6f9e8b66ebc0e8784b3061d3f70414d8c920e3eeab2c58fdb8b7c9.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../../../../../../index.html">Home</a>
</li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li class="dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Help
<b class="caret"></b>
</a>
<ul class="dropdown-menu">
<li><a href="https://questionbank.ibo.org/video_tour?locale=en">Video tour</a></li>
<li><a href="https://questionbank.ibo.org/instructions?locale=en">Detailed instructions</a></li>
<li><a target="_blank" href="https://ibanswers.ibo.org/">IB Answers</a></li>
</ul>
</li>
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
<div class="pull-right screen_only"><a class="btn btn-small btn-info" href="https://questionbank.ibo.org/updates?locale=en">Updates to Questionbank</a></div>
<p class="muted language_chooser">
User interface language:
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=en">English</a>
|
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=es">Español</a>
</p>
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="https://mirror.ibdocs.top/qb.png" alt="Ib qb 46 logo">
</div>
</div>
</div>
<h2>HL Paper 2</h2><div class="specification">
<p>An ice-skater is skating such that her position vector when viewed from above at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds can be modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>with respect to a rectangular coordinate system from a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, where the non-zero constants <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> can be determined. All distances are in metres.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the displacement of the ice-skater is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> and the velocity of the ice‑skater is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the velocity vector at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the magnitude of the velocity of the ice-skater at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></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>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the magnitude of the velocity of the ice-skater when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></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>At a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, the ice-skater is skating parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis for the first time.</p>
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OP</mtext></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>use of product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mover><mi>x</mi><mo>˙</mo></mover></mtd></mtr><mtr><mtd><mover><mi>y</mi><mo>˙</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi><mo>+</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="|" close="|"><mi mathvariant="bold-italic">v</mi></mfenced><mn>2</mn></msup><mo>=</mo><msup><mover><mi>x</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>+</mo><msup><mover><mi>y</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced open="[" close="]"><mrow><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi><mo>+</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> It is more likely that an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">v</mi></mfenced></math> is seen.<br> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mover><mi>x</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>+</mo><msup><mover><mi>y</mi><mo>˙</mo></mover><mn>2</mn></msup></msqrt></math> is not sufficient to award the <em><strong>M1</strong></em>, their part (a) must be substituted.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>+</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>t</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mn>2</mn><mi>b</mi><mi>t</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>=</mo><mn>1</mn></math> within a factorized expression that leads to the final answer <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mn>2</mn><mi>b</mi><mi>t</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>magnitude of velocity is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>=</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>t</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math> result from part (b) is an alternative approach.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn><mo>×</mo><mn>2</mn></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfenced></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>51</mn><mo> </mo><mo> </mo><mo>(</mo><mn>1</mn><mo>.</mo><mn>50504</mn><mo>…</mo><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mfenced><mrow><mi>b</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>t</mi><mo>=</mo><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>53</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>53086</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>correct substitution of their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>(M1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>697</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>696591</mn><mo>…</mo></mrow></mfenced></math> <strong>and </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>488</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>487614</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>use of Pythagoras / distance formula <em><strong>(M1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OP</mtext><mo>=</mo><mn>0</mn><mo>.</mo><mn>850</mn><mo> </mo><mtext>m</mtext><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>850297</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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>A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and a set of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axes.</p>
<p>In each case, the drone moves to a new position represented by the following transformations:</p>
<ul>
<li>a rotation anticlockwise of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> radians about <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math></li>
<li>a reflection in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><msqrt><mn>3</mn></msqrt></mfrac></math></li>
<li>a rotation clockwise of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> radians about <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</li>
</ul>
<p>All the movements are performed in the listed order.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a single matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> that defines a transformation that represents the overall change in position.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state what the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup></math> indicates for the possible movement of the drone.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Three drones are initially positioned at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>. After performing the movements listed above, the drones are positioned at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>′</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mo>′</mo></math> respectively.</p>
<p>Show that the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> is equal to the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> .</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a single transformation that is equivalent to the three transformations represented by matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></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><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>rotation anticlockwise <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr></mtable></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>(M1)A1</em></strong></p>
<p>reflection in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><msqrt><mn>3</mn></msqrt></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> <strong><em>(A1)</em></strong></p>
<p>matrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p>rotation clockwise <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>an attempt to multiply three matrices <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></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><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if final answer not resolved into the identity matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>if the overall movement of the drone is repeated <strong><em>A1</em></strong></p>
<p>the drone would return to its original position <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mtext>det</mtext><mo> </mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced><mo>=</mo><mfenced open="|" close="|"><mrow><mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced><mo>-</mo><mfenced><mfrac><mn>1</mn><mn>4</mn></mfrac></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <strong><em>A1</em></strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><mfenced open="|" close="|"><mrow><mtext>det</mtext><mo> </mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced></math> <strong><em>R1</em></strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> <strong><em>AG</em></strong></p>
<p><br><strong>Note:</strong> Award at most <em><strong>A1R0</strong></em> for responses that omit modulus sign.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>statement of fact that rotation leaves area unchanged <strong><em>R1</em></strong></p>
<p>statement of fact that reflection leaves area unchanged <strong><em>R1</em></strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext><mo>=</mo></math> area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></math> <strong><em>AG</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> For clarity, exact answers are used throughout this markscheme. However it is perfectly acceptable for candidates to write decimal values <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>e.g.</mtext><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>866</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>attempt to find angles associated with values of elements in matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>cos</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd><mtd><mi>sin</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mi>sin</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd><mtd><mo>-</mo><mi>cos</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math></p>
<p>reflection (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>tan</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mi>x</mi></math>) <strong><em>(M1)</em></strong></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p>reflection in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>tan</mi><mfenced><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>12</mn></mfrac></mrow></mfenced><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>268</mn><mi>x</mi></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </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;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem. In most cases these good attempts were undermined by a key lack of understanding. Whilst candidates were able to find the correct matrices in part (a)(i), they then invariably went onto multiply the matrices in the wrong order in part (a)(ii). Whilst follow through marks were readily available after this, the incorrect matrix for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> then caused issues in part (c). If these candidates had multiplied correctly, it seems that many of them could have gained close to full marks on this question. At the same time there was a lack of precision in the description of the transformation in part (c). As a general point, it would also help candidates if they resolved the trig ratios in the matrices; writing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></math> rather than, for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mfenced></math>. Finally, there were many attempts in part (c) that suggested candidates had a good knowledge and understanding of the concepts of matrices and affine transformations.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A ball is attached to the end of a string and spun horizontally. Its position relative to a given point, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, 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 the equation</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> where all displacements are in metres.</p>
</div>
<div class="specification">
<p>The string breaks when the magnitude of the ball’s acceleration exceeds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the ball is moving in a circle with its centre at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and state the radius of the circle.</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 an expression for the velocity of the ball at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></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>Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.</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>Find an expression for the acceleration of the ball at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at the instant the string breaks.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>How many complete revolutions has the ball completed from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> to the instant at which the string breaks?</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.iii.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">r</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn></math> <strong>R1</strong></p>
<p> </p>
<p><strong>Note:</strong> use of the identity needs to be explicitly stated.</p>
<p> </p>
<p>Hence moves in a circle as displacement from a fixed point is constant. <strong>R1</strong></p>
<p>Radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">v</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> <strong>M1A</strong><strong>1</strong></p>
<p> </p>
<p><strong>Note:</strong> <strong>M1</strong> is for an attempt to differentiate each term</p>
<p> </p>
<p><strong>[2 marks]</strong></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 mathvariant="bold-italic">v</mi><mo mathvariant="bold">∙</mo><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced><mo>∙</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> <strong>M1</strong></p>
<p> </p>
<p><strong>Note:</strong> <strong>M1</strong> is for an attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">v</mi><mo mathvariant="bold">∙</mo><mi mathvariant="bold-italic">r</mi></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>×</mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>×</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>A</strong><strong>1</strong></p>
<p>Hence velocity and position vector are perpendicular. <strong>AG</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> <strong>M1A1A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>400</mn></math> <strong>(M1)(A1)</strong></p>
<p> </p>
<p><strong>Note:</strong> <strong>M1</strong> is for an attempt to equate the magnitude of the acceleration to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>18</mn><mo>.</mo><mn>3</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>18</mn><mo>.</mo><mn>256</mn><mo>…</mo></mrow></mfenced><mo> </mo><mfenced><mtext>s</mtext></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Angle turned through is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>18</mn><mo>.</mo><msup><mn>256</mn><mn>2</mn></msup><mo>=</mo></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>33</mn><mo>.</mo><mn>329</mn><mo>…</mo></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>33</mn><mo>.</mo><mn>329</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>33</mn><mo>.</mo><mn>329</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><mo>=</mo><mn>5</mn><mo>.</mo><mn>30</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> complete revolutions <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">c.iii.</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.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">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A transformation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, of a plane is represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>′</mo><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">r</mi><mo>+</mo><mi mathvariant="bold-italic">q</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mo>×</mo><mo> </mo><mn>2</mn></math> matrix, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mo>×</mo><mo> </mo><mn>1</mn></math> vector, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi></math> is the position vector of a point in the plane and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>′</mo></math> the position vector of its image under <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<p>The triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OAB</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. Under T, these points are transformed to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced></math> respectively.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">S</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math> are matrices.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math> represents an enlargement with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math>, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math> represents a rotation about <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The transformation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> can also be described by an enlargement scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>)</mo></math>, followed by a rotation about the same centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>)</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, show that</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math>.</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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">S</mi></math> to find the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></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 find the angle and direction of the rotation represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an equation satisfied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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">[3]</div>
<div class="question_part_label">d.ii.</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"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </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><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>3</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>3</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong><em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">S</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><msup><mi mathvariant="bold-italic">S</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>is for an attempt at rearranging the matrix equation. Award even if the order of the product is reversed.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">R</mi><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo> </mo></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math></p>
<p>attempt to solve a system of equations <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>a</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>c</mi><mo>,</mo><mo> </mo><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>d</mi></math> <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for two correct equations, <em><strong>A2</strong></em> for all four equations correct.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>866</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>866025</mn><mo>…</mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>866025</mn><mo>…</mo></mtd></mtr></mtable></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The correct answer can be obtained from reversing the matrices, so do not award if incorrect product seen. If the given answer is obtained from the product <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi><mo>=</mo><msup><mi mathvariant="bold-italic">S</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="bold-italic">P</mi></math>, award <em><strong>(A1)(M1)(A0)A0</strong></em>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>clockwise <em><strong>A1</strong></em></p>
<p>arccosine or arcsine of value in matrix seen <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Both <em><strong>A1</strong></em> marks are dependent on the answer to part (c)(i) and should only be awarded for a valid rotation matrix.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.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"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold-italic">q</mi></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"><mfenced><mtable><mtr><mtd><mi>x</mi><mo>'</mo></mtd></mtr><mtr><mtd><mi>y</mi><mo>'</mo></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mi>x</mi><mo>-</mo><mi>a</mi></mtd></mtr><mtr><mtd><mi>y</mi><mo>-</mo><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note</strong>: Accept substitution of <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> (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>'</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo></math>) with particular points given in the question.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi mathvariant="bold-italic">q</mi></math> using simultaneous equations or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><msup><mfenced><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mi mathvariant="bold-italic">q</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>651</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>651084</mn><mo>…</mo></mrow></mfenced><mo>,</mo><mo> </mo><mo> </mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>48</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>47662</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mrow><mn>14</mn><mo>+</mo><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">P</mi><mfenced><mtable><mtr><mtd><mn>0</mn><mo>-</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>-</mo><mi>b</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> This line, with any of the points substituted, may be seen in part (d)(i) and if so the <em><strong>M1</strong></em> can be awarded there.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">P</mi></mrow></mfenced><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>651084</mn><mo>…</mo><mo>,</mo><mo> </mo><mo> </mo><mi>b</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>47662</mn><mo>…</mo><mo> </mo></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>+</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mrow><mn>14</mn><mo>+</mo><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow><mn>13</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (i) proved to be straightforward for most candidates. A common error in part (ii) was for candidates to begin with the matrix <strong><em>P</em></strong> and to show it successfully transformed the points to their images. This received no marks. For a ‘show that’ question it is expected that the work moves to rather than <em>from</em> the given answer.</p>
<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;">
<p>(b), (c) These two parts dealt generally with more familiar aspects of matrix transformations and were well done.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(b), (c) These two parts dealt generally with more familiar aspects of matrix transformations and were well done.</p>
<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;">
<p>The trick of recognizing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math> was invariant was generally not seen and as such the question could not be successfully answered.</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A sector of a circle, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math>, is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbsAAADfCAYAAABve33mAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAACN2SURBVHhe7d0LVFV12j/w78tyWROOr6mpkRGp5IhW6KtY6qRmguNrrmNarziKVGCmxghjmVKO4d0xcSgvr5fCu2OpZ1k5ilrYaBk6ZKkUUUZYDJWSQzCVy/95//vZZ2+5eIADnMve+3w/a7ng/M6Fc8H98Pv9nv08//F/ChAREVlYkPaViIjIshjsiIjI8hjsiIjI8hjsiIjI8hjsiIjI8hjsiIjI8hjsiIjI8hjsiMi3SuxICO2I0NBpsJdcUQbOw55wl3I5Bum55c7bUB0cKC/IRmbaZuTK21erMhRkZyIt8xTqvJmheO93gcGOiMhMSvZi+pDxmLP7e23AlSvK3xSzMSTuOez+waGNBTYGOyLys1thW/8xiooOILlXC22MyLMY7IjIiy6jJHcrUod1RWhoR3RLyMDhTy9q1+lcLV1Vv9/V+xaUadcrHCXI3ZyKYer1DyLVfhYF9mnO2yfYUaLc5EruckSqlzfgcOZkdJPHSc2G+ijV7i//opCQfhAF5TITkpmR/lgbkH04Awnd5DZdMWzOYZQ4lOd3Yk3lWKpdu58rlY8VmX4Yn1R9rNStyC25rN1OyOu2Iz0hyvmza95GloCjpiFLvi9dDlsneczcGsuU8vOmIyrJrl4qTR+JTlXfW3nd9uXac5B/ynu3+YTympxXu1aGgqvPu7bnrigvQHZmlfd0WCoyswtQuSDpxufqgqMkF/b0x9XPz/mzM5Fdz31qYrAjIi9xoDx3DeJtM7E5r0Idqchaikfj/uQ8WNfBUbANk6rcT6j3HbkM2WVyVL6E3L88CVvqRuSp136IzUmxeGrNKfXSNbL+hEfnvAl5tOvatsQN+BkFm/5Y5f6iGFnpj2LkonedwVCn3Dfu0aXIUp9KBfIypyB+0iTEj55fObZ5Jv74eoHyiutWmj4RMVUfS7nf7196T/t5+vs1DelZxeqIfhvb4KdhL64RWBpFe9+SlmvPQSjvXeooDJ5iR7HLF+BAWfYyjLz6vIX2vOLXIFcP8o4i2Gf8HnFzqryneRsxJ+5RLMq+oF6s/3N1ofwE/hIfi6T0A8pPFfKzn0PcyOcb9J4w2BGRl5Ti5K6d6oEvOHoRDuYVoagwB5viezqvrlU5Tr25HacQguEZ76Gw6GsUHn8Zw4OVqyq+QOG3ygGu7BR2rT2hDIQgOu0A8uQ2OemIQm37WD0RvylHeaxPsT/2DjS7koc3VxxRnthoZBw/h6KiczieMRrqjzhdiG+rHXf1+56BPbmPclk52GZdRFSNsVNHP8F36u3rEDwCaQfzqv+8t3LxuUzNHAV4fe5K5f1SXtOMncgp/BpFeYeRMUF5vyp2YdZKJSh2sGF9zsuIlsdqnQL7ua9xKrkXmsnlq5qhg20FcjJs6qXWyXtxTlsidhTYMTdded+CYzBj10nl+Rch7/DLmBARjIp9y7DyXWdQqu7f+PzDD5RXGIzItMPq51GUtwfJyn2QtxO7TpYqt1EC4rvrMWufEqSveY2F2Lz8TRQ43Phcr3EZxYe2YK3ySxQRv9H5nhTl4WDaCARX7Meav35UZdZYNwY7IvKOK0X48K1C5ZvumDRlFLq2UA43QSEYNOVx58G6Vi3QK/mAclA7hD+2+gh77Rvw/GMzsU/9s/4CfvjxChzfFuK0XG49FlPGd1fuoTx0h4GYMnmo3OharQfDNiBEOeC1QIcOyq2b9ULyKeXAeWIaWuXsgz3zBTyWtMs5cygqxY9Vg13kaEy4T+7bEl163qkGqMqxVrhr4G/RWr1h/YIfGouHurZUvmuOkHsGo79zWOX44n3sPqU8A3lNU/qhgxydW3SF7Snn+1Wx+23k1jb7ccvP+OLYQSXYKD9i0lRM6dNBef7KOxI+Ak+p71shdh88U31Wq2qGX9/YVvmqBPQ5IzE8dQPsh75Bl4VHlIB1FAsGyXWX8W3hF+r7V+012v6CTyQ47o1HeFD9n+u1yvDZiX8oj6v8gZE5EVFhsowZgaHqLF0Ze+cs/unmW8JgR0ReFo7bQ67Xvle0DUX3OqODvh8WgSFxTyIp6U/Vlr2E48dSFMk3vTsh5Oq0ppny0J1cB57Q1vh11aOdowQnMh5Ht4jBiEuahqSqS281tWul3TcIN7RsheuqjTWMcwnVNdevSaG/XxWluPTvpgS7K/jxB5m5tUbv22+qMhusfN8qvr2kzONquh7hcS9i14wYJdDLEuKflM9kGqbaeiNs2Hxkq/t2+mPX9Rrr/1yv9RMufVvH3lzNP0zqwGBHRN4RFIwbQ2UeVIAvi392jokLRTgrK1+1cXyJ/fPSkVURgujkZVhpP4nCc3uRrEaxtrjx180Q9OvWCJWLJ8+h+OqE4Iry0Ofg8qFrBCfHF/sxb9kBVATHIHnFethzzuGcPcUZKGsGRh9x/ZoU+vsV3BqtbmjKE9NnaKU4+eX3VZJaKt+34PatXAeqoA7ok7RBmaWdQ459PTIyliE5OgTIW4Mn1T1H/bGBXy6UuQiYCjc+12v9Cq3ayyyxNaIz3ldmhbKMWeXfqRT0cnU3FxjsiMg7gtog7M6blG/OYu2qPciXRAZHMbJXbag7QeW7T3BUlvNwC7oPHIYHe7XGd0f34UCVKBbUPgx3Shwt3YFVW86q+zaOkiNYteagen3druC7syfV5TzcdjcGRkejV7sLOGp/x3Wg9JGgm7tjsOyDyWta9Z4zO7I8H/aXnO9X8EP3o1dL5ZCtz/RKz6HogqulP1E5Wys9WwTnnKs5br6rLyJkbO1KrDpRAockxRS8iZfU9y0MDw3tAQkt1TjykTlSsiejkJBZgBa9hsFmG4aB3W9Rr3bOBpujfVhn5x7k7h3YnS+zMUm4yXBmZnabg+yCM/V+rtdqgVvCb1O+liJrzUbnLFISYSZLtmpXjMzMV36KexjsiMhL2uK+qTPUBISKrFkYGhGK0LAoxGV+qF1fi3bdMCBSDpsnkG7roRzUOiEqbo22zKjt7bS8B/EzByqXi5E1JwYRygE1LGopijuGqbeqWzO0694bkfJt3lLYaj6vBiyNeVSLuxH77AQlGCmvadkjzv2piCFI2qw8r+DRWDS1X41AZEdSVJiLUw9qyJqGKPXUg3+jReQYPCsJQhUHsGx0b4SFhiJiyDR1OTF4+AxMvc85O6smKBxj5k51Pi/tvQ4N7QGbJLooAXLC+AGQ3b+W9yVg0XBltlfxJuYMjVBuozy2banyuQUjYtKD6B3eo/7P9RrXI3zMdC0ZZg3iojopn1U/JKmJMMPwWPTtbgcxBjsi8pqgkOGYu3WJmu0ngqOfwaubXqg7QSWoK+LWvooZskwmIiZigf3vOJgmwa0Q73z8jfLXvHMfyb5gojpTcWYXrkfaiHC5VK+g8HFYu+s5RKtPSzkYT1gC+/F9SJODcek/8PGXVZZdfaY5Ogx6Bpn2l51LhCrtub3zZ9hCmjuHmvVA7LrK594B/w+unm2zyHFYp+6zCVmcVIKJJAjNXQd7Rop2f9ETExbswTurbAhxGRGClNnc5BrPS6F8LmmbXsUsNUFFERQK27Kt2JSmfSZClokztiPzD33Qwq3P1YUWffCHzO3ISNZfi/Kw8nu0d17le+KG//g/hfY9EZE5XMlFeu+RSC9VgkHyFryerBxMZYl07hPqDE3S7U9ek5JPgYzBjohM6AKyU22I2yynNtTUB8n2V5Hcq5V2mUjmp0REptMWg2a9Wn3JTBEcnYIM+2r8gYGOauDMjoiILI8zOyIisjwGOyIisjwGOyIisjzu2RlAaWkpLl6s7PH11VeFKC93XTMuPz8f//rXv7RLlSIiItCihevGl927d9e+A2644Qbccouz8gERUaBgsPOygoIC9evZs2fVrxKsLl68gM8//wInT+aoYw3xm990uyZY/fhjOXJyjmuX3BcaehvuuedeNG/eHFFRUerY7bffrgbEjh074le/+pU6RkRkdgx2HqDPzCSg/fOf/0RhYSGOHTuKoqKvtFtUqhqsunXrpn6VoNKhQwf1e9Gy5X8q/66pUNcov/zyC77/vrLL1qVL/8IPPziL0RUVnUdFRXmdwXLs2Fi0adMWXbt2VQNh27ZtOTMkItNhsGugb775Bl999RXOnTuH06dPY/v2rdo1lYYMeQDt2rVTgkQb3HxzCK6//jrcdFM7XHed2hzE0L7++rz69csvv8RPP/2kBm55nd999606rhs27Hfo2bMXwsO74LbbwpSv7pVpIiLyBwa7OsjB/rPPPlMP9q4Cmx7UZMlPZmZmCWiNJYFQZobFxd+4DIJ6ALz77ruVGexv0Lq12ruDiMjvGOyqkOXITz/9FB999BEOHTqEEyc+0K4BoqLuQefOndClSzjat2+vBDZpXUJlZWVKwPtOmel+oQbAw4cPadc49wQffHAk/uu/eqFbtwgufxKR3wR8sJPAJjOUd989gv37/6aNOmdtsqcm+1RWn7F5mswAS0pKkJ//GY4ff//q7K937ygMHTpUnfn17NmTCTBE5DMBF+z02ZsEt1WrVmqjzplbVFQfNbh17HirNkqe8P3336unU0jw27vXro0Cv/vdcDz88MOc9RGR1wVEsJMAl5v7D7z22mv429/2qWPt2rVXZm9D0KPHnbj11ls5c/Mhmfnl5eXh2LFjyh8en6hjffr0xahRozB48GAGPiLyOMsGO1cBTtL++/fvryZRcM/NGGTP77PP8nHkyLtXT39g4CMiT7NUsJPsSTm/rWqAk+XJgQPvwx13dPXYuWvkHRL4ioqK1H1UfblTD3zDhw9ndicRNZolgp0cHCW46XtwnMGZn6sZn+zxTZwYz+QWImqwhgW7smyk9n0C/5iktcHXhv1Blin37duH3bt3q2W39D243r17M8HEYiTwSVZn1T2+WbNS8cADD/BkdiJySwOC3WUU25/GkKRdqAh+DJs+mItBLX3fNKHmLE5OEejXr7960GOSifV9/vnnOHNGTvDfpl723WzvCkrs0xGVVJlNWk1wDJIXTMSI6AEIb8FmIkRG436wc+Qj0zYDR9tdRlZWBSZssmPBoLbald6l78WtXr1GPdFbn8Xdf/8Q7sMFKKn5efr0x3jjjTfU2V5oaCgmT57i/b099f/BSMxptwQ5621QK5qWF+DwusWYln4AGP4yDq+yIYTxjshQ3Ax2DpRlz0Xf5Z1gX9kK6dHTcGSg9/9T60uVq1evwvnzRepe3MMPP8JZHFUjs7233z58tXrLzJmzEB0d7aUlzvOwJ/w3kpBWGeyEHgRP9Uba4Q2ID79eu4KIjMDNUFWK3IMfYuBj9yP81gEY+1AYKva9hqwvftau9ywJcitXrkRk5F2YPftZ3HHHHXjxxeV44YU09OjRg4GOqunSpQsmTXoC69ZtQGzsOCxZskiZ+Q9Wgt7Mqy2WvC6oDcLuZDIUkVG5FewcBW9i+e6eGHt/R+UOrdFr6P0IxknsPvaVMufzHDkwLV68SA1ycsCSA5ccwORAxqQTqo8sadtso7Bp0xZMm/aUOtOToDdmzGi899572q28oQwF9gws3lyI4OEPI7ozZ3VERuPGMuYl5KY/gbk3psEe39UZHctPIH3MeKR/9T8eSVSRIPfKKxuwdesW9bIEOe7HUVPJvl5OzgfYuXOnWp8zKqovUlL+iH79+mm3aAxtGTPL2ROwGj8mbhFR3eoPdurpBuOxuUK7XE1YkxJVZLnyz39eqga59u07YMSIEWrnbAY58iQJevIH1Wuv7VSTWSToPf/8HLUgdcO52LNzlCB37zasmrUcWRiBNPtSxHfl7zCRkdQT7H5GQebjGFmQiA8WDELV/76OgkzYhjyHgglbrrmuPhLktm/fri5VCpnJSQo59+LI2/7+93evzvRiY3+PhISEBiay1JKgoidxxb2Cisj5OGyPRzgneESGUcd/RwfK819TAtIFTBrd+5pgFhQ+BI9Ft0bF5hexxJ6Pcm28LnIKwZYtW67Zk5N9FgY68oXf/vY+LFv2YrU9vVmznlX/ACMi66o12F3JXYEBQ2chq+Is0m39kGA/r10jqu5bfIjNSUMQkWBHifNKl+x2O4YOfUDNrhw50nY1yHHJknxN/rCqGvRkGV3+AFu9erX6B1mdykvw5de/aBd0l1GSux1LFv8VFQjB8MfuR2fO6ogMxf2TyhtJKp6kpaWpJ4NLtRM56ZeZlWQkUo5MTk6X4tNycvqMGc8of4jZtGt19VRQUQRHp2DB+AcRPSjcr6X0iOhaXgt2VZNPpOLJE09MVs+RIzIq6bP317/uVAtPSxLLokWLWXuTyCK8EuxeeeUVzJ07R/1elonkwME9OTKLM2fO4H//d42axDJ69Bg8/fQzCAkJ0a4lIjPyeLA7evQoxo0bq10iMr933jmCzp07a5eIyIy8MrNbtGgRgoKC1HJN5B2hoR1RVPS1dqnhmnr/QDB79mxERERg/Pjx2ggRmZVXcsamT5+Offve8l1dQiIPk8QqKTA9evRobYSIzMwrwU76isnmfmpqqjZCZB5y+kFS0lPq7y87ohNZg9fOBpL6g1KNXs6vIzKTXbt2oX//AY0sJ0ZERuTV8+y++eYb3HtvX5w69bF3G2oGIO7ZeQd/Z4msyat1Hm655RYsXLgYy5Yt00aIjE36KGZkvMxAR2QxXi9qJBv8x44dVTf8iYxMet5JUsq11VOIyOy8Huxkgz8j4yUsWLCg/rqDRH4iv5tSEFp+T4nIerwe7IRs9EuyyoEDB7QRImPJzMzE8OH/zfJgRBblk2AnZsyYgaSkaWoCAJGRyPmgO3ZsR2JiojZCRFbjs2AnG/6y8S8JAERGsnz5cjz//PNMSiGyMJ8FOxETE6MmAEgiAJERHDp0UP36wAND1a9EZE0+DXaSrCIJAJIIwGQV8jdpQzVv3jykpKRoI0RkVT4NdkISACQRQKpUEPnTunXrMHZsLJNSiAKAz4OdkESAdevWslA0+Y387kmx8vj4eG2EiKzMq+XC6iI1M+VUhNWrV2sj1BBS7qupArlc2COPPIJJkxK5V0cUIPwW7MSTTz6J0aMf4gGnEVjbsvHkD62cnBwsXLhQGyEiq/NrsJOlpISEx7F//wG2UmkgBrvGkaSUyMi78P77H6i1W4koMPhlz04niQGSIJCRkaGNEHmXFCWX4uQMdESBxa/BTkiCALuaky9IMXIpSs7u40SBx+/BTpYvpXoFu5qTN8l5nXKOpxQl55I5UeDxe7ATkqDSpk0bdjUnr5HMXylGzu7jRIHJrwkqVUmB6HHjYrFnj501Ct3ABBX3sfs4ERliZickYSAxcZJa1YLqx0DnPnYfJyLDBDshiQOSrMKu5uQpevdxKUJORIHLMMuYOjk4rVixAhs3bmQiATWJJKUMGxaD9es3sP4lUYAz1MxO9OvXj13NySOk2Di7jxORMNzMTuhVLphQQI2lV+dhwhMRCcPN7IQcnCShQKpdEDWGdB9PTk5hoCMilSGDnWBXc2os6T5+8eJF2Gw2bYSIAp0hlzF1kpWZlPQUC0WT25iUQkSuGHZmJ6TaBbuau+aJfnZWJEXF2X2ciGoydLATeldzqYJBVBd2Hyei2hg+2EmCgSQazJ8/Xxshck2KiUtRcS55E1FNhg92QhINJOFAEg+IXJEi4lJMnF3vicgVQyeoVMWu5tWxEHQlOS9z1Cgbtm3bzqasROSSKWZ2QhIOJFklMzNTGyFykuLhUkScgY6IamOaYCeSkpKwY8d2djWnq+T0FElKYfdxIqqLqYKdLF9KAoJUxyDSu48vWrSYS9tEVCdTBTuhJyCwqznp3celeDgRUV1Mk6BSFYv8MkGFxcKJqCFMGezEli1b1BPNZ86cqY1QIJk9ezaioqJY/5KI3GK6ZUyd3tWcySqBh93HiaihTDuzE+xqHnj0Qs8ZGS+ptVOJiNxh2pmdYFfzwCNFwfv3H8BAR0QNYuqZnZB9u3vv7ctEhQAgn/W4cbHsPk5EDWbqmZ2Qqhnsah4YpBg4u48TUWOYPtgJvau5VNMga2L3cSJqCtMvY+rY1dy62H2ciJrKEjM7IQkLkrhg+q7mjmJkz3kQoZHLkXtFG3MhkDqVS/FvKQLOQEdEjWWZYCdmzJiB2bOfNXFX88so3rsET2Z+qF0mOY9Sin9LEXAiosayVLCTxAVJVlm5cqU2Yi6O4n1Im3sGt90RrI2QFP1m93EiaipLBTshCQySrCInnJtK+VlsmmNH11fSMTnsOm0wsOnFvtl9nIiaynLBTkjbl1mznlUTG8zhEnLX/RlHBzyNxF43amOBTQo9p6cvR0pKijZCRNR4lgx25upq7kB57has+mEC5sd1RwttNNDp3ceZlEJEnmDJYCcSExNN0dXcUfIOlq4CpjwzGB0s+2k0jHxm7D5ORJ5kmfPsXJETkXft2o3Vq1drI0ZzBSX26YhKqq0RbWtEZ7yF9bZbtcuVrNrPTpaeJ06ciOnTp7MpKxF5jKXnEnpigwQ9Y2qGDraX1aBV+e99ZES3VuJcCuznPnYZ6KyM3ceJyBssv3AmCQ7z5s1TEx7I2OQzSkqahqlTp2ojRESeYflgJwkOY8fGqgkPVmLFJUwp5r1w4WK1uDcRkSdZes9Ox9qKxie1TeWUETbiJSJvCIhgJ/Su5jt37tRGyCjYfZyIvC1gkt31ruZ6VQ4yDnYfJyJvC5iZnWBXc+PhZ0JEvhBQpzFL4oMkQLCruXFI0W4p3s1AR0TeFFDBTkhVjmPHjrKruQHIPqoU7Wb3cSLyNmMHOzcbmaocRbBPjlIrizj/dcXIzHw4tKt1kukniRCS+WeeQtHWI++9FOuWz4GIyNsMHOwa0sjUgfJTdqwpHA97XpFWiSQfe+O7unyBkgghySpSrYP8g93HiciXDBvsGtTI1PE1Dq19C2GTbYhs4d5Lkq7mUq3DrF3NZfZqVnr3cSnWTUTkC8YMdg1qZOpA2bvrMWvfWexLGoNJ6a8hu6BMu652Zu9qbmZ693EmpRCRrxgw2DW0kWkQWg5KwydF55Bjn4sBFzYhbsgDSMg8i3LtFrWJiYkxZ1dzE9OLcrP7OBH5ksGCXVMamTZHh17DEb9gKw6m9cKxOc9hXe4l7TrXJFnFfF3NzUsKPUtRbnYfJyJfM1Sw80wj05boGpeMmZF52HjkHOpL4tS7mksVD/IuKcYtRbmZlEJEvmagYHcF3x3fg8ysxbBFhGqnD9yLpKxSZUqwHLZOdyHBfl67bT2C2iDszpu0C/WTRIl169Yavqu5mendx+Pj47URIiLfMVCw82AjU8dFFJ6+BRMHdlIetX6SKJGcnKImTpB3pKamqkkp7GhARP5gsD27RnCUIPcNO97ILXGeQF6eD/vzz2LH4BlI7NVKvYk79Coexu1qbl5SfFvOa2RSChH5iwmD3XnYE+5CaOg02EtkR86BshMbMNXWG2Gy9DlmEy4NXYrXk/s0MMGlsqs5k1U8h93HicgIAqrrgTtWr16NsrIyzJw5UxsxJtnTNEO38tmzZyMiIgLjx4/XRoiIfM/8y5geJgkUkkhh9GQVMwQ6KbYtRbel+DYRkT8x2NUgCRSSSCEJFdR4shQs5zBK0W0mpRCRvzHYuSCJFG3atGFX8yaQItuSlMLu40RkBNyzq4UUiB43LhZ79thZw7GB2H2ciIyGM7taSFfzxMRJatUPahh2Hycio2Gwq4MkVkiyCruau0/vPi5FtomIjILLmPWQg/eKFSuwceNGJlrUQ5JShg2Lwfr1G1j/kogMhTO7evTr149dzd0kxbTZfZyIjIgzOzdIFZDIyLsMlXBhtJPK5bzEhITHmdBDRIbEmZ0b5OAtCRfLli3TRqgmKaItxbQZ6IjIiBjs3MSu5rWT4tkXL168WkybiMhouIzZAJKVmZT0FPbvP+D3ZBWjLGMyKYWIzIAzuwaQaiDsal5dZmYmu48TkeEx2DWQ3tVcqoQEOklK2bFjO7uPE5HhMdg1kCRgSCLG/PnztZHAxe7jRGQWDHaNIIkYkpARyF3NpUi2FMtm93EiMgMmqDSSfl6Zv5JV/JmgIucdjhplw7Zt29UaokRERseZXSNJQoYkq0iCRqCR4thSJJuBjojMgjO7JgjEtHsjnX5BROQuzuyaQA72kqAh1UMCgd59fNGixQx0RGQqDHZNpCdoBEJXc737uBTHJiIyEy5jekAgFEE2YjFsIiJ3cWbnAbJfZ/Wu5lIEm93HicisGOw8RO9qLrM8q2H3cSIyOy5jepAVu5rrGacZGS+ptUGJiMyIMzsPsmJXcyl63b//AAY6IjI1zuw8TApE33tvX68ncviigoq8lnHjYtl9nIhMjzM7D5OqIlbpar5y5Up2HyciS2Cw8wK9q7lUGzErKXItr4Hdx4nIChjsvECSU6T9jZTVkgQPs5HnPG/ePLVaChGRFTDYeYkkdEhihxm7mktxaylyze7jRGQVTFDxIr3qyPvvf+DxDgHeSlDxd+siIiJv4MzOiySxQ5JVJNHDLKSoNbuPE5HVMNh5mSR4SKKHnHBudHoxa3YfJyKrYbDzAUn0mDXrWUMnq8iSa3r6cqSkpGgjRETWwWDnA2boaq53H2dSChFZEYOdjyQmJmLHju0eKxTtyeQUeU5SxFqKWRMRWRGDnY9IsooRu5rL0qqcE8ju40RkZQx2PqQnfkh1EqNg93EiCgQ8z87HjNTV3JvnARIRGQlndj4mCSBjx8Yaoqu5FKteuHAxAx0RWR6DnR/Ex8f7vau5FKmW8/+YlEJEgYDBzg8kEUQSQiQxxB8kKUXO/ZOfz6QUIgoEDHZ+onc116uW+JIUp5afze7jRBQomKDiR03pat7YQtC+6qRORGQknNn5kSSGSIKIL7uaS1FqKU7NQEdEgYTBzs8kQeTYsaM+6WouxajZfZyIAhGDnZ9JgkhGxktqwog3C0XLY0sxanYfJ6JAxGBnAJIoIgkjUs3EW9h9nIgCGRNUDKKh1UwakqBipKotRET+wJmdQUgQ8lZXc737OAMdEQUqBjsDiYmJ8XhXc73oNLuPE1Eg4zKmwehLjvv3H6izuok7y5iyNDpqlA3r12/gXh0RBTTO7AxG72ouVU6aSopNS9FpBjoiCnSc2RmQJ2Zk7s4QiYgCAWd2BiSJJMnJKU3qai5FniUphYGOiIjBzrD0KieN6WouxaXlvD0mpRAROXEZ08AasxSpL4Fu27adTVmJiDSc2RmY7NdJgklGRoY2Uj8pKp2YOImBjoioCgY7g2tIV3MpJi1Fpdl9nIioOgY7g5PlS0k0qa+rud59XIpKMymFiKg6BjsTkESTNm3a1NnVXIpIs/s4EZFrTFAxCekwPm5c7NVizlUrqLD7OBFR3TizMwlJOJHEE6mKUhO7jxMR1Y3BzkQk8USSVap2Nde7j0sRaSIico3LmCYjwW3FihU4fvw95OcXYNiwGBZ6JiKqB2d2JtOvXz81EUVIsWh2Hyciqh9ndiakdzW/9dZQvPHGm9yrIyKqB4OdSW3cuFE9n+6RRx7RRoiIqDYMdkREZHncsyMiIstjsCMiIstjsDOT8gJk716OhG4d1Qoq8q9bwnLszi5AuXYTIiK6FoOdKThQnr8ZCX0G48l9zRC7N08tFVZUeBJbB17E2rgRGDPnMEoc2s2JiKgaJqiYQfkJpI8Zj3RMhf31aejVourfKJdRbH8aQ5L247bkLXg9uQ9aaNcQEZETZ3aG9zMKXl+B9DwgcmwMIqsFOtEcIfePwkPBFchb+wZOlnF6R0RUE4Od4X2Ps0dPK19vwp1hbVx/YC07o0//1kDF2ziYW6oNEhGRjsHO6Bz/xqXvflG+CUbbltc7x2pVhm8v/aR9T0REOgY7owu6Aa3aXad8U4ELZT87x2p1M7qH3qh9T0REOgY7w7sJ3QfcqXz9HqcLL8LljlzZFzhxrFSZ/PVFzy43aINERKRjsDO869E5+mEMD67AqSWrsLf4sjauu4zit/dgd0UwImf+Hve15EdKRFQTj4wmEBQyEsvsixCNXZg1Zw0OF5Q5r3CUIHfzC3hMTjuIX4W1cV35gRIRucDz7EzEUZKLt/bvwso5G5GnjQVHp2DB+AcRPSic59cREdWCwY6IiCyPq15ERGR5DHZERGR5DHZERGR5DHZERGR5DHZERGR5DHZERGR5DHZERGRxwP8HBzDbq2Z7EREAAAAASUVORK5CYII="></p>
</div>
<div class="specification">
<p>A square field with side <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>m</mtext></math> has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> from the post.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p style="text-align: center;"><sup>[Source: mynamepong, n.d. Goat [image online] Available at: <a href="https://thenounproject.com/term/goat/1761571/">https://thenounproject.com/term/goat/1761571/</a></sup><br><sup>This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)</sup><br><sup><a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">https://creativecommons.org/licenses/by-sa/3.0/deed.en</a> [Accessed 22 April 2010] Source adapted.]</sup></p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> be the volume of grass eaten by the goat, in cubic metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> be the length of time, in hours, that the goat has been in the field.</p>
<p>The goat eats grass at the rate of <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><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>t</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded segment.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the field that can be reached by the goat.</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>t</mi></math> at which the goat is eating grass at the greatest rate.</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 goat is tied in the field for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> hours.</p>
<p>Find the total volume of grass eaten by the goat during this time.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>AÔB</mtext><mo>=</mo></mrow></mfenced><mo> </mo><mtext>arccos</mtext><mfenced><mfrac><mn>4</mn><mrow><mn>4</mn><mo>.</mo><mn>5</mn></mrow></mfrac></mfenced><mo>=</mo><mn>27</mn><mo>.</mo><mn>266</mn><mo>…</mo></math> <em><strong>(M1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext><mo>=</mo><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo><mo>≈</mo><mn>54</mn><mo>.</mo><mn>5</mn><mo>°</mo></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>951764</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>952</mn></math> radians) <em><strong>A1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Other methods may be seen; award <em><strong>(M1)(A1)</strong></em> for use of a correct trigonometric method to find an appropriate angle and then <em><strong>A1</strong> </em>for the correct answer.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding area of triangle</p>
<p><strong>EITHER</strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>×</mo><mi>sin</mi><mfenced><mrow><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correct substitution into formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>.</mo><mn>24621</mn><mo>…</mo><mo>≈</mo><mn>8</mn><mo>.</mo><mn>25</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext><mo>=</mo><mn>2</mn><mo>×</mo><msqrt><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>-</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><mo>=</mo><mn>4</mn><mo>.</mo><mn>1231</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p>area triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mo>.</mo><mn>1231</mn><mo>…</mo><mo>×</mo><mn>4</mn></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>.</mo><mn>24621</mn><mo>…</mo><mo>≈</mo><mn>8</mn><mo>.</mo><mn>25</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p>finding area of sector</p>
<p><strong>EITHER</strong></p>
<p>area of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo></mrow><mn>360</mn></mfrac><mo>×</mo><mi mathvariant="normal">π</mi><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>63661</mn><mo>…</mo><mo>≈</mo><mn>9</mn><mo>.</mo><mn>64</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p>area of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>9517641</mn><mo>…</mo><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>63661</mn><mo>…</mo><mo>≈</mo><mn>9</mn><mo>.</mo><mn>64</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>area of segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>63661</mn><mo>…</mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>24621</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>39</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>39040</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong> </em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>63</mn><mo>.</mo><mn>6172</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>39040</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo> </mo><mo> </mo><mo>(</mo><mn>5</mn><mo>.</mo><mn>56160</mn><mo>)</mo></math> <em><strong>(A1)</strong></em></p>
<p>subtraction of four segments from area of circle <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>58</mn><mo>.</mo><mn>1</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>58</mn><mo>.</mo><mn>055</mn><mo>…</mo><mo> </mo></mrow></mfenced></math> <em><strong>A1</strong> </em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>angle of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>90</mn><mo>-</mo><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mn>0</mn><mo>.</mo><mn>951764</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>area of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>90</mn><mo>-</mo><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo></mrow><mn>360</mn></mfrac><mo>×</mo><mi mathvariant="normal">π</mi><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mo>.</mo><mn>26771</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>area is made up of four triangles and four sectors <em><strong>(M1)</strong></em></p>
<p>total area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>4</mn><mo>×</mo><mn>8</mn><mo>.</mo><mn>2462</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>4</mn><mo>×</mo><mn>6</mn><mo>.</mo><mn>26771</mn><mo>…</mo></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>58</mn><mo>.</mo><mn>1</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>58</mn><mo>.</mo><mn>055</mn><mo>…</mo><mo> </mo></mrow></mfenced></math> <em><strong>A1</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>sketch of <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></math> <strong>OR</strong> <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><mn>0</mn><mo>.</mo><mn>110363</mn><mo>…</mo></math> <strong>OR </strong>attempt to find where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>V</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math> hour <em><strong>A1</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>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>d</mo><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>8</mn></msubsup><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>d</mo><mi>t</mi></math> <em><strong>(A1)</strong></em></p>
<p>volume eaten is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>299</mn><mo>…</mo><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>299094</mn><mo>…</mo></mrow></mfenced></math> <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;">
<p>Generally, this question was answered well but provided a good example of final marks being lost due to premature rounding. Some candidates gave a correct three significant figure intermediate answer of 27.3˚ for the angle in the right-angles triangle and then doubled it to get 54.6˚ as a final answer. This did not receive the final answer mark as the correct answer is 54.5˚ to three significant figures. Premature rounding needs to be avoided in all questions.</p>
<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;">
<p>Unfortunately, many candidates failed to see the connection to part (a). Indeed, the most common answer was to assume the goat could eat all the grass in a circle of radius 4.5m.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates completed this question successfully by graphing the function. A few tried to differentiate the function again and, in some cases, also managed to obtain the correct answer.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was a question that was pleasingly answered correctly by many candidates who recognized that integration was needed to find the answer. As in part (c) a few tried to do the integration ‘by hand’, and were largely unsuccessful.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math>.</p>
</div>
<div class="specification">
<p>The shape of a piece of metal can be modelled by the region bounded by the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math>, as shown in the following diagram. The units on the <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> axes are measured in metres.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by</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><mo>{</mo><mtable><mtr><mtd><msqrt><mi>x</mi></msqrt><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>16</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>16</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></math></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is obtained from the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> by:</p>
<ul>
<li>a stretch scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction,</li>
<li>followed by a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> direction,</li>
<li>followed by a translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> units to the right.</li>
</ul>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>825</mn><mo>)</mo></math>. Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> under the given transformations and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><mi>h</mi><mfenced><mi>x</mi></mfenced><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mi>b</mi><mo> </mo><mo> </mo></mtd><mtd><mi>a</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>p</mi></mtd></mtr></mtable></math></p>
</div>
<div class="specification">
<p>The area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> correct to six significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <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></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the equation of the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region on the diagram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</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"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math> <em><strong>(M1)</strong></em></p>
<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><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em> </p>
<p> </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>gradient at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>16</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><msqrt><mn>0</mn><mo>.</mo><mn>16</mn></msqrt></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>16</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></mfenced><mo>+</mo><mi>b</mi></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not allow working backwards from the given answer.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>45</mn><mo>,</mo><mo> </mo><mo> </mo><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4125</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>413</mn></math>) (accept " <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>45</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4125</mn><mo>)</mo></math> ") <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[2 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"><mfenced><mrow><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></msqrt></math> <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>if only two correct transformations are seen. </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn><mo>.</mo><mn>28</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>Correct substitution of their part (b) (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>28</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math>) into the given expression <strong><em>(M1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>.</mo><mn>25</mn><mo>×</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for transforming the equivalent expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> correctly.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>b</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing need to add two integrals <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></msubsup><msqrt><mi>x</mi></msqrt><mo>d</mo><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>d</mo><mi>x</mi></math> <strong><em>(A1)</em></strong></p>
<p><br><strong>Note:</strong> The second integral could be replaced by the formula for the area of a trapezoid <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>34</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>825</mn></mrow></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>251</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>250916</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>area of a trapezoid <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>05</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4125</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>825</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)(A1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>45</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>8</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)(A1)</em></strong></p>
<p><strong><br>Note:</strong> If the rounded answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>413</mn></math> from part (b) is used, the integral is <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>45</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>8</mn><mo>.</mo><mn>24</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>.</mo><mn>295</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>03095</mn></math> which would be awarded <strong><em>(M1)(A1)</em></strong>.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>shaded area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>250916</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>0627292</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for the subtraction of both <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo>…</mo></math> and their area for the trapezoid from their answer to (a)(i).</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>157</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>15725</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The differentiation using the power rule was well done. In part (ii) some candidates felt it was sufficient to refer to the equation being the same as the one generated by their calculator. Generally, for ‘show that’ questions an algebraic derivation is expected.</p>
<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;">
<p>The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function <em>h</em>. In most cases it was due to not appreciating the effect the horizontal transformations have on <em>x</em>.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function <em>h</em>. In most cases it was due to not appreciating the effect the horizontal transformations have on <em>x</em>.</p>
<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">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Part (i) was frequently done well using the inbuilt functionality of the GDC. Part (ii) was less structured, and candidates needed to create a clear diagram so they could easily see which areas needed to be subtracted. Most of those who were successful used the formula for the trapezoid for the area they needed to find, though others were also successful through finding the equation of the line AB.</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The cost adjacency matrix for the complete graph <em>K</em><sub>6</sub> is given below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">It represents the distances in kilometres along dusty tracks connecting villages on an island. Find the minimum spanning tree for this graph; in all 3 cases state the order in which the edges are added.</p>
</div>
<div class="specification">
<p>It is desired to tarmac some of these tracks so that it is possible to walk from any village to any other village walking entirely on tarmac.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Briefly explain the two differences in the application of Prim’s and Kruskal’s algorithms for finding a minimum spanning tree in a weighted connected graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using Kruskal’s algorithm.</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>Using Prim’s algorithm starting at vertex A.</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>Using Prim’s algorithm starting at vertex F.</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>State the total minimum length of the tracks that have to be tarmacked.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the tracks that are to be tarmacked.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>In Prim’s algorithm you start at a particular (given) vertex, whereas in Kruskal’s you start with the smallest edge. <em><strong>A1 </strong></em></p>
<p>In Prim’s as smallest edges are added (never creating a circuit) the created graph always remains connected, whereas in Kruskal’s this requirement to always be connected is not necessary. <em><strong>A1</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>Edges added in the order</p>
<p>AB EF AC AD AE <em><strong>A1A1</strong></em></p>
<p>[<strong>note</strong> <em><strong>A1</strong> </em>for the first 2 edges <em><strong>A1</strong> </em>for other 3]</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Edges added in the order</p>
<p>AB AC AD AE EF <em><strong>A1A1</strong></em></p>
<p>[<strong>note</strong> <em><strong>A1</strong> </em>for the first 2 edges <em><strong>A1</strong> </em>for other 3]</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>Edges added in the order</p>
<p>FE AE AB AC AD <em><strong>A1A1</strong></em></p>
<p>[<strong>note</strong> <em><strong>A1</strong> </em>for the first 2 edges <em><strong>A1</strong> </em>for other 3]</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2 + 3 + 4 + 5 = 15">
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>4</mn>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>15</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.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">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">b.iii.</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>
<br><hr><br><div class="specification">
<p>At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.</p>
<p>The path of the ball is modelled by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><msub><mi>u</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mi>y</mi></msub><mo>-</mo><mn>5</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is the horizontal displacement from the archer and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the vertical displacement from the ground, both measured in metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, in seconds, since the ball was launched.</p>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub></math> is the horizontal component of the initial velocity</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub></math> is the vertical component of the initial velocity.</li>
</ul>
<p>In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="specification">
<p>An archer releases an arrow from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>. The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and an angle of elevation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>°</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the initial speed of the ball.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle of elevation of the ball as it is launched.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum height reached by the ball.</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>Assuming that the ground is horizontal and the ball is not hit by the arrow, find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> coordinate of the point where the ball lands.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the path of the ball, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the two positions where the path of the arrow intersects the path of the ball.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</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"><msqrt><msup><mn>10</mn><mn>2</mn></msup><mo>+</mo><msup><mn>8</mn><mn>2</mn></msup></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>12</mn><mo>.</mo><mn>8062</mn><mo>…</mo><mo>,</mo><mo> </mo><msqrt><mn>164</mn></msqrt></mrow></mfenced><mo> </mo><mfenced><mrow><mtext>m</mtext><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>10</mn><mn>8</mn></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>896</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>3</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>896055</mn><mo>…</mo></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>3401</mn><mo>…</mo><mo>°</mo></math>) <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>897</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>4</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arcsin</mtext><mfenced><mfrac><mn>10</mn><mrow><mn>12</mn><mo>.</mo><mn>8</mn></mrow></mfrac></mfenced></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</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>t</mi><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mi>t</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>might be implied by a correct graph or use of the correct equation.</p>
<p> </p>
<p><strong>METHOD 1 – graphical Method</strong></p>
<p>sketch graph <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>might be implied by correct graph or correct maximum (eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>).</p>
<p><br>max occurs when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em><br><br></p>
<p><strong>METHOD 2 – calculus</strong><br><br>differentiating and equating to zero <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>10</mn><mo>-</mo><mn>10</mn><mi>t</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mo>=</mo><mn>1</mn><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3 – symmetry</strong></p>
<p>line of symmetry is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mo>=</mo><mn>1</mn><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p> </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>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mi>t</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"><mi>t</mi><mo>=</mo><mn>2</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><mo>+</mo><mn>8</mn><mo>×</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mo> </mo><mn>21</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em><br><br></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong> </em>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> is also seen.</p>
<p> </p>
<p><em><strong>[3 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>t</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></mfenced><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mo>×</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>21</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>13</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>5</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mn>5</mn><mrow><mfenced><mrow><mn>13</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>13</mn><mo>-</mo><mn>21</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>21</mn></mrow></mfenced></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>25</mn><mi>a</mi><mo>+</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi></math><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>169</mn><mi>a</mi><mo>+</mo><mn>13</mn><mi>b</mi><mo>+</mo><mi>c</mi></math><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>441</mn><mi>a</mi><mo>+</mo><mn>21</mn><mi>b</mi><mo>+</mo><mi>c</mi></math> <em><strong>M1A1</strong></em></p>
<p>solving simultaneously, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mo>,</mo><mo> </mo><mi>c</mi><mo>=</mo><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math>)</p>
<p> </p>
<p><strong>METHOD 4</strong><br><br>use quadratic regression on <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>13</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>21</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Question asks for expression; condone omission of "<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo></math>".</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>trajectory of arrow is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>10</mn><mo>+</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p>intersecting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>10</mn><mo>+</mo><mn>2</mn></math> and their answer to (d) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>8</mn><mo>.</mo><mn>66</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>53</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mfenced><mrow><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>52647</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>15</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>66</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mfenced><mrow><mn>15</mn><mo>.</mo><mn>0859</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>66006</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mtext>target</mtext></msub><mo>=</mo><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>,</mo><mo> </mo><mo> </mo><msub><mi>t</mi><mtext>target</mtext></msub><mo>=</mo><mfrac><mrow><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>457132</mn><mo>…</mo><mo> </mo><mtext>s</mtext></math> <em><strong>(A1)</strong></em></p>
<p>attempt to find the distance from point of release to intersection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>8</mn><mo>.</mo><mn>65705</mn><msup><mo>…</mo><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>3</mn><mo>.</mo><mn>52647</mn><mo>…</mo><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>8</mn><mo>.</mo><mn>79060</mn><mo>…</mo><mo> </mo><mtext>m</mtext></mrow></mfenced></math></p>
<p>time for arrow to get there is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>8</mn><mo>.</mo><mn>79060</mn><mo>…</mo></mrow><mn>60</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>146510</mn><mo>…</mo><mtext>s</mtext></math> <em><strong>(A1)</strong></em></p>
<p>so the arrow should be released when</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>311</mn><mo> </mo><mfenced><mtext>s</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>310622</mn><mo>…</mo><mo> </mo><mfenced><mtext>s</mtext></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question was found to be the most difficult on the paper. There were a good number of good solutions to parts (a) and part (b), frequently with answers just written down with no working. Part (c) caused some difficulties with confusing variables. The most significant difficulties started with part (d) and became greater to the end of the question. Few candidates were able to work through the final two parts.</p>
<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">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>
<br><hr><br><div class="specification">
<p>The matrices A and B are defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 2&4 \end{array}} \right)">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = \left( {\begin{array}{*{20}{c}} { - 1}&0 \\ 0&1 \end{array}} \right)">
<mi>B</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Triangle X is mapped onto triangle Y by the transformation represented by AB. The coordinates of triangle Y are (0, 0), (−30, −20) and (−16, 32).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe fully the geometrical transformation represented by B.</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 the coordinates of triangle X.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle X.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the area of triangle Y.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Matrix A represents a combination of transformations: </p>
<p style="padding-left:90px;">A stretch, with scale factor 3 and y-axis invariant;<br>Followed by a stretch, with scale factor 4 and x-axis invariant;<br>Followed by a transformation represented by matrix C.</p>
<p>Find matrix C.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>reflection in the y-axis <em><strong>A1</strong></em><em><strong>A1</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="X = {\left( {AB} \right)^{ - 1}}Y">
<mi>X</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mi>Y</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="AB = \left( {\begin{array}{*{20}{c}} { - 3}&{ - 2} \\ { - 2}&4 \end{array}} \right)">
<mi>A</mi>
<mi>B</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {AB} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} { - \frac{1}{4}}&{ - \frac{1}{8}} \\ { - \frac{1}{8}}&{\frac{3}{{16}}} \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>3</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X = {B^{ - 1}}{A^{ - 1}}Y">
<mi>X</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>B</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi>A</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mi>Y</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X = \left( {\begin{array}{*{20}{c}} 0&{10}&0 \\ 0&0&8 \end{array}} \right)">
<mi>X</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>So the coordinates are (0, 0), (10, 0) and (0, 8). <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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{10 \times 8}}{2} = 40">
<mfrac>
<mrow>
<mn>10</mn>
<mo>×</mo>
<mn>8</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mn>40</mn>
</math></span> units<sup>2</sup> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\det \left( {AB} \right) = - 16">
<mo movablelimits="true" form="prefix">det</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mi>B</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>16</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 40 \times 16 = 640">
<mo>=</mo>
<mn>40</mn>
<mo>×</mo>
<mn>16</mn>
<mo>=</mo>
<mn>640</mn>
</math></span> units<sup>2</sup> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A stretch, with scale factor 3 and y-axis invariant is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&0 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>A stretch, with scale factor 4 and x-axis invariant is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = A{\left( {\begin{array}{*{20}{c}} 3&0 \\ 0&1 \end{array}} \right)^{ - 1}}{\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&4 \end{array}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&{ - \frac{1}{2}} \\ {\frac{2}{3}}&1 \end{array}} \right)">
<mi>C</mi>
<mo>=</mo>
<mi>A</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[4 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.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>
<br><hr><br><div class="specification">
<p>Let <em>G</em> be the graph below.</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 total number of Hamiltonian cycles in <em>G</em>, starting at vertex A. Explain your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a minimum spanning tree for the subgraph obtained by deleting A from <em>G</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find a lower bound for the travelling salesman problem for <em>G</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>Give an upper bound for the travelling salesman problem for the graph above.</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 class="indent1" style="margin-top:12.0pt;">Show that the lower bound you have obtained is not the best possible for the solution to the travelling salesman problem for <em>G</em>.</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>Starting from vertex A there are 4 choices. From the next vertex there are three choices, etc… <em><strong> M1R1</strong></em></p>
<p>So the number of Hamiltonian cycles is 4! = 24. <em><strong>A1 N1 </strong></em></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>Start (for instance) at B, using Prim′s algorithm Then D is the nearest vertex <em><strong>M1 </strong></em></p>
<p>Next E is the nearest vertex <em><strong>A1 </strong></em></p>
<p>Finally C is the nearest vertex So a minimum spanning tree is B → D → E → C <em><strong>A1 N1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A lower bound for the travelling salesman problem is then obtained by adding the weights of AB and AE to the weight of the minimum <em><strong>M1 </strong></em></p>
<p>spanning tree (ie 20) <em><strong>A1 </strong></em></p>
<p>A lower bound is then 20 + 7 + 6 = 33 <em><strong>A1 N1</strong></em></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>ABCDE is an Hamiltonian cycle <em><strong> A1 </strong></em></p>
<p>Thus an upper bound is given by 7 + 9 + 9 + 8 + 6 = 39 <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>Eliminating C from <em>G</em> a minimum spanning tree is E → A → B → D <em><strong>M1 </strong></em></p>
<p>of weight 18 <em><strong>A1 </strong></em></p>
<p>Adding BC to CE(18 + 9 + 7) gives a lower bound of 34 > 33 <em><strong>A1 </strong></em></p>
<p>So 33 not the best lower bound. <em><strong>AG N0</strong></em></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.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>
<br><hr><br><div class="specification">
<p>The matrix A is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left( {\begin{array}{*{20}{c}} 3&0 \\ 0&2 \end{array}} \right)">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Pentagon, P, which has an area of 7 cm<sup>2</sup>, is transformed by A.</p>
</div>
<div class="specification">
<p>The matrix B is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = \frac{1}{2}\left( {\begin{array}{*{20}{c}} {3\sqrt 3 }&3 \\ { - 2}&{2\sqrt 3 } \end{array}} \right)">
<mi>B</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>B represents the combined effect of the transformation represented by a matrix X, followed by the transformation represented by A.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe fully the geometrical transformation represented by 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>Find the area of the image of P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the matrix X.</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>Describe fully the geometrical transformation represented by X.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>stretch <em><strong> A1</strong></em></p>
<p>scale factor 3, <em><strong> A1</strong></em></p>
<p>y-axis invariant (condone parallel to the x-axis) <em><strong> A1</strong></em></p>
<p>and</p>
<p>stretch, scale factor 2, <em><strong> A1</strong></em></p>
<p>x-axis invariant (condone parallel to the y-axis) <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="{\text{det}}\left( A \right) = 6">
<mrow>
<mtext>det</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>A</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>6</mn>
</math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7 \times 6 = 42\,{\text{c}}{{\text{m}}^2}">
<mn>7</mn>
<mo>×</mo>
<mn>6</mn>
<mo>=</mo>
<mn>42</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[2 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="B = AX">
<mi>B</mi>
<mo>=</mo>
<mi>A</mi>
<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="X = {A^{ - 1}}B">
<mi>X</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>A</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mi>B</mi>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X = \left( {\begin{array}{*{20}{c}} {0.866}&{0.5} \\ { - 0.5}&{0.866} \end{array}} \right)\left( { = \left( {\begin{array}{*{20}{c}} {\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}} \\ { - \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}} \end{array}} \right)} \right)">
<mi>X</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.866</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.5</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.866</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Rotation <em><strong>A1</strong></em></p>
<p>clockwise by 30° about the origin <em><strong>A1</strong></em></p>
<p><em><strong>[2 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>In triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PQR, PR}} = 12{\text{ cm, QR}} = p{\text{ cm, PQ}} = r{\text{ cm}}">
<mrow>
<mtext>PQR, PR</mtext>
</mrow>
<mo>=</mo>
<mn>12</mn>
<mrow>
<mtext> cm, QR</mtext>
</mrow>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mtext> cm, PQ</mtext>
</mrow>
<mo>=</mo>
<mi>r</mi>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{Q\hat PR}} = 30^\circ ">
<mrow>
<mrow>
<mi mathvariant="normal">Q</mi>
<mrow>
<mover>
<mi mathvariant="normal">P</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">R</mi>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mn>30</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the possible triangles with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{QR}} = 8{\text{ cm}}">
<mrow>
<mtext>QR</mtext>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the case where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, the length of QR is not fixed at 8 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the cosine rule to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} - 12\sqrt 3 r + 144 - {p^2} = 0">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mi>r</mi>
<mo>+</mo>
<mn>144</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the two corresponding values of PQ.</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, find the area of the smaller triangle.</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>Determine the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> for which it is possible to form two triangles.</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 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="{p^2} = {12^2} + {r^2} - 2 \times 12 \times r \times \cos (30^\circ )">
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>12</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>×</mo>
<mn>12</mn>
<mo>×</mo>
<mi>r</mi>
<mo>×</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} - 12\sqrt 3 r + 144 - {p^2} = 0">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mi>r</mi>
<mo>+</mo>
<mn>144</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>AG</em></strong></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><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} - 12\sqrt 3 r + 80 = 0">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mi>r</mi>
<mo>+</mo>
<mn>80</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p>using the sine rule <strong><em>(M1)</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PQ}} = 5.10{\text{ }}({\text{cm}})">
<mrow>
<mtext>PQ</mtext>
</mrow>
<mo>=</mo>
<mn>5.10</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> or <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PQ}} = 15.7{\text{ }}({\text{cm}})">
<mrow>
<mtext>PQ</mtext>
</mrow>
<mo>=</mo>
<mn>15.7</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></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="{\text{area}} = \frac{1}{2} \times 12 \times 5.1008 \ldots \times \sin (30^\circ )">
<mrow>
<mtext>area</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>12</mn>
<mo>×</mo>
<mn>5.1008</mn>
<mo>…</mo>
<mo>×</mo>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 15.3{\text{ }}({\text{c}}{{\text{m}}^2})">
<mo>=</mo>
<mn>15.3</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></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><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} - 12\sqrt 3 r + 144 - {p^2} = 0">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mi>r</mi>
<mo>+</mo>
<mn>144</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>discriminant <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {12\sqrt 3 } \right)^2} - 4 \times (144 - {p^2})">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mn>144</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<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=" = 4({p^2} - 36)">
<mo>=</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>36</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="({p^2} - 36) > 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>36</mn>
<mo stretchy="false">)</mo>
<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="p > 6">
<mi>p</mi>
<mo>></mo>
<mn>6</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>construction of a right angle triangle <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12\sin 30^\circ = 6">
<mn>12</mn>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mn>6</mn>
</math></span> <strong><em>M1(A1)</em></strong></p>
<p>hence for two triangles <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p > 6">
<mi>p</mi>
<mo>></mo>
<mn>6</mn>
</math></span> <strong><em>R1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p < 12">
<mi>p</mi>
<mo><</mo>
<mn>12</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="144 - {p^2} > 0">
<mn>144</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
</math></span> to ensure two positive solutions or valid geometric argument <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore 6 < p < 12">
<mo>∴</mo>
<mn>6</mn>
<mo><</mo>
<mi>p</mi>
<mo><</mo>
<mn>12</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>diagram showing two triangles <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12\sin 30^\circ = 6">
<mn>12</mn>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mn>6</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p>one right angled triangle when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 6">
<mi>p</mi>
<mo>=</mo>
<mn>6</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore p > 6">
<mo>∴</mo>
<mi>p</mi>
<mo>></mo>
<mn>6</mn>
</math></span> for two triangles <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p < 12">
<mi>p</mi>
<mo><</mo>
<mn>12</mn>
</math></span> for two triangles <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 < p < 12">
<mn>6</mn>
<mo><</mo>
<mi>p</mi>
<mo><</mo>
<mn>12</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[7 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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> that satisfy the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>k</mi>
<mo>−</mo>
<mn>12</mn>
<mo><</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B < \frac{1}{4}">
<mi>cos</mi>
<mo></mo>
<mi>B</mi>
<mo><</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span>, find the range of possible values for AB.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>k</mi>
<mo>−</mo>
<mn>12</mn>
<mo><</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(k - 4)(k + 3) < 0">
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<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=" - 3 < k < 4">
<mo>−</mo>
<mn>3</mn>
<mo><</mo>
<mi>k</mi>
<mo><</mo>
<mn>4</mn>
</math></span> <strong><em>A1</em></strong></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B = \frac{{{2^2} + {c^2} - {4^2}}}{{4c}}{\text{ }}({\text{or }}16 = {2^2} + {c^2} - 4c\cos B)">
<mi>cos</mi>
<mo></mo>
<mi>B</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mn>4</mn>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mi>c</mi>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>or </mtext>
</mrow>
<mn>16</mn>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>c</mi>
<mi>cos</mi>
<mo></mo>
<mi>B</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=" \Rightarrow \frac{{{c^2} - 12}}{{4c}} < \frac{1}{4}">
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
<mrow>
<mn>4</mn>
<mi>c</mi>
</mrow>
</mfrac>
<mo><</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {c^2} - c - 12 < 0">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>c</mi>
<mo>−</mo>
<mn>12</mn>
<mo><</mo>
<mn>0</mn>
</math></span></p>
<p>from result in (a)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{AB}} < 4">
<mn>0</mn>
<mo><</mo>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo><</mo>
<mn>4</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 < {\text{AB}} < 4">
<mo>−</mo>
<mn>3</mn>
<mo><</mo>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo><</mo>
<mn>4</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>but AB must be at least 2</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 2 < {\text{AB}} < 4">
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mo><</mo>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo><</mo>
<mn>4</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \leqslant {\text{AB}}">
<mo>⩽</mo>
<mrow>
<mtext>AB</mtext>
</mrow>
</math></span> for either of the final two <strong><em>A </em></strong>marks.</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="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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>area of segment <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times {0.5^2} \times (\theta - \sin \theta )">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = {\text{area of segment}} \times 10">
<mi>V</mi>
<mo>=</mo>
<mrow>
<mtext>area of segment</mtext>
</mrow>
<mo>×</mo>
<mn>10</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{5}{4}(\theta - \sin \theta )">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</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="\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{5}{4}(1 - \cos \theta )\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008 = \frac{5}{4}\left( {1 - \cos \frac{\pi }{3}} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mn>0.0008</mn>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128{\text{ }}({\text{rad}}\,{s^{ - 1}})">
<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>0.00128</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></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}}\theta }}{{{\text{d}}V}} \times \frac{{{\text{d}}V}}{{{\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>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}\theta }} = \frac{5}{4}(1 - \cos \theta )">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</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="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{4 \times 0.0008}}{{5\left( {1 - \cos \frac{\pi }{3}} \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>
<mfrac>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>0.0008</mn>
</mrow>
<mrow>
<mn>5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128\left( {\frac{4}{{3125}}} \right)({\text{rad }}{s^{ - 1}})">
<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>0.00128</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>4</mn>
<mrow>
<mn>3125</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad </mtext>
</mrow>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></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="specification">
<p>An aircraft’s position is given by the coordinates (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>), where <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> are the aircraft’s displacement east and north of an airport, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> is the height of the aircraft above the ground. All displacements are given in kilometres.</p>
<p>The velocity of the aircraft is given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ { - 20} \end{array}} \right)\,{\text{km}}\,{{\text{h}}^{ - 1}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>150</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>50</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>km</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mrow>
<mtext>h</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<p>At 13:00 it is detected at a position 30 km east and 10 km north of the airport, and at a height of 5 km. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> be the length of time in hours from 13:00.</p>
</div>
<div class="specification">
<p>If the aircraft continued to fly with the velocity given</p>
</div>
<div class="specification">
<p>When the aircraft is 4 km above the ground it continues to fly on the same bearing but adjusts the angle of its descent so that it will land at the point (0, 0, 0).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a vector equation for the displacement, <strong><em>r</em></strong> of the aircraft in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</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>verify that it would pass directly over the airport.</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>state the height of the aircraft at this point.</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>find the time at which it would fly directly over the airport.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time at which the aircraft is 4 km above the ground.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the direct distance of the aircraft from the airport at this point.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the velocity of the aircraft, after the adjustment of the angle of descent, is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right){\text{km}}\,{{\text{h}}^{ - 1}}"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>150</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>50</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>km</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mtext>h</mtext> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </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><em><strong>r </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {30} \\ {10} \\ 5 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ { - 20} \end{array}} \right)\,"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>30</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>150</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>50</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>20</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> </math></span> <em><strong>A1A1</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>when <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{{30}}{{150}} = 0.2"> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mn>30</mn> </mrow> <mrow> <mn>150</mn> </mrow> </mfrac> <mo>=</mo> <mn>0.2</mn> </math></span> <em><strong>M1</strong></em></p>
<p><em><strong>EITHER</strong></em></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{{10}}{{150}} = 0.2"> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mn>10</mn> </mrow> <mrow> <mn>150</mn> </mrow> </mfrac> <mo>=</mo> <mn>0.2</mn> </math></span> <em><strong>A1</strong></em></p>
<p>since the two values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> are equal the aircraft passes directly over the airport</p>
<p><em><strong>OR</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0.2"> <mi>t</mi> <mo>=</mo> <mn>0.2</mn> </math></span>, <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> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>height = 5 − 0.2 × 20 = 1 km <em><strong>A1</strong></em></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>time 13:12 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5 - 20t = 4 \Rightarrow t = \frac{1}{{20}}"> <mn>5</mn> <mo>−</mo> <mn>20</mn> <mi>t</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">⇒</mo> <mi>t</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>20</mn> </mrow> </mfrac> </math></span> (3 minutes) <em><strong>(M1)</strong></em></p>
<p>time 13:03 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>displacement is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {22.5} \\ {7.5} \\ 4 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>22.5</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>7.5</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>distance is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{22.5}^2} + {{7.5}^2} + {4^2}} "> <msqrt> <mrow> <msup> <mrow> <mn>22.5</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mn>7.5</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> </msqrt> </math></span> <em><strong>(M1)</strong></em></p>
<p>= 24.1 km <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>time until landing is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="12 - 3 = 9"> <mn>12</mn> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>9</mn> </math></span> minutes <em><strong>M1</strong></em></p>
<p>height to descend = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4\,{\text{km}}"> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>km</mtext> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{{ - 4}}{{\frac{9}{{60}}}}"> <mi>a</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mfrac> <mn>9</mn> <mrow> <mn>60</mn> </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=" = - 26.7"> <mo>=</mo> <mo>−</mo> <mn>26.7</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 150} \\ { - 50} \\ a \end{array}} \right) = s\left( {\begin{array}{*{20}{c}} {22.5} \\ {7.5} \\ 4 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>150</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>50</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>22.5</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>7.5</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> </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=" - 150 = 22.5\,s \Rightarrow s = - \frac{{20}}{3}"> <mo>−</mo> <mn>150</mn> <mo>=</mo> <mn>22.5</mn> <mspace width="thinmathspace"></mspace> <mi>s</mi> <mo stretchy="false">⇒</mo> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>20</mn> </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="a = - \frac{{20}}{3} \times 4"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> <mo>×</mo> <mn>4</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 26.7"> <mo>=</mo> <mo>−</mo> <mn>26.7</mn> </math></span> <em><strong>A1</strong></em></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.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.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>
<br><hr><br><div class="specification">
<p>Two submarines A and B have their routes planned so that their positions at time <em>t</em> hours, 0 ≤ <em>t</em> < 20 , would be defined by the position vectors <em><strong>r</strong><sub>A</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,2 \hfill \\ \,4 \hfill \\ - 1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 1 \hfill \\ \,1 \hfill \\ - 0.15 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.15</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> and <em><strong>r</strong><sub>B</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>3.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> relative to a fixed point on the surface of the ocean (all lengths are in kilometres).</p>
</div>
<div class="specification">
<p>To avoid the collision submarine B adjusts its velocity so that its position vector is now given by</p>
<p style="padding-left: 120px;"><em><strong>r</strong><sub>B</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>3.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.45</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.08</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.09</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the two submarines would collide at a point P and write down the coordinates of P.</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 the value of <em>t</em> when submarine B passes through P.</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>Find the value of <em>t</em> when the two submarines are closest together.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between the two submarines at this time.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</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><em><strong>r</strong><sub>A </sub>= <strong>r</strong><sub>B <strong>(M1)</strong></sub></em></p>
<p>2 − <em>t</em> = − 0.5t ⇒ <em>t</em> = 4 <strong>A1</strong></p>
<p>checking <em>t</em> = 4 satisfies 4 + <em>t</em> = 3.2 + 1.2<em>t</em> and − 1 − 0.15<em>t</em> = − 2 + 0.1<em>t <strong>R1</strong></em></p>
<p>P(−2, 8, −1.6) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award final <em><strong>A1</strong></em> if answer given as column vector.</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="\left( \begin{gathered} \, - 0.45t \hfill \\ 3.2 + 1.08t \hfill \\ - 2 + 0.09t \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 2 \hfill \\ \,8 \hfill \\ - 1.6 \hfill \\ \end{gathered} \right)"> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mn>0.45</mn> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mn>3.2</mn> <mo>+</mo> <mn>1.08</mn> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>0.09</mn> <mi>t</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mspace width="thinmathspace"></mspace> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1.6</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> The <strong><em>M1</em></strong> can be awarded for any one of the resultant equations.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow t = \frac{{40}}{9} = 4.44 \ldots "> <mo stretchy="false">⇒</mo> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <mn>40</mn> </mrow> <mn>9</mn> </mfrac> <mo>=</mo> <mn>4.44</mn> <mo>…</mo> </math></span> <em><strong>A1</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>minimum when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}D}}{{{\text{d}}t}} = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>D</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>t</em> = 3.83 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.511 (km) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</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.ii.</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">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A canal system divides a city into six land masses connected by fifteen bridges, as shown in the diagram below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>State with reasons whether or not this graph has</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw a graph to represent this map.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the adjacency matrix of the graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>List the degrees of each of the vertices.</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>an Eulerian circuit.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>an Eulerian trail.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of walks of length 4 from E to F.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A2</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><em><strong>M</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\begin{array}{*{20}{c}} {} \\ {\text{A}} \\ {\text{B}} \\ {\text{C}} \\ {\text{D}} \\ {\text{E}} \\ {\text{F}} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\text{A}}&{\text{B}}&{\text{C}}&{\text{D}}&{\text{E}}&{\text{F}} \end{array}} \\ {\left( {\begin{array}{*{20}{c}} 0&1&2&1&2&2 \\ 1&0&0&0&1&2 \\ 2&0&0&1&0&1 \\ 1&0&1&0&1&0 \\ 2&1&0&1&0&1 \\ 2&2&1&0&1&0 \end{array}} \right)} \end{array}"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>A</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>B</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>C</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>D</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>E</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>F</mtext> </mrow> </mtd> </mtr> </mtable> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mtext>A</mtext> </mrow> </mtd> <mtd> <mrow> <mtext>B</mtext> </mrow> </mtd> <mtd> <mrow> <mtext>C</mtext> </mrow> </mtd> <mtd> <mrow> <mtext>D</mtext> </mrow> </mtd> <mtd> <mrow> <mtext>E</mtext> </mrow> </mtd> <mtd> <mrow> <mtext>F</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </math></span> <em><strong>A2</strong></em></p>
<p><strong>Note: </strong>Award A1 for one error or omission, A0 for more than one error or omission. Two symmetrical errors count as one error.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A B C D E F</p>
<p>(8, 4 4, 3 5, 6) <em><strong>A2</strong></em></p>
<p><strong>Note: </strong>Award no more than A1 for one error, A0 for more than one error.</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>no, because there are odd vertices <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>yes, because there are exactly two odd vertices <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>M</strong></em><sup>4</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\begin{array}{*{20}{c}} {} \\ {\text{A}} \\ {\text{B}} \\ {\text{C}} \\ {\text{D}} \\ {\text{E}} \\ {\text{F}} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\text{A}}&{}&{\text{B}}&{}&{\text{C}}&{}&{\text{D}}&{}&{\text{E}}&{}&{\text{F}} \end{array}} \\ {\left( {\begin{array}{*{20}{c}} {309\,}&{174}&{140}&{118}&{170}&{214} \\ {174}&{117}&{106}&{70}&{122}&{132} \\ {140\,}&{106}&{117}&{66}&{134}&{138} \\ {118}&{70}&{66}&{53}&{80}&{102} \\ {170}&{122}&{134}&{80}&{157}&{170} \\ {214}&{132}&{138}&{102}&{170}&{213} \end{array}} \right)} \end{array}"> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>A</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>B</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>C</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>D</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>E</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mtext>F</mtext> </mrow> </mtd> </mtr> </mtable> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mtext>A</mtext> </mrow> </mtd> <mtd> <mrow> </mrow> </mtd> <mtd> <mrow> <mtext>B</mtext> </mrow> </mtd> <mtd> <mrow> </mrow> </mtd> <mtd> <mrow> <mtext>C</mtext> </mrow> </mtd> <mtd> <mrow> </mrow> </mtd> <mtd> <mrow> <mtext>D</mtext> </mrow> </mtd> <mtd> <mrow> </mrow> </mtd> <mtd> <mrow> <mtext>E</mtext> </mrow> </mtd> <mtd> <mrow> </mrow> </mtd> <mtd> <mrow> <mtext>F</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>309</mn> <mspace width="thinmathspace"></mspace> </mrow> </mtd> <mtd> <mrow> <mn>174</mn> </mrow> </mtd> <mtd> <mrow> <mn>140</mn> </mrow> </mtd> <mtd> <mrow> <mn>118</mn> </mrow> </mtd> <mtd> <mrow> <mn>170</mn> </mrow> </mtd> <mtd> <mrow> <mn>214</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>174</mn> </mrow> </mtd> <mtd> <mrow> <mn>117</mn> </mrow> </mtd> <mtd> <mrow> <mn>106</mn> </mrow> </mtd> <mtd> <mrow> <mn>70</mn> </mrow> </mtd> <mtd> <mrow> <mn>122</mn> </mrow> </mtd> <mtd> <mrow> <mn>132</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>140</mn> <mspace width="thinmathspace"></mspace> </mrow> </mtd> <mtd> <mrow> <mn>106</mn> </mrow> </mtd> <mtd> <mrow> <mn>117</mn> </mrow> </mtd> <mtd> <mrow> <mn>66</mn> </mrow> </mtd> <mtd> <mrow> <mn>134</mn> </mrow> </mtd> <mtd> <mrow> <mn>138</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>118</mn> </mrow> </mtd> <mtd> <mrow> <mn>70</mn> </mrow> </mtd> <mtd> <mrow> <mn>66</mn> </mrow> </mtd> <mtd> <mrow> <mn>53</mn> </mrow> </mtd> <mtd> <mrow> <mn>80</mn> </mrow> </mtd> <mtd> <mrow> <mn>102</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>170</mn> </mrow> </mtd> <mtd> <mrow> <mn>122</mn> </mrow> </mtd> <mtd> <mrow> <mn>134</mn> </mrow> </mtd> <mtd> <mrow> <mn>80</mn> </mrow> </mtd> <mtd> <mrow> <mn>157</mn> </mrow> </mtd> <mtd> <mrow> <mn>170</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>214</mn> </mrow> </mtd> <mtd> <mrow> <mn>132</mn> </mrow> </mtd> <mtd> <mrow> <mn>138</mn> </mrow> </mtd> <mtd> <mrow> <mn>102</mn> </mrow> </mtd> <mtd> <mrow> <mn>170</mn> </mrow> </mtd> <mtd> <mrow> <mn>213</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>number of walks of length 4 is 170</p>
<p class="indent1"><strong>Note: </strong>The complete matrix need not be shown. Only one of the FE has to be shown.</p>
<p><em><strong>[2 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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</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 following table shows the costs in US dollars (US$) of direct flights between six cities. Blank cells indicate no direct flights. The rows represent the departure cities. The columns represent the destination cities.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The following table shows the least cost to travel between the cities.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>A travelling salesman has to visit each of the cities, starting and finishing at city A.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show the direct flights between the cities as a graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the adjacency matrix for this graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your answer to part (b), find the number of different ways to travel from and return to city A in exactly 6 flights.</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>State whether or not it is possible to travel from and return to city A in exactly 6 flights, having visited each of the other 5 cities exactly once. Justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <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>.</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>Use the nearest neighbour algorithm to find an upper bound for the cost of the trip.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By deleting vertex A, use the deleted vertex algorithm to find a lower bound for the cost of the trip.</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A2</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>attempt to form an adjacency matrix <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&1&1&0&0&0 \\ 1&0&1&1&1&0 \\ 1&1&0&0&0&0 \\ 0&1&0&0&1&1 \\ 0&1&0&1&0&1 \\ 0&0&0&1&1&0 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <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>
<div class="question" style="padding-left: 20px;">
<p>raising the matrix to the power six <em><strong>(M1)</strong></em></p>
<p>50 <em><strong>A</strong><strong>1</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>not possible <em><strong>A1</strong></em></p>
<p>because you must pass through B twice <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Do not award<em><strong> A1R0</strong></em>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 230"> <mi>a</mi> <mo>=</mo> <mn>230</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 340"> <mi>b</mi> <mo>=</mo> <mn>340</mn> </math></span> <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A → B → D → E → F → C → A <em><strong>(M1)</strong></em></p>
<p>90 + 70 + 100 + 210 + 330 + 150 <em><strong>(A1)</strong></em></p>
<p>(US$) 950 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding weight of minimum spanning tree <em><strong>M1</strong></em></p>
<p>70 + 80 + 100 + 180 = (US$) 430 <em><strong>A1</strong></em></p>
<p>adding in two edges of minimum weight <em><strong>M1</strong></em></p>
<p>430 + 90 + 150 = (US$) 670 <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">g.</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>
<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>
<br><hr><br><div class="specification">
<p>The adjacency matrix of the graph <em>G</em>, with vertices P, Q, R, S, T is given by:</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\begin{array}{*{20}{c}} {} \\ {\text{P}} \\ {\text{Q}} \\ {\text{R}} \\ {\text{S}} \\ {\text{T}} \end{array}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\text{P}}&{\text{Q}}&{\text{R}}&{\text{S}}&{\text{T}} \end{array}} \\ {\left( {\begin{array}{*{20}{c}} 0&2&1&1&0 \\ 2&1&1&1&0 \\ 1&1&1&0&2 \\ 1&1&0&0&0 \\ 0&0&2&0&0 \end{array}} \right)} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>P</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>Q</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>R</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>S</mtext>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>T</mtext>
</mrow>
</mtd>
</mtr>
</mtable>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtext>P</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>Q</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>R</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>S</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>T</mtext>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw the graph of <em>G</em>.</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>Define an Eulerian circuit.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an Eulerian circuit in <em>G</em> starting at P.</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>Define a Hamiltonian cycle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why it is not possible to have a Hamiltonian cycle in <em>G.</em></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of walks of length 5 from P to Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Which pairs of distinct vertices have more than 15 walks of length 3 between them?</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A3</strong></em></p>
<p> </p>
<p class="indent1"><strong>Note: </strong>Award A2 for one missing or misplaced edge, <strong> </strong></p>
<p class="indent1"> A1 for two missing or misplaced edges.</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>an Eulerian circuit is one that contains every edge of the graph exactly once <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>a possible Eulerian circuit is</p>
<p>P → Q → S → P → Q → Q → R → T → R → R → P <em><strong>A2</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>a Hamiltonian cycle passes through each vertex of the graph <em><strong>A1</strong></em></p>
<p>exactly once <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>to pass through T, you must have come from R and must return to R. <em><strong>R3</strong></em></p>
<p>hence there is no Hamiltonian cycle</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using the adjacency matrix <em><strong>A</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} 0&2&1&1&0 \\ 2&1&1&1&0 \\ 1&1&1&0&2 \\ 1&1&0&0&0 \\ 0&0&2&0&0 \end{array}} \right)}"> <mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> </math></span>, <em><strong>(M1)</strong></em></p>
<p>we need the entry in the first row second column of the matrix <em><strong>A</strong></em><sup>5</sup> <em><strong>(M1)</strong></em></p>
<p><em><strong>A</strong></em><sup>5</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {245}&{309}&{274}&{143}&{126} \\ {309}&{363}&{322}&{168}&{156} \\ {274}&{322}&{295}&{141}&{164} \\ {143}&{168}&{141}&{77}&{72} \\ {126}&{156}&{164}&{72}&{72} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>245</mn> </mrow> </mtd> <mtd> <mrow> <mn>309</mn> </mrow> </mtd> <mtd> <mrow> <mn>274</mn> </mrow> </mtd> <mtd> <mrow> <mn>143</mn> </mrow> </mtd> <mtd> <mrow> <mn>126</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>309</mn> </mrow> </mtd> <mtd> <mrow> <mn>363</mn> </mrow> </mtd> <mtd> <mrow> <mn>322</mn> </mrow> </mtd> <mtd> <mrow> <mn>168</mn> </mrow> </mtd> <mtd> <mrow> <mn>156</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>274</mn> </mrow> </mtd> <mtd> <mrow> <mn>322</mn> </mrow> </mtd> <mtd> <mrow> <mn>295</mn> </mrow> </mtd> <mtd> <mrow> <mn>141</mn> </mrow> </mtd> <mtd> <mrow> <mn>164</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>143</mn> </mrow> </mtd> <mtd> <mrow> <mn>168</mn> </mrow> </mtd> <mtd> <mrow> <mn>141</mn> </mrow> </mtd> <mtd> <mrow> <mn>77</mn> </mrow> </mtd> <mtd> <mrow> <mn>72</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>126</mn> </mrow> </mtd> <mtd> <mrow> <mn>156</mn> </mrow> </mtd> <mtd> <mrow> <mn>164</mn> </mrow> </mtd> <mtd> <mrow> <mn>72</mn> </mrow> </mtd> <mtd> <mrow> <mn>72</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>hence there are 309 ways <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>A</strong></em><sup>3</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {13}&{21}&{17}&{10}&6 \\ {21}&{22}&{19}&{11}&8 \\ {17}&{19}&{18}&7&{14} \\ {10}&{11}&7&5&4 \\ 6&8&{14}&4&4 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>13</mn> </mrow> </mtd> <mtd> <mrow> <mn>21</mn> </mrow> </mtd> <mtd> <mrow> <mn>17</mn> </mrow> </mtd> <mtd> <mrow> <mn>10</mn> </mrow> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>21</mn> </mrow> </mtd> <mtd> <mrow> <mn>22</mn> </mrow> </mtd> <mtd> <mrow> <mn>19</mn> </mrow> </mtd> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>17</mn> </mrow> </mtd> <mtd> <mrow> <mn>19</mn> </mrow> </mtd> <mtd> <mrow> <mn>18</mn> </mrow> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mrow> <mn>14</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> </mrow> </mtd> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mrow> <mn>14</mn> </mrow> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>hence the pairs of vertices are PQ, PR and QR <em><strong>A1A1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.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">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.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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find AM.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region.</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><strong>METHOD 1</strong></p>
<p>PC <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 3 }}{2}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> or 0.8660 <em><strong>(M1)</strong></em></p>
<p>PM <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span>PC <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 3 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> or 0.4330 <strong>(A1)</strong></p>
<p>AM <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {\frac{1}{4} + \frac{3}{{16}}} "> <mo>=</mo> <msqrt> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mn>3</mn> <mrow> <mn>16</mn> </mrow> </mfrac> </msqrt> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 7 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> or 0.661 (m) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using the cosine rule</p>
<p>AM<sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {1^2} + {\left( {\frac{{\sqrt 3 }}{4}} \right)^2} - 2 \times \frac{{\sqrt 3 }}{4} \times {\text{cos}}\left( {30^\circ } \right)"> <mo>=</mo> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mo>×</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mn>30</mn> <mo>∘</mo> </msup> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p>AM <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 7 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> or 0.661 (m) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</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="\frac{1}{2}{\text{A}}{{\text{M}}^2}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - {\text{sin}}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}} \right)} \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>A</mtext> </mrow> <mrow> <msup> <mrow> <mtext>M</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mover> <mrow> <mtext>M</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>P</mtext> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mover> <mrow> <mtext>M</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>P</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>OR </strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{\text{A}}{{\text{M}}^2} \times 2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - = \frac{{\sqrt 3 }}{8}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>A</mtext> </mrow> <mrow> <msup> <mrow> <mtext>M</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>×</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mover> <mrow> <mtext>M</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>P</mtext> </mrow> <mo>−</mo> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>8</mn> </mfrac> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>= 0.158(m<sup>2</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to calculate area of a sector minus area of a triangle.</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.i.</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>The diagram shows two circles with centres at the points A and B and radii <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2r">
<mn>2</mn>
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, respectively. The point B lies on the circle with centre A. The circles intersect at the points C and D.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.29.37.png" alt="N16/5/MATHL/HP2/ENG/TZ0/09"></p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α<!-- α --></mi>
</math></span> be the measure of the angle CAD and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> be the measure of the angle CBD in radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the shaded area in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">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="\alpha = 4\arcsin \frac{1}{4}"> <mi>α</mi> <mo>=</mo> <mn>4</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> given that the shaded area is equal to 4.</p>
<div class="marks">[3]</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="A = 2(\alpha - \sin \alpha ){r^2} + \frac{1}{2}(\theta - \sin \theta ){r^2}"> <mi>A</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mi>α</mi> <mo stretchy="false">)</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </math></span> <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A1A1 </em></strong>for alternative correct expressions <em>eg</em>. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 4\left( {\frac{\alpha }{2} - \sin \frac{\alpha }{2}} \right){r^2} + \frac{1}{2}\theta {r^2}"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>θ</mi> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>consider for example triangle ADM where M is the midpoint of BD <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\alpha }{4} = \frac{1}{4}"> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\alpha }{4} = \arcsin \frac{1}{4}"> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}"> <mi>α</mi> <mo>=</mo> <mn>4</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempting to use the cosine rule (to obtain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - \cos \frac{\alpha }{2} = \frac{1}{8}"> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </math></span>) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\alpha }{4} = \frac{1}{4}"> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> (obtained from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\alpha }{4} = \sqrt {\frac{{1 - \cos \frac{\alpha }{2}}}{2}} "> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </mrow> <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{\alpha }{4} = \arcsin \frac{1}{4}"> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}"> <mi>α</mi> <mo>=</mo> <mn>4</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left( {\frac{\pi }{2} - \frac{\alpha }{4}} \right) = 2\sin \frac{\alpha }{2}"> <mi>sin</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\theta }{2} = \frac{\pi }{2} - \frac{\alpha }{4}"> <mfrac> <mi>θ</mi> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \frac{\alpha }{4} = 4\sin \frac{\alpha }{4}\cos \frac{\alpha }{4}"> <mi>cos</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mi>cos</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>either for use of the double angle formula or the conversion from sine to cosine.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4} = \sin \frac{\alpha }{4}"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\alpha }{4} = \arcsin \frac{1}{4}"> <mfrac> <mi>α</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}"> <mi>α</mi> <mo>=</mo> <mn>4</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(from triangle ADM), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \pi - \frac{\alpha }{2}{\text{ }}\left( { = \pi - 2\arcsin \frac{1}{4} = 2\arcsin \frac{1}{4} = 2.6362 \ldots } \right)"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mn>2</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mn>2</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mn>2.6362</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p>attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(\alpha - \sin \alpha ){r^2} + \frac{1}{2}(\theta - \sin \theta ){r^2} = 4"> <mn>2</mn> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mi>α</mi> <mo stretchy="false">)</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>4</mn> </math></span></p>
<p>with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}"> <mi>α</mi> <mo>=</mo> <mn>4</mn> <mi>arcsin</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \pi - \frac{\alpha }{2}{\text{ }}\left( { = 2\arccos \frac{1}{4}} \right)"> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <mi>arccos</mi> <mo></mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 1.69"> <mi>r</mi> <mo>=</mo> <mn>1.69</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 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.</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>In triangle ABC, AB = 5, BC = 14 and AC = 11.</p>
<p>Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.</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 to apply cosine rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,{\text{A}} = \frac{{{5^2} + {{11}^2} - {{14}^2}}}{{2 \times 5 \times 11}} = - 0.4545 \ldots "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>5</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mn>11</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mrow> <mn>14</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>11</mn> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>0.4545</mn> <mo>…</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{A}} = 117.03569 \ldots ^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <mn>117.03569</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{A}} = 117.0^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <msup> <mn>117.0</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1</strong></em></p>
<p>attempt to apply sine rule or cosine rule: <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{sin}}\,117.03569 \ldots ^\circ }}{{14}} = \frac{{{\text{sin}}\,{\text{B}}}}{{11}}"> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>117.03569</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </mrow> <mrow> <mn>14</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>B</mtext> </mrow> </mrow> <mrow> <mn>11</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{B}} = 44.4153 \ldots ^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>B</mtext> </mrow> <mo>=</mo> <mn>44.4153</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{B}} = 44.4^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>B</mtext> </mrow> <mo>=</mo> <msup> <mn>44.4</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}} = 180^\circ - {\text{A}} - {\text{B}}"> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> <mo>−</mo> <mrow> <mtext>A</mtext> </mrow> <mo>−</mo> <mrow> <mtext>B</mtext> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}} = 18.5^\circ "> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <msup> <mn>18.5</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Candidates may attempt to find angles in any order of their choosing.</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>In a triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}">
<mrow>
<mtext>ABC, AB</mtext>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<mtext> cm, BC</mtext>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{B\hat AC}} = \frac{\pi }{9}">
<mrow>
<mrow>
<mi mathvariant="normal">B</mi>
<mrow>
<mover>
<mi mathvariant="normal">A</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>9</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the cosine rule to find the two possible values for AC.</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 difference between the areas of the two possible triangles ABC.</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><strong>METHOD 1</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = x"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{3^2} = {x^2} + {4^2} - 8x\cos \frac{\pi }{9}"> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>8</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> <strong><em>M1A1</em></strong></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.09,{\text{ }}6.43"> <mi>x</mi> <mo>=</mo> <mn>1.09</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6.43</mn> </math></span> <strong><em>A1A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = x"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mi>x</mi> </math></span></p>
<p>using the sine rule to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C"> <mi>C</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^2} = {x^2} + {3^2} - 6x\cos (152.869 \ldots ^\circ ) \Rightarrow x = 1.09"> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>152.869</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>1.09</mn> </math></span> <strong><em>(M1)A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^2} = {x^2} + {3^2} - 6x\cos (27.131 \ldots ^\circ ) \Rightarrow x = 6.43"> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>27.131</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>6.43</mn> </math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = x"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mi>x</mi> </math></span></p>
<p>using the sine rule to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B"> <mi>B</mi> </math></span> and a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C"> <mi>C</mi> </math></span> <strong><em>M1</em></strong></p>
<p>obtaining <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = 132.869 \ldots ^\circ ,{\text{ }}7.131 \ldots ^\circ "> <mi>B</mi> <mo>=</mo> <mn>132.869</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>7.131</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 27.131 \ldots ^\circ ,{\text{ }}152.869 \ldots ^\circ "> <mi>C</mi> <mo>=</mo> <mn>27.131</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>152.869</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(B = 2.319 \ldots ,{\text{ }}0.124 \ldots "> <mo stretchy="false">(</mo> <mi>B</mi> <mo>=</mo> <mn>2.319</mn> <mo>…</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0.124</mn> <mo>…</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 0.473 \ldots ,{\text{ }}2.668 \ldots )"> <mi>C</mi> <mo>=</mo> <mn>0.473</mn> <mo>…</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2.668</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span></p>
<p>attempting to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> using the cosine rule <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.09,{\text{ }}6.43"> <mi>x</mi> <mo>=</mo> <mn>1.09</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6.43</mn> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A0(M1)A1A0 </em></strong>for one correct value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span></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="\frac{1}{2} \times 4 \times 6.428 \ldots \times \sin \frac{\pi }{9}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>6.428</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 4 \times 1.088 \ldots \times \sin \frac{\pi }{9}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>1.088</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4.39747 \ldots "> <mn>4.39747</mn> <mo>…</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.744833 \ldots "> <mn>0.744833</mn> <mo>…</mo> </math></span>)</p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D"> <mi>D</mi> </math></span> be the difference between the two areas</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = \frac{1}{2} \times 4 \times 6.428 \ldots \times \sin \frac{\pi }{9} - \frac{1}{2} \times 4 \times 1.088 \ldots \times \sin \frac{\pi }{9}"> <mi>D</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>6.428</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>1.088</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(D = 4.39747 \ldots - 0.744833 \ldots )"> <mo stretchy="false">(</mo> <mi>D</mi> <mo>=</mo> <mn>4.39747</mn> <mo>…</mo> <mo>−</mo> <mn>0.744833</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3.65{\text{ (c}}{{\text{m}}^2})"> <mo>=</mo> <mn>3.65</mn> <mrow> <mtext> (c</mtext> </mrow> <mrow> <msup> <mrow> <mtext>m</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> </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="question">
<p>Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></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 style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">use of cosine rule <em><strong>(M1)</strong></em></p>
<p style="text-align: left;">CÂB = arccos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{49 + 100 - 25}}{{2 \times 7 \times 10}}} \right) = 0.48276 \ldots \left( { = 27.660 \ldots ^\circ } \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>49</mn> <mo>+</mo> <mn>100</mn> <mo>−</mo> <mn>25</mn> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.48276</mn> <mo>…</mo> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>27.660</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p style="text-align: left;">C<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\text{B}}\limits^ \wedge "> <mover> <mrow> <mtext>B</mtext> </mrow> <mo>∧</mo> </mover> </math></span>A = arccos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{25 + 100 - 49}}{{2 \times 5 \times 10}}} \right) = 0.70748 \ldots \left( { = 40.535 \ldots ^\circ } \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>25</mn> <mo>+</mo> <mn>100</mn> <mo>−</mo> <mn>49</mn> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.70748</mn> <mo>…</mo> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>40.535</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p style="text-align: left;">attempt to subtract triangle area from sector area <em><strong>(M1)</strong></em></p>
<p style="text-align: left;">area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times 49\left( {2{\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{B}} - {\text{sin}}\,{\text{2C}}\mathop {\text{A}}\limits^ \wedge {\text{B}}} \right)\, + \frac{1}{2} \times 25\left( {2{\text{C}}\mathop {\text{B}}\limits^ \wedge {\text{A}} - {\text{sin}}\,{\text{2C}}\mathop {\text{B}}\limits^ \wedge {\text{A}}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>49</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mtext>C</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>B</mtext> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>2C</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>B</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>25</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mtext>C</mtext> </mrow> <mover> <mrow> <mtext>B</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>A</mtext> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>2C</mtext> </mrow> <mover> <mrow> <mtext>B</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>A</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="text-align: left;">= 3.5079… + 5.3385… <em><strong>(A1)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award this <em><strong>A1</strong></em> for either of these two values.</p>
<p style="text-align: left;">= 8.85 (km<sup>2</sup>) <em><strong>A1</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Accept all answers that round to 8.8 or 8.9.</p>
<p style="text-align: left;"> </p>
<p style="text-align: left;"><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>