File "SL-paper1.html"
Path: /IB QUESTIONBANKS/4 Fourth Edition - PAPER/HTML/Further Mathematics/Topic 1/SL-paper1html
File size: 39.55 KB
MIME-type: text/x-tex
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-212ef6a30de2a281f3295db168f85ac1c6eb97815f52f785535f1adfaee1ef4f.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-13d27c3a5846e837c0ce48b604dc257852658574909702fa21f9891f7bb643ed.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../index.html">Home</a>
</li>
<li class="active dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Questionbanks
<b class="caret"></b>
</a><ul class="dropdown-menu">
<li>
<a href="../../geography.html" target="_blank">DP Geography</a>
</li>
<li>
<a href="../../physics.html" target="_blank">DP Physics</a>
</li>
<li>
<a href="../../chemistry.html" target="_blank">DP Chemistry</a>
</li>
<li>
<a href="../../biology.html" target="_blank">DP Biology</a>
</li>
<li>
<a href="../../furtherMath.html" target="_blank">DP Further Mathematics HL</a>
</li>
<li>
<a href="../../mathHL.html" target="_blank">DP Mathematics HL</a>
</li>
<li>
<a href="../../mathSL.html" target="_blank">DP Mathematics SL</a>
</li>
<li>
<a href="../../mathStudies.html" target="_blank">DP Mathematical Studies</a>
</li>
</ul></li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="images/logo.jpg" alt="Ib qb 46 logo">
</div>
</div>
</div><h2>SL Paper 1</h2><div class="specification">
<p>Consider the simultaneous linear equations</p>
<p style="padding-left: 180px;">\(x + z = - 1\)<br>\(3x + y + 2z = 1\)<br>\(2x + ay - z = b\)</p>
<p>where \(a\) and \(b\) are constants.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using row reduction, find the solutions in terms of \(a\) and \(b\) when \(a\) ≠ 3 .</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the equations have no unique solution when \(a\) = 3.</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>Find all the solutions to the equations when \(a\) = 3, \(b\) = 10 in the form <em><strong>r</strong></em> = <em><strong>s</strong></em> + \(\lambda \)<em><strong>t</strong></em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrix <em><strong>M</strong></em> = \(\left[ {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 1} <br>\end{array}\,\,\,\begin{array}{*{20}{c}}<br> { - 4} \\ <br> { - 1} <br>\end{array}} \right]\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the linear transformation represented by <em><strong>M</strong></em> transforms any point on the line \(y = x\) to a point on the same line.</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>Explain what happens to points on the line \(4y + x = 0\) when they are transformed by <strong><em>M</em></strong>.</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>State the two eigenvalues of <strong><em>M</em></strong>.</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 two eigenvectors of <em><strong>M</strong></em> which correspond to the two eigenvalues.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A matrix <span class="s1"><strong><em>M </em></strong></span>is called idempotent if <span class="s1"><strong><em>M</em></strong></span><span class="s2">\(^2 = \) </span><span class="s1"><strong><em>M</em></strong></span>.</p>
</div>
<div class="specification">
<p class="p1">The idempotent matrix <span class="s1"><strong><em>N </em></strong></span>has the form</p>
<p class="p1" style="text-align: center;"><strong><em>N</em></strong> \( = \left( {\begin{array}{*{20}{c}} a&{ - 2a} \\ a&{ - 2a} \end{array}} \right)\)</p>
<p class="p1" style="text-align: left;">where \(a \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Explain why <span class="s1"><strong><em>M </em></strong></span>is a square matrix.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the set of possible values of <span class="s1">det(<strong><em>M</em></strong>).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(a\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the eigenvalues of <span class="s1"><strong><em>N</em></strong></span><span class="s2">.</span></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find corresponding eigenvectors.</p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em><strong>A</strong></em><sup>2</sup> = 2<em><strong>A</strong></em> + <em><strong>I</strong></em> where <em><strong>A</strong></em> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em><strong>A</strong></em><sup>4</sup> = 12<em><strong>A</strong></em> + 5<em><strong>I</strong></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>Let <em><strong>B</strong></em> = \(\left[ {\begin{array}{*{20}{c}}<br> 4&2 \\ <br> 1&{ - 3} <br>\end{array}} \right]\).</p>
<p>Given that <em><strong>B</strong></em><sup>2</sup> – <em><strong>B</strong></em> – 4<em><strong>I</strong></em> = \(\left[ {\begin{array}{*{20}{c}}<br> k&0 \\ <br> 0&k <br>\end{array}} \right]\), find the value of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the system of equations</p>
<p>\[\left[ {\begin{array}{*{20}{l}} 1&2&1&3 \\ 2&1&3&1 \\ 5&1&8&0 \\ 3&3&4&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \\ {{x_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ 3 \\ \lambda \\ \mu \end{array}} \right]\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of \(\lambda \) and the value of \(\mu \) for which the equations are consistent.</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>For these values of \(\lambda \) and \(\mu \), solve the equations.</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>State the rank of the matrix of coefficients, justifying your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The non-zero vectors <strong><em>v</em></strong><sub>1</sub>, <strong><em>v</em></strong><sub>2</sub>, <strong><em>v</em></strong><sub>3</sub> form an orthogonal set of vectors in \({\mathbb{R}^3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering \({\alpha _1}\)<strong><em>v</em></strong>\(_1 + {\alpha _2}\)<strong><em>v</em></strong>\(_2 + {\alpha _3}\)<strong><em>v</em></strong>\(_3 = 0\), show that <strong><em>v</em></strong>\(_1\), <strong><em>v</em></strong>\(_2\), <strong><em>v</em></strong>\(_3\) are linearly independent.</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>Explain briefly why <strong><em>v</em></strong>\(_1\), <strong><em>v</em></strong>\(_2\), <strong><em>v</em></strong>\(_3\) form a basis for vectors in \({\mathbb{R}^3}\).</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>Show that the vectors</p>
<p>\[\left[ {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right];{\text{ }}\left[ {\begin{array}{*{20}{c}} { - 1} \\ 1 \\ 1 \end{array}} \right];{\text{ }}\left[ {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right]\]</p>
<p>form an orthogonal basis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express the vector</p>
<p>\[\left[ {\begin{array}{*{20}{c}} 2 \\ 8 \\ 0 \end{array}} \right]\]</p>
<p>as a linear combination of these vectors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In this question, \(x\)<span class="s1">, </span>\(y\) and \(z\) denote the coordinates of a point in three-dimensional Euclidean space with respect to fixed rectangular axes with origin O. The vector space of position vectors relative to <span class="s1">O </span>is denoted by \({\mathbb{R}^3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the set of position vectors of points whose coordinates satisfy \(x - y - z = 1\) <span class="s1">does not form a vector subspace of \({\mathbb{R}^3}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that the set of position vectors of points whose coordinates satisfy \(x - y - z = 0\) forms a vector subspace, \(V\), of \({\mathbb{R}^3}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Determine an orthogonal basis for \(V\) <span class="s1">of which one member is \(\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)\).</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>Augment this basis with an orthogonal vector to form a basis for \({\mathbb{R}^3}\).</p>
<p class="p1">(iv) <span class="Apple-converted-space"> </span>Express the position vector of the point with coordinates \((4,{\text{ }}0,{\text{ }} - 2)\) as a linear combination of these basis vectors.</p>
<div class="marks">[13]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The matrix <strong><em>A</em></strong> is given by <strong><em>A</em></strong> = \(\left( {\begin{array}{*{20}{c}}1&2&1\\1&1&2\\2&3&1\end{array}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that <strong><em>A</em></strong>\(^3\) can be expressed in the form <strong><em>A</em></strong>\(^3 = a\)<strong><em>A</em></strong>\(^2 = b\)<strong><em>A</em></strong> \( + c\)<strong><em>I</em></strong>, determine the values of the constants \(a\), \(b\), \(c\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Hence express <strong><em>A</em></strong>\(^{ - 1}\) in the form <strong><em>A</em></strong>\(^{ - 1} = d\)<strong><em>A</em></strong>\(^2 = e\)<strong><em>A</em></strong> \( + f\)<strong><em>I</em></strong> where \(d,{\text{ }}e,{\text{ }}f \in \mathbb{Q}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Use this result to determine <strong><em>A</em></strong>\(^{ - 1}\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A transformation \(T\) is a linear mapping from \({\mathbb{R}^3}\) to \({\mathbb{R}^4}\), represented by the matrix</p>
<p class="p1">\[M = \left( {\begin{array}{*{20}{c}} 1&2&1 \\ 2&7&5 \\ { - 3}&1&4 \\ 1&5&4 \end{array}} \right)\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the row rank of \(M\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise find the kernel of \(T\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the column rank of \(M\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the basis for the range of this transformation.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em><strong>S</strong></em> be the set of matrices given by</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\left[ \begin{array}{l}<br>a\\<br>c<br>\end{array} \right.\left. \begin{array}{l}<br>b\\<br>d<br>\end{array} \right]\) ; \(a,b,c,d \in \mathbb{R}\), \(ad - bc = 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relation \(R\) is defined on \(S\) as follows. Given \(\boldsymbol{A}\) , </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{B} \in S\)</span> , \(\boldsymbol{ARB}\) if and only if there </span><span style="font-size: medium;"><span style="font-family: times new roman,times;">exists </span></span><span style="font-size: medium;"><span style="font-family: times new roman,times;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{X} \in S\)</span></span> </span></span><span style="font-size: medium;"><span style="font-family: times new roman,times;">such that </span></span><span style="font-family: times new roman,times; font-size: medium;">\(\boldsymbol{A} = \boldsymbol{BX}\)</span><span style="font-size: medium;"><span style="font-family: times new roman,times;"> .<br></span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(R\) is an equivalence relation.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State, with </span><span style="font-family: times new roman,times; font-size: medium;">a reason, whether or not there are any non-zero values of \(n\) , other than \(1\), </span><span style="font-family: times new roman,times; font-size: medium;">for which \(R\) is an equivalence relation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The matrix <strong><em>M </em></strong>is defined by <strong><em>M</em></strong> = \(\left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).</span><span style="background-color: #f7f7f7; line-height: normal;"><span style="font-family: 'times new roman', times; font-size: medium;"><br></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="background-color: #f7f7f7; line-height: normal;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The eigenvalues of <strong><em>M </em></strong>are denoted by \({\lambda _1},{\text{ }}{\lambda _2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \({\lambda _1} + {\lambda _2} = a + d\) and \({\lambda _1}{\lambda _2} = \det \)(<strong><em>M</em></strong>).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that \(a + b = c + d = 1\), show that 1 is an eigenvalue of <strong><em>M</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find eigenvectors for the matrix \(\left( {\begin{array}{*{20}{c}}2&{ - 1}\\3&{ - 2}\end{array}} \right)\).</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The matrix \(\boldsymbol{A}\) is given by \[\boldsymbol{A} = \left( {\begin{array}{*{20}{c}}<br>0&1&0\\<br>2&4&1\\<br>4&{ - 11}&{ - 2}<br>\end{array}} \right) .\]<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the matrices \({\boldsymbol{A}^2}\) and </span><span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{A}^3}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , and verify that \({{\boldsymbol{A}}^3} = 2{{\boldsymbol{A}}^2} - {\boldsymbol{A}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Deduce that \({{\boldsymbol{A}}^4} = 3{{\boldsymbol{A}}^2} - 2{\boldsymbol{A}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Suggest a similar expression for </span><span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{A}^n}\)</span><span style="font-family: times new roman,times; font-size: medium;"> in terms of \(\boldsymbol{A}\) and </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\boldsymbol{A}^2}\)</span> , valid for \(n \ge 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Use mathematical induction to prove the validity of your suggestion.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the system of equations \[\left( {\begin{array}{*{20}{c}}<br>1&{ - 1}&2\\<br>2&2&{ - 1}\\<br>3&5&{ - 4}\\<br>3&1&1<br>\end{array}} \right)\left( \begin{array}{l}<br>x\\<br>y\\<br>z<br>\end{array} \right) = \left( \begin{array}{l}<br>5\\<br>3\\<br>1\\<br>k<br>\end{array} \right) .\]<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">By reducing the augmented matrix to row echelon form,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) find the rank of the coefficient matrix;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) find the value of \(k\) for which the system has a solution.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For this value of \(k\) , determine the solution.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that the following vectors form a basis for the vector space \({\mathbb{R}^3}\) .\[\left( \begin{array}{l}<br>1\\<br>2\\<br>3<br>\end{array} \right);\left( \begin{array}{l}<br>2\\<br>3\\<br>1<br>\end{array} \right);\left( \begin{array}{l}<br>5\\<br>2\\<br>5<br>\end{array} \right)\]</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express the following vector as a linear combination of the above vectors.\[\left( \begin{array}{l}<br>12\\<br>14\\<br>16<br>\end{array} \right)\]</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The set \(S\) contains the eight matrices of the form\[\left( {\begin{array}{*{20}{c}}<br>a&0&0\\<br>0&b&0\\<br>0&0&c<br>\end{array}} \right)\]where \(a\), \(b\), \(c\) can each take one of the values \( + 1\) or \( - 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that any matrix of this form is its own inverse.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(S\) forms an Abelian group under matrix multiplication.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Giving a reason, state whether or not this group is cyclic.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>By considering the images of the points (1, 0) and (0, 1),</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>determine the 2 × 2 matrix <em><strong>P</strong></em> which represents a reflection in the line \(y = \left( {{\text{tan}}\,\theta } \right)x\).</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>determine the 2 × 2 matrix <em><strong>Q</strong></em> which represents an anticlockwise rotation of <em>θ</em> about the origin.</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>Describe the transformation represented by the matrix <em><strong>PQ</strong></em>.</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>A matrix <em>M</em> is said to be orthogonal if <strong><em>M </em></strong><em><sup>T</sup></em><strong><em>M</em></strong> = <strong><em>I</em></strong> where <strong><em>I</em></strong> is the identity. Show that <em><strong>Q</strong></em> is orthogonal.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The transformations <em>T</em><sub>1</sub>, <em>T</em><sub>2</sub>, <em>T</em><sub>3</sub>, <em>T</em><sub>4</sub>, in the plane are defined as follows:</p>
<p><em>T</em><sub>1</sub> : A rotation of 360° about the origin<br><em>T</em><sub>2</sub> : An anticlockwise rotation of 270° about the origin<br><em>T</em><sub>3</sub> : A rotation of 180° about the origin<br><em>T</em><sub>4</sub> : An anticlockwise rotation of 90° about the origin.</p>
</div>
<div class="specification">
<p>The transformation <em>T</em><sub>5</sub> is defined as a reflection in the \(x\)-axis.</p>
</div>
<div class="specification">
<p>The transformation <em>T</em> is defined as the composition of <em>T</em><sub>3</sub> followed by <em>T</em><sub>5</sub> followed by <em>T</em><sub>4</sub>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Copy and complete the following Cayley table for the transformations of <em>T</em><sub>1</sub>, <em>T</em><sub>2</sub>, <em>T</em><sub>3</sub>, <em>T</em><sub>4</sub>, under the operation of composition of transformations.</p>
<p><img src="data:image/png;base64,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"></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><em>Show that T</em><sub>1</sub>, <em>T</em><sub>2</sub>, <em>T</em><sub>3</sub>, <em>T</em><sub>4 </sub>under the operation of composition of transformations form a group. Associativity may be assumed.</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>Show that this group is cyclic.</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>Write down the 2 × 2 matrices representing <em>T</em><sub>3</sub>, <em>T</em><sub>4</sub> and <em>T</em><sub>5</sub>.</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 2 × 2 matrix representing <em>T</em>.</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>Give a geometric description of the transformation <em>T</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br>