File "HL-paper1.html"
Path: /IB QUESTIONBANKS/4 Fourth Edition - PAPER/HTML/Mathematics HL/Topic 4/HL-paper1html
File size: 101.68 KB
MIME-type: application/octet-stream
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-212ef6a30de2a281f3295db168f85ac1c6eb97815f52f785535f1adfaee1ef4f.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-13d27c3a5846e837c0ce48b604dc257852658574909702fa21f9891f7bb643ed.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../index.html">Home</a>
</li>
<li class="active dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Questionbanks
<b class="caret"></b>
</a><ul class="dropdown-menu">
<li>
<a href="../../geography.html" target="_blank">DP Geography</a>
</li>
<li>
<a href="../../physics.html" target="_blank">DP Physics</a>
</li>
<li>
<a href="../../chemistry.html" target="_blank">DP Chemistry</a>
</li>
<li>
<a href="../../biology.html" target="_blank">DP Biology</a>
</li>
<li>
<a href="../../furtherMath.html" target="_blank">DP Further Mathematics HL</a>
</li>
<li>
<a href="../../mathHL.html" target="_blank">DP Mathematics HL</a>
</li>
<li>
<a href="../../mathSL.html" target="_blank">DP Mathematics SL</a>
</li>
<li>
<a href="../../mathStudies.html" target="_blank">DP Mathematical Studies</a>
</li>
</ul></li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="images/logo.jpg" alt="Ib qb 46 logo">
</div>
</div>
</div><h2>HL Paper 1</h2><div class="question">
<p class="p1">A point \(P\), relative to an origin \(O\), has position vector \(\overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}} {1 + s} \\ {3 + 2s} \\ {1 - s} \end{array}} \right),{\text{ }}s \in \mathbb{R}\).</p>
<p class="p1">Find the minimum length of \(\overrightarrow {{\text{OP}}} \).</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Three distinct non-zero vectors are given by \(\overrightarrow {{\text{OA}}} \) = <strong><em>a</em></strong>, \(\overrightarrow {{\text{OB}}} \) = <strong><em>b</em></strong>, and \(\overrightarrow {{\text{OC}}} \) = <strong><em>c</em></strong> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If \(\overrightarrow {{\text{OA}}} \) is perpendicular to \(\overrightarrow {{\text{BC}}} \) and \(\overrightarrow {{\text{OB}}} \) is perpendicular to \(\overrightarrow {{\text{CA}}} \) , show that \(\overrightarrow {{\text{OC}}} \) is perpendicular to \(\overrightarrow {{\text{AB}}} \).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the vectors \(\overrightarrow {{\text{OA}}} \) = <strong><em>a</em></strong>, \(\overrightarrow {{\text{OB}}} \) = <strong><em>b</em></strong> and \(\overrightarrow {{\text{OC}}} \) = <strong><em>a</em></strong> + <strong><em>b</em></strong>. Show that if \(|\)<strong><em>a</em></strong>\(|\) = \(|\)<strong><em>b</em></strong>\(|\) then (<strong><em>a</em></strong> + <strong><em>b</em></strong>)\( \cdot \)(<strong><em>a</em></strong> − <strong><em>b</em></strong>) = 0. Comment on what this tells us about the parallelogram OACB.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the plane with equation \(4x - 2y - z = 1\) and the line given by the parametric equations</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(x = 3 - 2\lambda \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(y = (2k - 1) + \lambda \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(z = - 1 + k\lambda .\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the line is perpendicular to the plane, find</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) the value of <em>k</em>;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the coordinates of the point of intersection of the line and the plane.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Two boats, <em>A </em>and <em>B </em>, move so that at time <em>t </em>hours, their position vectors, in kilometres, are <em><strong>r</strong></em>\(_A\) = (9<em>t</em>)<em><strong>i</strong></em> + (3 – 6<em>t</em>)<em><strong>j</strong></em> and <strong><em>r</em></strong>\(_B\) = (7 – 4<em>t</em>)<em><strong>i</strong></em> + (7<em>t</em> – 6)<em><strong>j</strong></em> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the common point of the paths of the two boats.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the boats do not collide.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">O, A, B and C are distinct points such that \(\overrightarrow {{\text{OA}}} = \) <strong><em>a</em></strong>, \(\overrightarrow {{\text{OB}}} = \) <strong><em>b </em></strong>and \(\overrightarrow {{\text{OC}}} = \) <strong><em>c</em></strong><span class="s1">.</span></p>
<p class="p1"><span class="s1">It is given that </span><strong><em>c </em></strong>is perpendicular to \(\overrightarrow {{\text{AB}}} \) <span class="s1">and </span><strong><em>b </em></strong>is perpendicular to \(\overrightarrow {{\text{AC}}} \).</p>
<p class="p1"><span class="s1">Prove that </span><strong><em>a </em></strong>is perpendicular to \(\overrightarrow {{\text{BC}}} \).</p>
</div>
<br><hr><br><div class="specification">
<p>The points A, B, C and D have position vectors <em><strong>a</strong></em>, <em><strong>b</strong></em>, <em><strong>c</strong></em> and <em><strong>d</strong></em>, relative to the origin O.</p>
<p>It is given that \(\mathop {{\text{AB}}}\limits^ \to = \mathop {{\text{DC}}}\limits^ \to \).</p>
</div>
<div class="specification">
<p>The position vectors \(\mathop {{\text{OA}}}\limits^ \to \), \(\mathop {{\text{OB}}}\limits^ \to \), \(\mathop {{\text{OC}}}\limits^ \to \) and \(\mathop {{\text{OD}}}\limits^ \to \) are given by</p>
<p style="padding-left: 150px;"><em><strong>a</strong></em> = <em><strong>i</strong></em> + 2<em><strong>j</strong></em> − 3<em><strong>k</strong></em></p>
<p style="padding-left: 150px;"><em><strong>b</strong></em> = 3<em><strong>i</strong></em> − <em><strong>j</strong></em> + <em>p<strong>k</strong></em></p>
<p style="padding-left: 150px;"><em><strong>c</strong></em> = <em>q<strong>i</strong></em> + <em><strong>j</strong></em> + 2<em><strong>k</strong></em></p>
<p style="padding-left: 150px;"><em><strong>d</strong></em> = −<em><strong>i</strong></em> + <em>r<strong>j</strong></em> − 2<em><strong>k</strong></em></p>
<p>where <em>p</em> , <em>q</em> and <em>r</em> are constants.</p>
</div>
<div class="specification">
<p>The point where the diagonals of ABCD intersect is denoted by M.</p>
</div>
<div class="specification">
<p>The plane \(\Pi \) cuts the <em>x</em>, <em>y</em> and <em>z</em> axes at X , Y and Z respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why ABCD is a parallelogram.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using vector algebra, show that \(\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to \).</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 <em>p</em> = 1, <em>q</em> = 1 and <em>r</em> = 4.</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 the parallelogram ABCD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector equation of the straight line passing through M and normal to the plane \(\Pi \) containing ABCD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of \(\Pi \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of X, Y and Z.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find YZ.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the points \({\text{O}}(0,{\text{ }}0,{\text{ }}0)\), \({\text{ A}}(6,{\text{ }}0,{\text{ }}0)\), \({\text{B}}({6,{\text{ }}- \sqrt {24} ,{\text{ }}\sqrt {12} })\), \({\text{C}}({0,{\text{ }}- \sqrt {24} ,{\text{ }}\sqrt {12}})\) form a square.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of M, the mid-point of [OB].</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that an equation of the plane </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\({\mathit{\Pi }}\), containing the square OABC, is \(y + \sqrt 2 z = 0\).</span></span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find a vector equation of the line \(L\), through M, perpendicular to the plane <span style="background-color: #f7f7f7;">\({\mathit{\Pi }}\)</span>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of D, the point of intersection of the line \(L\) with the plane whose equation is \(y = 0\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of E, the reflection of the point D in the plane <span style="background-color: #f7f7f7;">\({\mathit{\Pi }}\)</span>.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the angle \({\rm{O\hat DA}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State what this tells you about the solid OABCDE.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the two planes</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{\pi _1}:x + 2y - z = 1\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{\pi _2}:x + z = - 2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">are perpendicular.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the equation of the plane \({\pi _3}\) that passes through the origin and is perpendicular to both \({\pi _1}\) and \({\pi _2}\).</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">The following system of equations represents three planes in space.</p>
<p class="p1">\[x + 3y + z = - 1\]</p>
<p class="p1">\[x + 2y - 2z = 15\]</p>
<p class="p1">\[2x + y - z = 6\]</p>
<p class="p1">Find the coordinates of the point of intersection of the three planes.</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The position vectors of the points \(A\), \(B\) and \(C\) are \(a\), \(b\) and \(c\) respectively, relative to an origin \(O\). The following diagram shows the triangle \(ABC\) and points \(M\), \(R\), \(S\) and \(T\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-08_om_15.53.19.png" alt></p>
<p>\(M\) is the midpoint of [\(AC\)].</p>
<p>\(R\) is a point on [\(AB\)] such that \(\overrightarrow {{\text{AR}}} = \frac{1}{3}\overrightarrow {{\text{AB}}} \).</p>
<p>\(S\) is a point on [\(AC\)] such that \(\overrightarrow {{\text{AS}}} = \frac{2}{3}\overrightarrow {{\text{AC}}} \).</p>
<p>\(T\) is a point on [\(RS\)] such that \(\overrightarrow {{\text{RT}}} = \frac{2}{3}\overrightarrow {{\text{RS}}} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Express \(\overrightarrow {{\text{AM}}} \) in terms of <em>\(a\)</em> and \(c\).</p>
<p>(ii) Hence show that \(\overrightarrow {{\text{BM}}} = \frac{1}{2}\)\(a\) – \(b\)\( + \frac{1}{2}c\).</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>(i) Express \(\overrightarrow {{\text{RA}}} \) in terms of <em>\(a\) </em>and <em>\(b\)</em>.</p>
<p>(ii) Show that \(\overrightarrow {RT} = - \frac{2}{9}a - \frac{2}{9}b + \frac{4}{9}c\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(T\) lies on [\(BM\)].</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of <em>x </em>for which the vectors \(\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> {2\cos x} \\ <br> 0 <br>\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> {2\sin x} \\ <br> 1 <br>\end{array}} \right)\) are perpendicular, \(0 \leqslant x \leqslant \frac{\pi }{2}\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Consider the vectors </span><strong><em>a</em></strong> \( = \) <strong><em>i</em></strong> \( - {\text{ }}3\)<strong><em>j</em></strong> \( - {\text{ }}2\)<strong><em>k</em></strong>, <strong><em>b</em></strong> \( = - {\text{ }}3\)<strong><em>j</em></strong> \( + {\text{ }}2\)<strong><em>k</em></strong><span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find </span><strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong><span class="s2">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the Cartesian equation of the plane containing the vectors <span class="s1"><strong><em>a </em></strong></span>and <span class="s1"><strong><em>b</em></strong></span>, and passing through the point \((1,{\text{ }}0,{\text{ }} - 1)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The points A(1, 2, 1) , B(−3, 1, 4) , C(5, −1, 2) and D(5, 3, 7) are the vertices of a tetrahedron.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the vectors \(\overrightarrow {{\text{AB}}} \)</span> <span style="font-family: times new roman,times; font-size: medium;">and \(\overrightarrow {{\text{AC}}} \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the Cartesian equation of the plane \(\prod \) that contains the face ABC.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the points \({\text{A(1, 0, 0)}}\), \({\text{B(2, 2, 2)}}\) and \({\text{C(0, 2, 1)}}\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">A third plane \({\Pi _3}\) is defined by the Cartesian equation \(16x + \alpha y - 3z = \beta \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the vector \(\overrightarrow {{\text{CA}}} \times \overrightarrow {{\text{CB}}} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find an exact value for the area of the triangle ABC.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the Cartesian equation of the plane \({\Pi _1}\), containing the triangle ABC, is \(2x + 3y - 4z = 2\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A second plane \({\Pi _2}\) is defined by the Cartesian equation \({\Pi _2}:4x - y - z = 4\). \({L_1}\) is the line of intersection of the planes \({\Pi _1}\) and \({\Pi _2}\).</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;">Find a vector equation for \({L_1}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the value of \(\alpha \) if all three planes contain \({L_1}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find conditions on \(\alpha \) and \(\beta \) if the plane \({\Pi _3}\) does <strong>not </strong>intersect with \({L_1}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-29_om_08.02.43.png" alt></p>
<p>Consider the triangle \(ABC\). The points \(P\), \(Q\) and \(R\) are the midpoints of the line segments [\(AB\)], [\(BC\)] and [\(AC\)] respectively.</p>
<p>Let \(\overrightarrow {{\text{OA}}} = {{a}}\), \(\overrightarrow {{\text{OB}}} = {{b}}\) and \(\overrightarrow {{\text{OC}}} = {{c}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\overrightarrow {{\text{BR}}} \) <span class="s1">in terms of </span>\({{a}}\), \({{b}}\) <span class="s1">and </span>\({{c}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Find a vector equation of the line that passes through \(B\) and \(R\) in terms of \({{a}}\), \({{b}}\) and \({{c}}\) and a parameter \(\lambda \).</p>
<p>(ii) Find a vector equation of the line that passes through \(A\) and \(Q\) in terms of \({{a}}\), \({{b}}\) and \({{c}}\) and a parameter \(\mu \).</p>
<p>(iii) Hence show that \(\overrightarrow {{\text{OG}}} = \frac{1}{3}({{a}} + {{b}} + {{c}})\) given that \(G\) is the point where [\(BR\)] and [\(AQ\)] intersect.</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 class="p1">Show that the line segment <span class="s1">[\(CP\)] </span>also includes the point <span class="s1">\(G\)</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The coordinates of the points <span class="s1">\(A\)</span>, <span class="s1">\(B\)</span> and <span class="s1">\(C\)</span> are \((1,{\text{ }}3,{\text{ }}1)\), \((3,{\text{ }}7,{\text{ }} - 5)\) and \((2,{\text{ }}2,{\text{ }}1)\) respectively.</p>
<p class="p1">A point <span class="s1">\(X\)</span> is such that [<span class="s1">\(GX\)</span>] is perpendicular to the plane <span class="s1">\(ABC\)</span>.</p>
<p class="p1">Given that the tetrahedron <span class="s1">\(ABCX\)</span> has volume \({\text{12 unit}}{{\text{s}}^{\text{3}}}\), find possible coordinates</p>
<p class="p1">of <span class="s1">\(X\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the lines \({l_1}\) and \({l_2}\) defined by</p>
<p class="p1">\({l_1}:\) <em><strong>r</strong></em> \( = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ a \end{array}} \right) + \beta \left( {\begin{array}{*{20}{c}} 1 \\ 4 \\ 2 \end{array}} \right)\) and \({l_2}:\frac{{6 - x}}{3} = \frac{{y - 2}}{4} = 1 - z\) where \(a\) is a constant.</p>
<p class="p1">Given that the lines \({l_1}\) and \({l_2}\) intersect at a point P,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">find the value of \(a\);</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">determine the coordinates of the point of intersection <span class="s1">P</span><span class="s2">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Consider the plane \({\mathit{\Pi} _1}\), parallel to both lines \({L_1}\) and \({L_2}\). Point C lies in the plane \({\mathit{\Pi} _1}\).</span></p>
</div>
<div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_3}\) has vector equation </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\(\boldsymbol{r} = \left( \begin{array}{l}3\\0\\1\end{array} \right) + \lambda \left( \begin{array}{c}k\\1\\ - 1\end{array} \right)\).</span></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 plane \({\mathit{\Pi} _2}\) has Cartesian equation \(x + y = 12\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The angle between the line \({L_3}\) and the plane \({\mathit{\Pi} _2}\) is 60°.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given the points A(1, 0, 4), B(2, 3, −1) and C(0, 1, − 2) , </span><span style="font-family: 'times new roman', times; font-size: medium;">find the vector equation of the line \({L_1}\) passing through the points A and B.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) has Cartesian equation \(\frac{{x - 1}}{3} = \frac{{y + 2}}{1} = \frac{{z - 1}}{{ - 2}}\).</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;">Show that \({L_1}\) and \({L_2}\) are skew lines.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the Cartesian equation of the plane \({\Pi _1}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the value of \(k\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the point of intersection P of the line \({L_3}\) and the plane \({\mathit{\Pi} _2}\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given any two non-zero vectors <strong><em>a</em></strong> and <strong><em>b</em></strong> , show that \(|\)<strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong>\({|^2}\) = \(|\)<strong><em>a</em></strong>\({|^2}\)\(|\)<strong><em>b</em></strong>\({|^2}\) – (<strong><em>a</em></strong> \( \cdot \) <strong><em>b</em></strong>)\(^2\).</span></p>
</div>
<br><hr><br><div class="specification">
<p>The points A and B are given by \({\text{A}}(0,{\text{ }}3,{\text{ }} - 6)\) and \({\text{B}}(6,{\text{ }} - 5,{\text{ }}11)\).</p>
<p>The plane <em>Π</em> is defined by the equation \(4x - 3y + 2z = 20\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation of the line <em>L </em>passing through the points A and B.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point of intersection of the line <em>L </em>with the plane <em>Π</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The three vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\) and \(\boldsymbol{c}\) are given by\[{\boldsymbol{a}} = \left( {\begin{array}{*{20}{c}}<br> {2y} \\ <br> { - 3x} \\ <br> {2x} <br>\end{array}} \right),{\text{ }}{\boldsymbol{b}}{\text{ }} = \left( {\begin{array}{*{20}{c}}<br> {4x} \\ <br> y \\ <br> {3 - x} <br>\end{array}} \right),{\text{ }}{\boldsymbol{c}}{\text{ }} = \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> { - 7} \\ <br> 6 <br>\end{array}} \right){\text{ where }}x,y \in \mathbb{R}{\text{ }}{\text{.}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) If <strong><em>a</em></strong> + 2<strong><em>b</em></strong> − <strong><em>c</em></strong> = 0, find the value of <em>x</em> and of <em>y</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the exact value of \(|\)<strong><em>a</em></strong> + 2<strong><em>b</em></strong>\(|\).</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a circle with centre O. The points A, B, C lie on the </span><span style="font-family: times new roman,times; font-size: medium;">circumference of the circle and [AC] is a diameter.</span></p>
<p><br><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></span><br><span style="font-family: times new roman,times; font-size: medium;">Let \(\overrightarrow {{\text{OA}}} = {\boldsymbol{a}}\) and \(\overrightarrow {{\text{OB}}} = {\boldsymbol{b}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down expressions for \(\overrightarrow {{\text{AB}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">and \(\overrightarrow {{\text{CB}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">in terms of the vectors \({\boldsymbol{a}}\) and \({\boldsymbol{b}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence prove that angle \({\text{A}}\hat {\rm{B}}{\text{C}}\) is a right angle.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Two planes have equations</p>
<p class="p1">\[{\Pi _1}:{\text{ }}4x + y + z = 8{\text{ and }}{\Pi _2}:{\text{ }}4x + 3y - z = 0\]</p>
</div>
<div class="specification">
<p class="p1">Let \(L\) be the line of intersection of the two planes.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">B </span>is the point on \({\Pi _1}\) <span class="s1">with coordinates \((a,{\text{ }}b,{\text{ }}1)\).</span></p>
</div>
<div class="specification">
<p class="p1">The point P <span class="s1">lies on \(L\) </span>and \({\rm{A\hat BP}} = 45^\circ \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the cosine of the angle between the two planes in the form \(\sqrt {\frac{p}{q}} \) where \(p,{\text{ }}q \in \mathbb{Z}\).</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) Show that \(L\) <span class="s1">has direction \(\left( {\begin{array}{*{20}{c}} { - 1} \\ 2 \\ 2 \end{array}} \right)\).</span></p>
<p class="p4">(ii) <span class="Apple-converted-space"> </span>Show that the point \({\text{A }}(1,{\text{ }}0,{\text{ }}4)\) <span class="s2">lies on both planes.</span></p>
<p class="p3">(iii) <span class="Apple-converted-space"> </span>Write down a vector equation of \(L\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given the vector \(\overrightarrow {{\text{AB}}} \) </span>is perpendicular to \(L\) find the value of \(a\) and the value of \(b\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \({\text{AB}} = 3\sqrt 2 \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of the two possible positions of \(P\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>In the following diagram, \(\overrightarrow {{\text{OA}}} \) = <strong><em>a</em></strong>, \(\overrightarrow {{\text{OB}}} \) = <strong><em>b</em></strong>. C is the midpoint of [OA] and \(\overrightarrow {{\text{OF}}} = \frac{1}{6}\overrightarrow {{\text{FB}}} \).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.26.10.png" alt="N17/5/MATHL/HP1/ENG/TZ0/09"></p>
</div>
<div class="specification">
<p>It is given also that \(\overrightarrow {{\text{AD}}} = \lambda \overrightarrow {{\text{AF}}} \) and \(\overrightarrow {{\text{CD}}} = \mu \overrightarrow {{\text{CB}}} \), where \(\lambda ,{\text{ }}\mu \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <strong><em>a </em></strong>and <strong><em>b </em></strong>\(\overrightarrow {{\text{OF}}} \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <strong><em>a </em></strong>and <strong><em>b </em></strong>\(\overrightarrow {{\text{AF}}} \).</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 an expression for \(\overrightarrow {{\text{OD}}} \) in terms of <strong><em>a</em></strong>, <strong><em>b </em></strong>and \(\lambda \);</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for \(\overrightarrow {{\text{OD}}} \) in terms of <strong><em>a</em></strong>, <strong><em>b </em></strong>and \(\mu \).</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>Show that \(\mu = \frac{1}{{13}}\), and find the value of \(\lambda \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce an expression for \(\overrightarrow {{\text{CD}}} \) in terms of <strong><em>a </em></strong>and <strong><em>b </em></strong>only.</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>Given that area \(\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})\), find the value of \(k\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">The vertices of a triangle ABC have coordinates given by A(−1, 2, 3), B(4, 1, 1) and C(3, −2, 2).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the lengths of the sides of the triangle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find \(\cos {\rm{B\hat AC}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{CA}}} = \) −7<strong><em>i</em></strong> − 3<strong><em>j</em></strong> − 16<strong><em>k</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, show that the area of the triangle ABC is \(\frac{1}{2}\sqrt {314} \).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the Cartesian equation of the plane containing the triangle ABC.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find a vector equation of (AB).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point D on (AB) is such that \(\overrightarrow {{\text{OD}}} \) is perpendicular to \(\overrightarrow {{\text{BC}}} \) where O is the origin.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the coordinates of D.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that D does not lie between A and B.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A line \(L\) has equation \(\frac{{x - 2}}{p} = \frac{{y - q}}{2} = z - 1\) where \(p,{\text{ }}q \in \mathbb{R}\).</p>
<p class="p1">A plane \(\Pi \) has equation \(x + y + 3z = 9\).</p>
</div>
<div class="specification">
<p class="p1">Consider the different case where the acute angle between \(L\) and \(\Pi \) is \(\theta \)</p>
<p class="p1">where \(\theta = \arcsin \left( {\frac{1}{{\sqrt {11} }}} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(L\) is not perpendicular to \(\Pi \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(L\) lies in the plane \(\Pi \), find the value of \(p\) and the value of \(q\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(p = - 2\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>If \(L\) intersects \(\Pi \) at \(z = - 1\), find the value of \(q\).</p>
<div class="marks">[11]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">For non-zero vectors </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\boldsymbol{a}}\)</span> and </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\boldsymbol{b}}\)</span>, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) if \(\left| {{\boldsymbol{a}} - {\boldsymbol{b}}} \right| = \left| {{\boldsymbol{a}} + {\boldsymbol{b}}} \right|\), then \({\boldsymbol{a}}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> and </span><span style="font-family: 'times new roman', times; font-size: medium;">\({\boldsymbol{b}}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> are perpendicular;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\left| {{\boldsymbol{a}} \times {\boldsymbol{b}}} \right|^2} = {\left| {\boldsymbol{a}} \right|^2}{\left| {\boldsymbol{b}} \right|^2} - {({\boldsymbol{a}} \cdot {\boldsymbol{b}})^2}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The points A, B and C have position vectors \({\boldsymbol{a}}\), \({\boldsymbol{b}}\) and \({\boldsymbol{c}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that the area of triangle ABC is \(\frac{1}{2}\left| {{\boldsymbol{a}} \times {\boldsymbol{b}} + {\boldsymbol{b}} \times {\boldsymbol{c}} + {\boldsymbol{c}} \times {\boldsymbol{a}}} \right|\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, show that the shortest distance from B to AC is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{\left| {{\boldsymbol{a}} \times {\boldsymbol{b}} + {\boldsymbol{b}} \times {\boldsymbol{c}} + {\boldsymbol{c}} \times {\boldsymbol{a}}} \right|}}{{\left| {{\boldsymbol{c}} - {\boldsymbol{a}}} \right|}}{\text{.}}\]</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the points A(1, −1, 4), B (2, − 2, 5) and O(0, 0, 0).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Calculate the cosine of the angle between \(\overrightarrow {{\text{OA}}} \) and \(\overrightarrow {{\text{AB}}} \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find a vector equation of the line \({L_1}\) which passes through A and B.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) has equation <strong><em>r</em></strong> = 2<strong><em>i</em></strong> + 4<strong><em>j</em></strong> + 7<strong><em>k</em></strong> + t(2<strong><em>i</em></strong> + <strong><em>j</em></strong> + 3<strong><em>k</em></strong>), where \(t \in \mathbb{R}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that the lines \({L_1}\) and \({L_2}\) intersect and find the coordinates of their point of intersection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the Cartesian equation of the plane which contains both the line \({L_2}\) and the point A.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">ABCD </span>is a parallelogram, where \(\overrightarrow {{\text{AB}}} \) = –<strong><em>i</em></strong> + 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong> and \(\overrightarrow {{\text{AD}}} \) = 4<strong><em>i</em></strong> – <strong><em>j</em></strong> – 2<strong><em>k</em></strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the parallelogram ABCD.</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>By using a suitable scalar product of two vectors, determine whether \({\rm{A\hat BC}}\) is acute or obtuse.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The points A, B, C have position vectors <strong><em>i</em></strong> + <strong><em>j</em></strong> + 2<strong><em>k</em></strong> , <strong><em>i</em></strong> + 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong> , 3<strong><em>i</em></strong> + <strong><em>k</em></strong> respectively and lie in the plane \(\pi \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the area of the triangle ABC;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the shortest distance from C to the line AB;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) the cartesian equation of the plane \(\pi \) . </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The line <em>L</em> passes through the origin and is normal to the plane \(\pi \) , it intersects \(\pi \) at the</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">point D.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the coordinates of the point D;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) the distance of \(\pi \) from the origin.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the points A(1, 2, 3), B(1, 0, 5) and C(2, −1, 4).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the area of the triangle ABC.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the diagram below, [AB] is a diameter of the circle with centre O. Point C is on the circumference of the circle. Let \(\overrightarrow {{\text{OB}}} = \boldsymbol{b} \) and \(\overrightarrow {{\text{OC}}} = \boldsymbol{c}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARgAAAD+CAIAAABuho3wAAAXKUlEQVR4nO2dT2jk1pbG7y6PNzNJL968ftWTeTbPkE2cbII2A3nlzh9nSBZapfNn4UUjSDrD2wRk6DwmzEYMfgZvBGkYcIgmDzowiARmpTTFQBANA4ZqyKKRV2FUeBOUYBpuCKG4szhtdXVVuVx175V0r/T9lrarSlbpu/ec8x0dMQHAypxmgy/2dzYZsbV7OLg/Gvz7zcH3TR9YY7CmDwDYxng0uLnN2Pbu4SB7MBZCjE+O4v2dHtvYif+v6YNrDAgJrMSPR/tvMLZ5Pf5uPP3zNyEkAJbjdLDbY6x3c3A6nvrNePTVv352f/qnnQFCAsszPh3c7DHW2x2cNn0opgEhgeX55SR+jzG2sX/0S9OHYhoQkokURZGdD+e8oeN6KCTsSLNASHXDOc+ybDgcxnEcx7F/BpPCcRx6eRAE9IYktooOfpwdbjPGtg+zziZD5wAhVQvtLXEcB0EwpRYSQBRFJIDhcFjuOXmeL3jPyd0pTVN6eRiGvu97njelsTAMkyTRto+Nv4uvb86r2onxyeDg8OiBhs+wEghJM5xz2m1833cch65pz/NoxyC11HAMpcZ833ddlw7DdV06DJVjGJ98fXOrx9j27uF/H538LIQQ4jT7+uD69c/uP+juPgUhaaAoijRNwzAsL1nP86IoStN08d5SJ1mWJUkye5DD4XDlzerMgn1Ib2c/PjrproiEgJBUyLIsiqIymvJ9X3Gxrw3aNqcOPkkSc2RvHRDSanDOafOhsI0WdSvEcx4kqnKncl2Xtqnyt0mSUFDa7HEaDoS0LMPhMAgCWr+DIEjTtCiKpg9KM3meJ0lC25TjOAcHB6+++uqlS5fovz44OGj6AM0FQrqALMvK/Yf005yNUx9FUSRJ8sorrzDGLj351KUnnyItIfY7DwhpPhTSULTjeV6SJF3QzxQffvhh7/LljbX1jbX13uXLjLHbt283fVCGAiFNQ1sQxTZRFHV2DR6NRq+89PLG2vrTvStP965srK3TvkSnpX1hrSIQ0iPSNCXP1PO8NE2bPpwm+erLLzfW1vsv/vGtt95ijP3qiSf+8Pu1117d/uGHH+I4po06DEOrqyx6gZCEECJNU1wcxGg0+uDGjY219T9/9BFtO3meZ1n2zTffbKyt/2Vvj/6sXHR83+/4GSO6LqRJCSFcuXPnztV+/2q/f+fOndnf/vXzzzfW1u/du1f+JMsyyInorpBIQoj4iaIo/vzRRxtr6x/cuDEajeb+Def87WvXrvb7U3WXPM8pq+yynLoopMldqIO1uFnu3bt3td/fWFv/6ssvF//l8fExRX2zvyqKosu7U7eElGUZuY0I5AjO+V/29jbW1t++du28jWgKqkPMjf3ERLAXBEGnznBXhFQUBfUl+L7f2Yr2FMfHx7QR/fXzz1famT+4ceNqv79AeFmW0Z4fx3FH9vxOCCmOY8dxXNftYMgxF845VQ6u9vvHx8ervnw0Gl3t9z+4cWPxnyVJQqe9C316LRcSLY2O48Rx3PSxmMJoNHr72jWqZUtvF3fu3Fkmp+Kcl3WIdkd6rRUS5zyKoi58hStBGc7Vfn+yii0HVfmWyawoNW33ctZOIQ2HQ8dxHMfpQlCxJLNOqyJFUVzt99++dm3JbS2OY+oaaWWO2jYhlbFEEAQdSXOXgZzWBdU2Oe7du0e1iiX/Ps9zqpomSaLxMEygVULK85wyImxEJZzzC51WFah6vlLFgramloXc7RFS+fVgIypZ3mmVhnNOXUUrnfb2LXltEBLnnEzA9gUM0kw6rRIF7pWgdoeyn3VJyiA8iqKKDqxOrBdSnudkVrQyhZXj+PiYCtyrOq3SkCt19+7dVV+YpqnjOJ7n2R7m2S2kJElQV5hk0mlVL3Cv9LnU7iChB6pAOI5jtV1usZAoMGixNbEqZYFbxWlV+fTz+lkvhHNODVz23k9ppZA457SGtSZVVad0WiXiK73HIF1hp3JRGIZ6j6oe7BMSFXyQFJUURaHXaVWBjkS6zk4pk42lV8uERKUFz/OsO9EVUZHTKg21O1zYz7qA8itufFFYCZuElKYp7sYrqdpplYb6WZdvd5ilKAoK3S0KOqwRUqmipg/ECGpwWlWQaHeYokyDbdGSHUIiFaFAJ+p1WqWhdofl+1nPe5MgCGwpi1sgJCpz21sY1Uj9Tqs01M9665Nbiu9jy7dvupBsOY9V05TTqsKtT25Nje+Sw4prwGghWXEGa6BZp1Wa88Z3SWD+lWCukOj+VpPPXT2Y4LRKI9fPOhfDtWSokKi6YOxZqwejnFZpKCLVYnPR83XMrOOZKCSoSJjntKog3c86i7FaMk5IpKJ23KMih7FOqzTUz6rS7jCJmf6SWUKi9pAuu66GO63SLDm+axnIq3Vd16i6i0FCIhUFQdD0gTSDFU6rCsuP77oQzrnruka1XJoiJFpmjDo1dWKR0ypN2c+q5R80LXgxRUi+7zuO09ZraAE2Oq3S3L17V7GfdZIsy8xJp40QkrGlmKqx1GlVQb2fdRJzCrzNC8mcc1EzVjut0siN71qAIatww0LK89yc3bk22uG0SqOx3YEwoYjXpJCo9uL7foPHUD9tclqlmX0crQqc88brvU0Kie426UhuINrotEpT9rPq2pCp8NDghNDGhEQj6ay4Z0sLbXVapVEZ3zUXGkLUVLLUjJAoNerIHa+td1qlURzfNYvv+00lS80IibzXRj66ZrrgtKpw4eNoV6IoCsdxGqldNSAkeqJr66tVnXJapVnycbTLMxwOGWP1Tw6tW0iNJ4X10EGnVRqN/axEI0WsuoXUhXp3N51WFaiYqSuBbKQaXquQWh/UddxplUbL+K5JKMCrsyZcn5CoUtfioA5OqwqrPo72QoIgqLOCV5+QfN9va6UOTqsWyCTQ2+5Qm8VSk5CoM7XxzsIqgNOqC43juwgy/eu56uoQEq0N7etMhdOqnSr6WespbtUhpCiK2tdTB6e1IjSO7xJndksNtlLlQiqKomW3G8FprRqN47uEEGEYuq6r5a0WULmQqP2p6k+pDTitNaC3n5X6hqquOlQrJNpYW9PiDae1NuhU66rfkIFZ6cJXrZB8329HHwOc1vrRO76r6k2pQiG1ZjuC09oI6o+jnaTqTalCIbVgO4LT2izqj6OdxHXd6ubgVSUk2o6sjoLgtJqAxvFd1BVQ0aZUlZB837d3+DCcVnPQ28/qum5FmVIlQrI6Oyqd1luf3EKB2wSon1VLu0N1mVIlQrI3O4LTaia6xndVV77TLyS6XcK67QhOq8lo7GelTUnLUU2iX0j1dGTopXRaUeA2FupnVW934JxX0bOmWUgVHWV1FEVBBW44reaja3xXFWu9ZiFVtG9WROm0osBtC1rGd1WRfWgWkuu6Vtx3VBa44bTaha7xXb7v6zVndQqJJk6YHyCVTqvGCQGgNrSM79JuzuoUUhAEhle9Oee3PrkFp9V2tPSzOo6jcRSPNiGZX2aA09oaqJ9Vsd1Bb8lBm5CSJDG5zACntWWoj++ikoOu0SjahOR5njmPmJ4ETmtbUe9n1Vgb0yMkGsxgYDcDnNYWo/442iRJdEV3eoSk8YB0Aae1CyiO76INQMuMIT1C8jzPKPsITmt3oOxXeoqGrpREg5BI1oZMUYXT2jU45yrju3QFUxqEZE5cB6e1m6iM79K1DWgQkgn1OjitHUeln1VL7U5VSOTD1v+kwUngtAIhBJkcEvF8FEXqz0lRFRL1LCm+iQpwWgEhPb5Ly6AeVQ2EYdhUfx2cVjCF9Pgux3EUu9tUheS6biMP4YPTCuYi1+4QBIFinq8kpEYK33BawQLkxnepV56VhJSmac2NqnBawYVQP+utT24t/xJqYFVZl5WEFIZhbVMg4bSC5SEvZKX6k2KapCSk6uZWTgGnFayExPguxTRJXkjkIFXd8Q2nFcix6viuOI5V3CR5IVH1XfrlywCnFaiw0uNo6XqWvszklaCo4AuB0wrUWb6fVTHCkheSeun9POC0Al1QP+uS7Q4qOb+8kCqyYuG0Ar0sP75LZW+QFFIVlQY4raAilhzfpZKtSApJMTObBU4rqI6yn3XxFUsTTuU+QvJlGnsa4LSCGrh79+6FPqTKgC5JIcVxrKXpG04rqI1l+lmlExZJIfm+r3hTIZxWUDPLjO/yPE+ucCcvJJXmIDitoBEuHN8lvUPIplYKt5fDaQUNsvhxtNI5i7yQJEJJOK2gccp+1rkWi3QFXEZIcrVvOK3AEBaM75KugMsLafm/h9MKTOO88V3Srdgyr1nJRILTCsxk7vgu6ekJMkJaMiGD0wpM5rzH0crl/1UJCU4rMJ+5/az1CSmKogVCgtMKLIKy98kLVW54g4yQFrixcFqBXcyO75JrNtApJDitwEamHkfbpJDgtAKroaoYbQCNCQlOK7CdyfFdWoT0/WD3BbZ1cPRgfN4L8jx3XTcIAs556bR+cOMGnFZgNWU/6xwhncQ7bJbNnf0vBtkp/cljQhqP4us9xtgLu4Pv537YYDAo3+bll17qv/hHOK1VkyTJ+++/v7e3Z8jDRVvMp59+urG2/tvf/IYx9qc//enxJOWn7PBNxt6LT34RQghxmn19sNNjrPcv8ehn8biQfsoO393aeXeLsd7u4HTmY2hOA2PsV0888XTvysba+j+/9hqc1krZ29ubXAOhpUo5ODhYe/of//D7tb/59a8ZY3uP3W3xy0n83oSQxJm0etuH98ePCel0sNt78/D+/fj6ZqmzSagNiTH2D7/rbayt9y5ffu6553xQJVPBxOuvv970EbWZK1euPPm3f7extv67v/8tnfCFQhqfDm72ZoQ0Ph3c3Lgej8aP/XoSzvkzzzxDO9KlJ59ijH388ccxqIwwDKeE9PLLLzd9UG3m2WefZYzRdsQYe/HFF88X0vhBFu9u9WZCu/H9w+1/epganQ52e4xtH2YzFQd60CXxzjvvoNJdNfTVEpcuXWrkmW7dIUmSyWXr9u3bE78kIU3xqJpAQho/ODrYeqScH4/23ziv5FAUxbvvvtv4Y8w7Qp7nW1tb9KUdHBw0fTjtJ03T559/vt/vz3QJTe1IP58cxfs7m4xt7hx+++BMSN8Pdl+YLe/NLTkI5YENYFXyPMfmXxvnXN6zOdJZ7MbePMx+YkKIcXa43bs5OJ2I5Mb3D7d7bOqHF3wSAG1AVkgPvj3c2dzYP/plzst6W7txNmPOQkigxZxzeVO+80hI45P//Wx3mzHWux6PxoLtb5xFcjvxycPXTOVVj6sQQgKtZs7lPb+zgbHezv5//Q/tNJK9dorTIQEwFrkZkRXeag6UOc0GX+zvbM6uf6A6mFG3mgNVHnx7uLPJejv7X2cPhBDi55Oj/9zd6rGtfxucTDedAI1ASG2CUts39o9+nPzpw67ihe35QJH6hKTyGBmwDOPscJvNbS4hx+/c9nygiPSDv+oYEAlW5GHVdJ4hvuBXQAO1DohUeR4TWALqz2cz5p545Ew88iqATqSfoFfrEH2wHBfvSPM0BjQgnf9LCklu9hdYEuRITRGGYRAEEi9s5kFj4ALG38XXNxnbvB5/Nymlh1W7eXe4AC1IX9iSQpIWLlgW+EhN4DiO3E1fDT+MGSxifHL01X/sbvXQ2VAb0sm/pJBgJYH2IW0iCWkhUQUcs+xAm5CufQtpIQm15zEDYCAqCYu8kOS6zQEwFpX7g+SFhMIdaBkq7qi8kJIkcV1X+uUAGIX002MJeSFRiQP1BtAOaGaj9MuVStioN4DWsPiBrheiJCQMbwCtQbF4piSkKIo8z1N5BwBMgJ60ohJeKQmJ+hswBBTYjvqVrCQkdR0DYALqsZVqv5zneUiTgO2oX8aqQorjGG4SsBotgZWqkOAmAdtRdJAIDbdCSN8LBYAJaGl20yAkNN0Bq9EygESDkGhnRBEc2AgVvtVzEw1ColwNQ4WAjYRhqKWpQM/t4kEQILoDNqIrw9cjJER3wEZ0xXVCl5AQ3QEb0RXXCV1CEkKEYYgBXcAuNDo32oSkcZcEoAb05iM6Z9M5joNxKMAWfN/XWCHTKaQoitB3B6yAJjRovHFBp5C0HxwAFaG92Vrz2GG92yUAFaE9DdEsJErgUHIAJlPFVap/EL7rurjVD5iM53lhGOp9T/1CiuPYcRx0OQAzoTvotD+4Vb+QOOe4QwkYSxAEVXQOVPKMI9TBgZlUV1iuREh0uGi9A6YRhmFFS3xVT92r7ogBkKPS9b0qIWFTAqZR6eJe4XNgsSkBc6h6Za9QSNiUgDlUvaxX+2TyKIrgKYHGoWeHV7qmVysk8pRwbwVoFt/3q77rtFohCTQ6gKapqJVhisqFxDl3XVd7axMAS+K6bg13JFQuJHHWbFv1kgDALEmS1HM7Qh1CEkL4vo9n+4GaqTNFr0lIVDZBJyuokyAIXNetJz+vSUgCpXBQL1RjqG3wQX1CoqoDbkQHNVD/xVafkETtiwToLPWHP7UKSQgRBAECPFAptF7XnJDXLSQqpCDAAxVBQV3907PrFpJAgAeqhIK6+udYNSAk0dx/C9oNDaBvZI1uRkicc8/zYNECjVDW0FQzWjNCEkLkee44DibgAV34vl+b/TpLY0ISZz14SJaAOnEcM8byPG/qAJoUkhAiDEMkS0ARSo2avRe7YSGVyRKcJSAH5QiN36fTsJCEMScC2Ig5C3HzQhJnWzN6w8GqUKOMCamBEUISZzdgYeQQWJ4wDJstMExiipDEWeHBkPMCDIdKvuasvAYJSQjh+z60BC6EVGTUdCqzhGRO7giMxczqlFlCEmfdu9ASmAupyMC7B4wTkjg7WdASmMLkC8NEIQmzTxloBOpJNfaSMFRIwtRQGDSC+cmzuUISEwGxsacP1ID5KhKGC0kgxus8tlwApgtJ2HMqgXYs+uotEJKw6oQCXdj1pdshJCFEnucUKKPvoQsMh0O70mNrhCTOkk70ELUe6gCyq2Brk5CEEJxz6pw3p1sR6CWKIsaYdcM8LBMSQf3zRvUsAnVolTSqp3t5rBSSmNj9bYmhwWKKorA6brdVSEKILMuoqmPpqQcl5Vdpwr2uclgsJDGxjGGml73QJC3bgwu7hSSE4JxTymRdego4577vt2Nch/VCItI0tT026BoUzrmu247IvCVCEmeOreM4LVje2g3nnGrcFvmtF9IeIRHt+4ZaRrne2VjjXkDbhCSEyLLMdV1UIAyE6gq+77cvAm+hkMRE8NDK78xGytWtrYF3O4VElF8eeiAapCyrtntRa7OQCAonPM/LsqzpY+kcVE3tQpjdfiEJIYqiIL8iCIIWL4pGkWWZ53nk73Wh8NMJIREU6VG3axe+2qYoioJ6T33fb4dHtAwdEhKRJAkFG5CTdoqioHTIdd2uBdKdE5IQgnMexzHJqWVuRlOUp9R13W6e0i4KiaDvnpZP7E7SFEURRRGtSm0tbS9Dd4VETO5OkNNKTAZy3dyFJum6kIhSTtTPj8reYrIsoyooJFQCIT2Cc56mKVX2fN/HJTIF5zxJkvL8tN4aWgkIaQ5ZllEB13GcKIq6U8M9jyzLKIqjaezYsWeBkM6lKIo4jmkB9jwvSZKuXUB5nkdRVJ6BNE2RQ54HhHQxtB5TBuX7fusVled5kiTUl+C6bhRF7f5/tQAhrcBwOCwVRXtUm6K+LMvK/Yf006b/rmogJBlIUeU1F4ZhmqY2Ltu0+dDMTVodoB85ICQlyguRMTYpKpOvxSzLJsVD9QNLFwJzgJC0kWVZHMe+79MFSglVFEVpmjbYeMY5J+WEYUhpDwk+CIKWhabNAiFVQp7naZpGUTSpK9d1SVpxHGdZpl1dpJnhcBjHcRiGsx8dx/FwOMTOUwUQUh0URUH7FUmr3BkIx3H8M+Klobci2OOUb0WbIWrWNQAhNQZtIBR3zWpjGUpRkWC6dueCUUBIAGjg/wEE4CoOU5eNowAAAABJRU5ErkJggg==" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \(\overrightarrow {{\text{CB}}} \) and for \(\overrightarrow {{\text{AC}}} \) in terms of \(\boldsymbol{b}\) and \(\boldsymbol{c}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence prove that \({\rm{A\hat CB}}\) is a right angle.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Find the coordinates of the point of intersection of the planes defined by the equations \(x + y + z = 3,{\text{ }}x - y + z = 5\) and \(x + y + 2z = 6\).</p>
</div>
<br><hr><br><div class="specification">
<p>The following figure shows a square based pyramid with vertices at O(0, 0, 0), A(1, 0, 0), B(1, 1, 0), C(0, 1, 0) and D(0, 0, 1).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The Cartesian equation of the plane \({\Pi _2}\), passing through the points B , C and D , is \(y + z = 1\).</p>
</div>
<div class="specification">
<p>The plane \({\Pi _3}\) passes through O and is normal to the line BD.</p>
</div>
<div class="specification">
<p>\({\Pi _3}\) cuts AD and BD at the points P and Q respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of the plane \({\Pi _1}\), passing through the points A , B and D.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle between the faces ABD and BCD.</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 Cartesian equation of \({\Pi _3}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that P is the midpoint of AD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the triangle OPQ.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">PQRS is a rhombus. Given that \(\overrightarrow {{\text{PQ}}} = \) \(\boldsymbol{a}\) and \(\overrightarrow {{\text{QR}}} = \) \(\boldsymbol{b}\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) express the vectors \(\overrightarrow {{\text{PR}}} \) and \(\overrightarrow {{\text{QS}}} \) in terms of \(\boldsymbol{a}\) and \(\boldsymbol{b}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) hence show that the diagonals in a rhombus intersect at right angles.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The vectors <strong><em>a</em></strong> , <strong><em>b</em></strong> , <strong><em>c</em></strong> satisfy the equation <strong><em>a</em></strong> + <strong><em>b</em></strong> + <strong><em>c</em></strong> = <strong>0</strong> . Show that <strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong> = <strong><em>c</em></strong> \( \times \) <strong><em>a</em></strong> .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the vectors <strong><em>a</em></strong> = 6<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 2<strong><em>k</em></strong>, <strong><em>b</em></strong> = −3<strong><em>j</em></strong> + 4<strong><em>k</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the cosine of the angle between vectors <strong><em>a</em></strong> and <strong><em>b</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find <strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) <strong>Hence</strong> find the Cartesian equation of the plane \(\prod \) containing the vectors <strong><em>a</em></strong> and <strong><em>b</em></strong> and passing through the point (1, 1, −1).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) The plane \(\prod \) intersects the <em>x-y</em> plane in the line <em>l</em>. Find the area of the finite triangular region enclosed by <em>l</em>, the <em>x</em>-axis and the <em>y</em>-axis.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given two vectors <strong><em>p</em></strong> and <strong><em>q</em></strong>,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) show that <strong><em>p</em></strong>\( \cdot \)<strong><em>p</em></strong> = \(|\)<strong><em>p</em></strong>\({|^2}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) hence, or otherwise, show that \(|\)<strong><em>p</em></strong> + <strong><em>q</em></strong>\({|^2}\) = \(|\)<strong><em>p</em></strong>\({|^2}\) + 2<strong><em>p</em></strong>\( \cdot \)<strong><em>q</em></strong> + \(|\)<strong><em>q</em></strong>\({|^2}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) deduce that \(|\)<strong><em>p</em></strong> + <strong><em>q</em></strong>\(|\) ≤ \(|\)<strong><em>p</em></strong>\(|\) + \(|\)<strong><em>q</em></strong>\(|\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that a Cartesian equation of the line, \({l_1}\), containing points A(1, −1, 2) and B(3, 0, 3) has the form \(\frac{{x - 1}}{2} = \frac{{y + 1}}{1} = \frac{{z - 2}}{1}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) An equation of a second line, \({l_2}\), has the form \(\frac{{x - 1}}{1} = \frac{{y - 2}}{2} = \frac{{z - 3}}{1}\). Show that the lines \({l_1}\) and \({l_2}\) intersect, and find the coordinates of their point of intersection.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given that direction vectors of \({l_1}\) and \({l_2}\) are <strong><em>d</em></strong>\(_1\) and <strong><em>d</em></strong>\(_2\) respectively, determine <strong><em>d</em></strong>\(_1 \times \) <strong><em>d</em></strong>\(_2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Show that a Cartesian equation of the plane, \(\prod \), that contains \({l_1}\) and \({l_2}\) is \( - x - y + 3z = 6\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find a vector equation of the line \({l_3}\) which is perpendicular to the plane \(\prod \) and passes through the point T(3, 1, −4).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) (i) Find the point of intersection of the line \({l_3}\) and the plane \(\prod \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the coordinates of \({{\text{T}}}'\), the reflection of the point T in the plane \(\prod \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Hence find the magnitude of the vector \(\overrightarrow {{\text{TT}}'} \).</span></p>
</div>
<br><hr><br><div class="question">
<p>The acute angle between the vectors 3<em><strong>i</strong></em> − 4<em><strong>j</strong></em> − 5<em><strong>k</strong></em> and 5<em><strong>i</strong></em> − 4<em><strong>j</strong></em> + 3<em><strong>k</strong></em> is denoted by <em>θ</em>.</p>
<p>Find cos <em>θ</em>.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\alpha \) be the angle between the unit vectors <strong><em>a</em></strong> and <strong><em>b</em></strong>, where \(0 \leqslant \alpha \leqslant \pi \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Express \(|\)<strong><em>a</em></strong> − <strong><em>b</em></strong>\(|\) and \(|\)<strong><em>a</em></strong> + <strong><em>b</em></strong>\(|\) in terms of \(\alpha \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence determine the value of \(\cos \alpha \) for which \(|\)<strong><em>a</em></strong> + <strong><em>b</em></strong>\(|\) = 3 \(|\)<strong><em>a</em></strong> − <strong><em>b</em></strong>\(|\).</span></p>
</div>
<br><hr><br><div class="question">
<p>A triangle has vertices A(1, −1, 1), B(1, 1, 0) and C(−1, 1, −1) .</p>
<p>Show that the area of the triangle is \(\sqrt 6 \) .</p>
</div>
<br><hr><br>