File "HL-paper3.html"
Path: /IB QUESTIONBANKS/5 Fifth Edition - PAPER/HTML/Math AA/Topic 1/HL-paper3html
File size: 211.44 KB
MIME-type: text/html
Charset: utf-8
<!DOCTYPE html>
<html>
<meta http-equiv="content-type" content="text/html;charset=utf-8">
<head>
<meta charset="utf-8">
<title>IB Questionbank</title>
<link rel="stylesheet" media="all" href="css/application-02ef852527079acf252dc4c9b2922c93db8fde2b6bff7c3c7f657634ae024ff1.css">
<link rel="stylesheet" media="print" href="css/print-6da094505524acaa25ea39a4dd5d6130a436fc43336c0bb89199951b860e98e9.css">
<script src="js/application-9717ccaf4d6f9e8b66ebc0e8784b3061d3f70414d8c920e3eeab2c58fdb8b7c9.js"></script>
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML-full"></script>
<!--[if lt IE 9]>
<script src='https://cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv.min.js'></script>
<![endif]-->
<meta name="csrf-param" content="authenticity_token">
<meta name="csrf-token" content="iHF+M3VlRFlNEehLVICYgYgqiF8jIFlzjGNjIwqOK9cFH3ZNdavBJrv/YQpz8vcspoICfQcFHW8kSsHnJsBwfg==">
<link href="favicon.ico" rel="shortcut icon">
</head>
<body class="teacher questions-show">
<div class="navbar navbar-fixed-top">
<div class="navbar-inner">
<div class="container">
<div class="brand">
<div class="inner"><a href="http://ibo.org/">ibo.org</a></div>
</div>
<ul class="nav">
<li>
<a href="../../../../../../../index.html">Home</a>
</li>
<!-- - if current_user.is_language_services? && !current_user.is_publishing? -->
<!-- %li= link_to "Language services", tolk_path -->
</ul>
<ul class="nav pull-right">
<li class="dropdown">
<a class="dropdown-toggle" data-toggle="dropdown" href="#">
Help
<b class="caret"></b>
</a>
<ul class="dropdown-menu">
<li><a href="https://questionbank.ibo.org/video_tour?locale=en">Video tour</a></li>
<li><a href="https://questionbank.ibo.org/instructions?locale=en">Detailed instructions</a></li>
<li><a target="_blank" href="https://ibanswers.ibo.org/">IB Answers</a></li>
</ul>
</li>
<li>
<a href="https://06082010.xyz">IB Documents (2) Team</a>
</li></ul>
</div>
</div>
</div>
<div class="page-content container">
<div class="row">
<div class="span24">
<div class="pull-right screen_only"><a class="btn btn-small btn-info" href="https://questionbank.ibo.org/updates?locale=en">Updates to Questionbank</a></div>
<p class="muted language_chooser">
User interface language:
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=en">English</a>
|
<a href="https://questionbank.ibo.org/en/users/set_user_locale?new_locale=es">Español</a>
</p>
</div>
</div>
<div class="page-header">
<div class="row">
<div class="span16">
<p class="back-to-list">
</p>
</div>
<div class="span8" style="margin: 0 0 -19px 0;">
<img style="width: 100%;" class="qb_logo" src="https://mirror.ibdocs.top/qb.png" alt="Ib qb 46 logo">
</div>
</div>
</div>
<h2>HL Paper 3</h2><div class="specification">
<p>This question will investigate power series, as an extension to the Binomial Theorem for negative and fractional indices.</p>
<p>A power series in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is defined as a function of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> where the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i} \in \mathbb{R}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>It can be considered as an infinite polynomial.</p>
</div>
<div class="specification">
<p>This is an example of a power series, but is only a finite power series, since only a finite number of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
</math></span> are non-zero.</p>
</div>
<div class="specification">
<p>We will now attempt to generalise further.</p>
<p>Suppose <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span> can be written as the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^5}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</math></span> using the Binomial Theorem.</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>Consider the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - x + {x^2} - {x^3} + {x^4} - ...">
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span></p>
<p>By considering the ratio of consecutive terms, explain why this series is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 1}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which this equality is true.</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>Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 2}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Repeat this process to find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 3}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, by recognising the pattern, deduce the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - n}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</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>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0}">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating both sides of the expression and then substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_1}">
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Repeat this procedure to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2}">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_3}">
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the first four terms in what is called the Extended Binomial Theorem for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 + {x^2}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, using integration, find the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, giving the first four non-zero terms.</p>
<div class="marks">[4]</div>
<div class="question_part_label">k.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to investigate conditions for the existence of complex roots of polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">3</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">4</mtext></math>.</strong></p>
<p> <br>The cubic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mi>r</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Noah believes that if <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>≥</mo><mn>3</mn><mi>q</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> are all real.</p>
</div>
<div class="specification">
<p>Now consider polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p>
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>,</mo><mo> </mo><mi>s</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math>.</p>
<p>In a similar way to the cubic equation, it can be shown that:</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>β</mi><mi>δ</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mi>δ</mi></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, has one integer root.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> show that:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math>, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> cannot all be real.</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>Using the result from part (c), show that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>17</mn></math>, this equation has at least one complex root.</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>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> in the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the smallest positive integer value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> required to show that Noah is incorrect.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the equation will have at least one real root for all values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state a condition in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> that would imply <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your result from part (f)(ii) to show that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what the result in part (f)(ii) tells us when considering this equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the integer root of this equation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn></math> as a product of one linear and one cubic factor, prove that the equation has at least one complex root.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question will explore connections between complex numbers and regular polygons.</p>
<p>The diagram below shows a sector of a circle of radius 1, with the angle subtended at the centre <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="O">
<mi>O</mi>
</math></span> being <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha {\text{,}}\,\,0 < \alpha < \frac{\pi }{2}">
<mi>α<!-- α --></mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>α<!-- α --></mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>. A perpendicular is drawn from point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> to intersect the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q">
<mi>Q</mi>
</math></span>. The tangent to the circle at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> intersects the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the area of two triangles and the area of the sector show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\alpha \,{\text{sin}}\,\alpha < \alpha < \frac{{{\text{sin}}\,\alpha }}{{{\text{cos}}\,\alpha }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo><</mo> <mi>α</mi> <mo><</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{\alpha \to 0} \frac{\alpha }{{{\text{sin}}\,\alpha }} = 1"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>α</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mi>α</mi> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^n} = 1{\text{,}}\,\,z \in \mathbb{C}{\text{,}}\,\,n \in \mathbb{N}{\text{,}}\,\,n \geqslant 5"> <mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>⩾</mo> <mn>5</mn> </math></span>. Working in modulus/argument form find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> solutions to this equation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Represent these <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> solutions on an Argand diagram. Let their positions be denoted by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{\text{,}}\,\,{P_1}{\text{,}}\,\,{P_2}{\text{,}}\, \ldots {P_{n - 1}}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></span> placed in order in an anticlockwise direction round the circle, starting on the positive <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Show the positions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{\text{,}}\,\,{P_1}{\text{,}}\,\,{P_2}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_{n - 1}}"> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></span>.</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>Show that the length of the line segment <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{P_1}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{sin}}\frac{\pi }{n}"> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the total length of the perimeter of the regular <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> sided polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_0}{P_1}{P_2} \ldots {P_{n - 1}}{P_0}"> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> <mo>…</mo> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using part (b) find the limit of this perimeter as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total area of this <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> sided polygon.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using part (b) find the limit of this area as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to \infty "> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will be exploring the strategies required to solve a system of linear differential equations.</strong></p>
<p> </p>
<p>Consider the system of linear differential equations of the form:</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>y</mi></math>,</p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is a parameter.</p>
<p>First consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>From previous cases, we might conjecture that a solution to this differential equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is a constant.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation in part (a)(ii) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math>.</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>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the two values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let the two values found in part (c)(ii) be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math>.</p>
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> is a solution to the differential equation in (c)(i),where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> is a constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A <strong>Gaussian integer</strong> is a complex number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>. In this question, you are asked to investigate certain divisibility properties of Gaussian integers.</p>
</div>
<div class="specification">
<p>Consider two Gaussian integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><mtext>i</mtext></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mi>α</mi><mi>β</mi></math> for some Gaussian integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
</div>
<div class="specification">
<p>Now consider two Gaussian integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mn>11</mn><mo>+</mo><mn>2</mn><mtext>i</mtext></math>.</p>
</div>
<div class="specification">
<p>The norm of a complex number <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>z</mi></mfenced></math>, is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><msup><mfenced open="|" close="|"><mi>z</mi></mfenced><mn>2</mn></msup></math>. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>2</mn><mo>+</mo><mn>3</mn><mtext>i</mtext></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mtext>i</mtext></mrow></mfenced><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>=</mo><mn>13</mn></math>.</p>
</div>
<div class="specification">
<p>A <strong>Gaussian prime</strong> is a Gaussian integer, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math>, that <strong>cannot</strong> be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>α</mi><mi>β</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> are Gaussian integers with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>,</mo><mo> </mo><mi>N</mi><mfenced><mi>β</mi></mfenced><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>The positive integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is a prime number, however it is not a Gaussian prime.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> be Gaussian integers.</p>
</div>
<div class="specification">
<p>The result from part (h) provides a way of determining whether a Gaussian integer is a Gaussian prime.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</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>Determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>γ</mi><mi>α</mi></mfrac></math> is a Gaussian integer.</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>On an Argand diagram, plot and label all Gaussian integers that have a norm less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>α</mi></mfenced><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math>.</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>By expressing the positive integer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math> as a product of two Gaussian integers each of norm <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is not a Gaussian prime.</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>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is not a Gaussian prime.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down another prime number of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mn>2</mn></msup><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></math> that is not a Gaussian prime and express it as a product of two Gaussian integers.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mo>=</mo><mi>N</mi><mfenced><mi>α</mi></mfenced><mi>N</mi><mfenced><mi>β</mi></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><mtext>i</mtext></math> is a Gaussian prime.</p>
<div class="marks">[3]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use proof by contradiction to prove that a prime number, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, that is not of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> is a Gaussian prime.</p>
<div class="marks">[6]</div>
<div class="question_part_label">j.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will explore some of the properties of special functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">g</mi></math> and their relationship with the trigonometric functions, sine and cosine.</strong></p>
<p><br>Functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>,</mo><mo> </mo><mi>u</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math>, find expressions, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>u</mi></math>, for</p>
</div>
<div class="specification">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi></math> are known as circular functions as the general point (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>,</mo><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math>) defines points on the unit circle with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>.</p>
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> are known as hyperbolic functions, as the general point ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>θ</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math> ) defines points on a curve known as a hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>. This hyperbola has two asymptotes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> satisfies the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>u</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find, and simplify, an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>, stating the coordinates of any axis intercepts and the equation of each asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> can be rotated to coincide with the curve defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to investigate and prove a geometric property involving the roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> for integers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</strong></p>
<p><br>The roots of the equation <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math></strong> where <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math></strong> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><msup><mtext>e</mtext><mfrac><mrow><mn>2</mn><mi>πi</mi></mrow><mi>n</mi></mfrac></msup></math>. Each root can be represented by a point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math>, respectively, on an Argand diagram.</p>
<p>For example, the roots of the equation <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math></strong> where <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math></strong> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>. On an Argand diagram, the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> can be represented by a point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub></math> and the root <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> can be represented by a point <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>1</mn></msub></math>.</p>
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math>.</p>
<p>The roots of the equation <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn></math></strong> where <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math></strong> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>2</mn></msup></math>. On the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>2</mn></msub></math> lie on a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Line segments <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo></math> are added to the Argand diagram in part (a) and are shown on the following Argand diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math>is the length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math> is the length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo></math>.</p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>.</p>
<p>The roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>3</mn></msup></math>.</p>
</div>
<div class="specification">
<p>On the following Argand diagram, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>3</mn></msub></math> lie on a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>]</mo></math> are line segments.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>For the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>, the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>5</mn></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>3</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>4</mn></msup></math>.</p>
<p>It can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>4</mn></msub><mo>=</mo><mn>5</mn></math>.</p>
<p>Now consider the general case for integer values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</p>
<p>The roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>=</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mi>ω</mi><mo>,</mo><mo> </mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msup><mi>ω</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>. On an Argand diagram, these roots can be represented by the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math> respectively where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>]</mo><mo>,</mo><mo> </mo><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>]</mo><mo>,</mo><mo> </mo><mo>…</mo><mo>,</mo><mo> </mo><mo>[</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>]</mo></math> are line segments. The roots lie on a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mn>1</mn><mo>-</mo><mi>ω</mi><mo>|</mo></math>.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mi>n</mi></msup><mo>-</mo><mn>1</mn><mo>=</mo><mo>(</mo><mi>z</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo> </mo></math>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>ω</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>(</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>=</mo><mn>3</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By factorizing <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>4</mn></msup><mo>-</mo><mn>1</mn></math>, or otherwise, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>+</mo><mi>ω</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub><mo>=</mo><mn>4</mn></math>.</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>Suggest a value for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>1</mn></msub><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub><mo>×</mo><mo> </mo><mo>…</mo><mo> </mo><mo>×</mo><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mn>3</mn></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>.</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>Hence, write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>P</mtext><mn>0</mn></msub><msub><mtext>P</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mi>z</mi><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mo> </mo><mo>…</mo><mo> </mo><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn></math> as a product of linear factors over the set <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, using the part (g)(i) and part (f) results, or otherwise, prove your suggested result to part (e).</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore some properties of polygonal numbers and to determine and prove interesting results involving these numbers.</strong></p>
<p><br>A polygonal number is an integer which can be represented as a series of dots arranged in the shape of a regular polygon. Triangular numbers, square numbers and pentagonal numbers are examples of polygonal numbers.</p>
<p>For example, a triangular number is a number that can be arranged in the shape of an equilateral triangle. The first five triangular numbers are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math>.</p>
<p>The following table illustrates the first five triangular, square and pentagonal numbers respectively. In each case the first polygonal number is one represented by a single dot.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>For an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>-sided regular polygon, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th polygonal number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<p style="text-align: left;">Hence, for square numbers, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>4</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>4</mn><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup></math>.</p>
</div>
<div class="specification">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th pentagonal number can be represented by the arithmetic series</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo>+</mo><mo>…</mo><mo>+</mo><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For triangular numbers, verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The number <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>351</mn></math> is a triangular number. Determine which one it is.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.</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>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>, sketch a diagram clearly showing your answer to part (b)(ii).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> is the square of an odd number for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable table of values or otherwise, determine the smallest positive integer, greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, that is both a triangular number and a pentagonal number.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A polygonal number, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math>, can be represented by the series</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>.</p>
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> <strong>and corresponding cubic equations with one real root and two complex roots of the form </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>z</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p> </p>
</div>
<div class="specification">
<p>In parts (a), (b) and (c), let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>z</mi><mo>+</mo><mn>17</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>On the Cartesian plane, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mo>-</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> represent the real and imaginary parts of the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<p><br>The following diagram shows a particular curve of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> and the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mn>80</mn></mrow></mfenced></math>. The curve and the tangent both intersect the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math> are also shown.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mi>r</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mo>(</mo><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> are as defined in part (d)(ii). The curve has a point of inflexion at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>Consider the special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> are roots of the equation, write down the third root.</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>Verify that the mean of the two complex roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and the tangent to the curve at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, clearly showing where the tangent crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math>.</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>Hence, or otherwise, prove that the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow></mfenced></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce from part (d)(i) that the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this diagram to determine the roots of the corresponding equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math>.</p>
<p>You are <strong>not</strong> required to demonstrate a change in concavity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence describe numerically the horizontal position of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> relative to the horizontal positions of the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>, state in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the coordinates of points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question investigates some applications of differential equations to modeling population growth.</p>
<p>One model for population growth is to assume that the rate of change of the population is proportional to the population, i.e. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = kP">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>P</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mi>P</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is the time (in years) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> is the population</p>
</div>
<div class="specification">
<p>The initial population is 1000.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
</math></span>, use your answer from part (a) to find</p>
</div>
<div class="specification">
<p>Consider now the situation when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but a function of time.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003 + 0.002t">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
<mo>+</mo>
<mn>0.002</mn>
<mi>t</mi>
</math></span>, find</p>
</div>
<div class="specification">
<p>Another model for population growth assumes</p>
<ul>
<li>there is a maximum value for the population, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</li>
<li>that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but is proportional to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - \frac{P}{L}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mfrac>
<mi>P</mi>
<mi>L</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</li>
</ul>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the general solution of this differential equation is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = A{{\text{e}}^{kt}}"> <mi>P</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A \in \mathbb{R}"> <mi>A</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the population after 10 years</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>the number of years it will take for the population to triple.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{t \to \infty } P"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mi>P</mi> </math></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the solution of the differential equation, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = f\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{R}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = g\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the initial population is 1000, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = 10000"> <mi>L</mi> <mo>=</mo> <mn>10000</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 0.003"> <mi>m</mi> <mo>=</mo> <mn>0.003</mn> </math></span>, find the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>In parts (b) and (c), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {abc \ldots } \right)_n}">
<mrow>
<msub>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mi>b</mi>
<mi>c</mi>
<mo>…<!-- … --></mo>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msub>
</mrow>
</math></span> denotes the number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{abc \ldots }">
<mrow>
<mi>a</mi>
<mi>b</mi>
<mi>c</mi>
<mo>…<!-- … --></mo>
</mrow>
</math></span> written in base <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>. For example, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {359} \right)_n} = 3{n^2} + 5n + 9">
<mrow>
<msub>
<mrow>
<mo>(</mo>
<mrow>
<mn>359</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mi>n</mi>
<mo>+</mo>
<mn>9</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State Fermat’s little theorem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the remainder when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{15^{1207}}"> <mrow> <msup> <mn>15</mn> <mrow> <mn>1207</mn> </mrow> </msup> </mrow> </math></span> is divided by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="13"> <mn>13</mn> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Convert <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {7A2} \right)_{16}}"> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>7</mn> <mi>A</mi> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>16</mn> </mrow> </msub> </mrow> </math></span> to base <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5"> <mn>5</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( A \right)_{16}} = {\left( {10} \right)_{10}}"> <mrow> <msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> <mrow> <mn>16</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>10</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>10</mn> </mrow> </msub> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}"> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>1251</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>30</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>504</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mrow> <mo>(</mo> <mrow> <mn>504</mn> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the remainder when <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>2022</mn></msup></math> is divided by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math>.</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>Use Fermat’s little theorem to find the remainder when <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>14</mn><mn>2022</mn></msup></math> is divided by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>17</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that a number in base <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> if, and only if, the sum of its digits is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The base <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> number <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mi>y</mi><mn>93</mn><mi>y</mi><mn>25</mn></math> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>. Find the possible values of the digit <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br>