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<h2>HL Paper 1</h2><div class="specification">
<p>An electric circuit has two power sources. The voltage, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>1</mn></msub></math>, provided by the first power source, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, is modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><mtext>Re</mtext><mo>(</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mo>)</mo></math>.</p>
<p>The voltage, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub></math>, provided by the second power source is modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mtext>Re</mtext><mo>(</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn><mo>)</mo><mtext>i</mtext></mrow></msup><mo>)</mo></math>.</p>
<p>The total voltage in the circuit, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub></math>, is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><msub><mi>V</mi><mn>1</mn></msub><mo>+</mo><msub><mi>V</mi><mn>2</mn></msub></math>.</p>
</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"><msub><mi>V</mi><mi>T</mi></msub></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo><mi>cos</mi><mo>(</mo><mi>B</mi><mi>t</mi><mo>+</mo><mi>C</mi><mo>)</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> are real constants.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the maximum voltage in the circuit.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following Argand diagram shows a circle centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> with a radius of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> units.</p>
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"></p>
<p>A set of points, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close="}"><msub><mi>z</mi><mi>θ</mi></msub></mfenced></math>, on the Argand plane are defined by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mi>θ</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>θ</mi><msup><mtext>e</mtext><mrow><mi>θ</mi><mtext>i</mtext></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Plot on the Argand diagram the points corresponding to</p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi>z</mi><mi>θ</mi></msub></mfenced><mo>=</mo><mn>4</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mi mathvariant="normal">π</mi></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>, plot the approximate position of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mi>θ</mi></msub></math> on the Argand diagram.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms of a sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ {u_n}\} ">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_n} = 3{n^2} - 2n">
<mrow>
<msub>
<mi>S</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<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>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</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>Prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ {u_n}\} ">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</math></span> is an arithmetic sequence, stating clearly its common difference.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>C</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2}&4 \\ 1&7 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>D</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5&2 \\ { - 1}&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p class="question" style="margin-top: 12.0pt; tab-stops: 1.0cm 36.0pt 72.0pt 108.0pt 144.0pt 180.0pt 216.0pt 252.0pt 288.0pt 324.0pt 360.0pt 396.0pt 432.0pt 468.0pt;">The 2 × 2 matrix <strong><em>Q</em></strong> is such that 3<strong><em>Q</em></strong> = 2<strong><em>C</em></strong> – <strong><em>D</em></strong></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>Q</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>CD</em></strong>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>D</em></strong><sup>–1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = a{{\text{e}}^{\frac{\pi }{4}{\text{i}}}}">
<mi>w</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</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>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 2,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^2}"> <mrow> <msup> <mi>w</mi> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3}"> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> </math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^4}"> <mrow> <msup> <mi>w</mi> <mn>4</mn> </msup> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>draw <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^2}"> <mrow> <msup> <mi>w</mi> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3}"> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> </math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^4}"> <mrow> <msup> <mi>w</mi> <mn>4</mn> </msup> </mrow> </math></span> on the following Argand diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{w}{{2 - {\text{i}}}}"> <mi>z</mi> <mo>=</mo> <mfrac> <mi>w</mi> <mrow> <mn>2</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> </mfrac> </math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> for which successive powers of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z"> <mi>z</mi> </math></span> lie on a circle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The geometric sequence <em>u</em><sub>1</sub>, <em>u</em><sub>2</sub>, <em>u</em><sub>3</sub>, … has common ratio <em>r.</em></p>
<p>Consider the sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left\{ {{a_n} = {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|{\text{:}}\,n \in {\mathbb{Z}^ + }} \right\}">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>A</em> is an arithmetic sequence, stating its common difference<em> d</em> in terms of <em>r</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A particular geometric sequence has <em>u</em><sub>1</sub> = 3 and a sum to infinity of 4.</p>
<p>Find the value of <em>d</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mtext>i</mtext><mi>z</mi><mo>+</mo><mn>1</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="specification">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> when</p>
</div>
<div class="specification">
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> on the Argand diagram can be transformed to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> by two transformations.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>2</mn><mtext>i</mtext></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn><mo>+</mo><mtext>i</mtext></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>Describe these two transformations and give the order in which they are applied.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mn>2</mn><mo>−</mo><mtext>i</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} b&3 \\ 7&8 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 9&5 \\ { - 2}&7 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4&8 \\ a&{15} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>b</mi>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mn>15</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {\begin{array}{*{20}{c}} { - 4}&8 \\ 2&1 \end{array}} \right) - 5\left( {\begin{array}{*{20}{c}} 2&0 \\ q&{ - 4} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 22}&{24} \\ 9&{23} \end{array}} \right)">
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>q</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>22</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mrow>
<mn>23</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p class="indent2">Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows a sector, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OAB</mtext></math>, of a circle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext><mo>=</mo><mi>θ</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Sam measured the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mtext>cm</mtext></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math>.</p>
</div>
<div class="specification">
<p>It is found that Sam’s measurements are accurate to only one significant figure.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Sam’s measurements to calculate the area of the sector. Give your answer to four significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the upper bound and lower bound of the area of the sector.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, with justification, the largest possible percentage error if the answer to part (a) is recorded as the area of the sector.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>In an arithmetic sequence, the sum of the 3rd and 8th terms is 1.</p>
<p>Given that the sum of the first seven terms is 35, determine the first term and the common difference.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong>, <strong><em>B</em> </strong>and <strong><em>C</em> </strong>be non-singular 2×2 matrices, <strong><em>I</em> </strong>the 2×2 identity matrix and <em>k</em> a scalar. The following statements are <strong>incorrect</strong>. For each statement, write down the correct version of the right hand side.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">(<strong><em>A</em> </strong>+ <strong><em>B</em></strong>)<sup>2</sup> = <strong><em>A</em></strong><sup>2</sup> + 2<strong><em>AB</em> </strong>+ <strong><em>B</em></strong><sup>2</sup></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"> (<strong><em>A</em> </strong>– <em>k<strong>I</strong></em>)<sup>3</sup><strong> </strong>= <strong><em>A</em></strong><sup>3</sup> – 3<em>k<strong>A</strong></em><sup>2</sup><strong> </strong>+ 3<em>k</em><sup>2</sup><strong><em>A</em> </strong>– <em>k</em><sup>3 </sup></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 class="indent1" style="margin-top:12.0pt;"><strong><em>CA</em></strong> = <strong><em>B</em></strong> <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{B}{A}">
<mfrac>
<mi>B</mi>
<mi>A</mi>
</mfrac>
</math></span></p>
<div class="marks">[2]</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>5</mn><mtext>i</mtext></math> in exponential form.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="text-align:center;"><img 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"></p>
<p style="text-align:left;">An equilateral triangle is to be drawn on the Argand plane with one of the vertices at the point corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>5</mn><mtext>i</mtext></math> and all the vertices equidistant from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>.</p>
<p style="text-align:left;">Find the points that correspond to the other two vertices. Give your answers in Cartesian form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An infinite geometric sequence, with terms <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub></math>, is such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msub><mi>u</mi><mi>k</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, for the sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The 1st, 4th and 8th terms of an arithmetic sequence, with common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>, are the first three terms of a geometric sequence, with common ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>. Given that the 1st term of both sequences is 9 find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 2\left( {{\text{cos}}\frac{\pi }{3} + {\text{i}}\,{\text{sin}}\frac{\pi }{3}} \right)">
<mi>w</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>These four points form the vertices of a quadrilateral, <em>Q</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup> in modulus-argument form.</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>Sketch on an Argand diagram the points represented by <em>w</em><sup>0</sup> , <em>w</em><sup>1</sup> , <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup>.</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 the area of the quadrilateral <em>Q</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{21\sqrt 3 }}{2}"> <mfrac> <mrow> <mn>21</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2\left( {{\text{cos}}\frac{\pi }{n} + {\text{i}}\,{\text{sin}}\frac{\pi }{n}} \right),\,\,n \in {\mathbb{Z}^ + }"> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>. The points represented on an Argand diagram by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}"> <mrow> <msup> <mi>z</mi> <mn>0</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>1</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></span> form the vertices of a polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<p>Show that the area of the polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span> can be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {{b^n} - 1} \right){\text{sin}}\frac{\pi }{n}"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>b</mi> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,\,\,b\, \in \mathbb{R}"> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrices</p>
<p style="text-align: center;"><em><strong>A</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 5&{ - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>B</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&3 \\ 2&{ - 2} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find <strong><em>BA</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Calculate det (<strong><em>BA</em></strong>)<em>.</em></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 class="indent1" style="margin-top:12.0pt;"> Find <strong><em>A</em></strong>(<strong><em>A</em></strong><sup>–1</sup><strong><em>B</em></strong> + 2<strong><em>A</em></strong><sup>–1</sup>)<strong><em>A</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The strength of earthquakes is measured on the Richter magnitude scale, with values typically between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> is the most severe.</p>
<p>The Gutenberg–Richter equation gives the average number of earthquakes per year, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>, which have a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. For a particular region the equation is</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>N</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>M</mi></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>This region has an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> earthquakes per year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The equation for this region can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><mi>b</mi><msup><mn>10</mn><mi>M</mi></msup></mfrac></math>.</p>
</div>
<div class="specification">
<p>Within this region the most severe earthquake recorded had a magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The number of earthquakes in a given year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> can be modelled by a Poisson distribution, with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>. The number of earthquakes in one year is independent of the number of earthquakes in any other year.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> be the number of years between the earthquake of magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> and the next earthquake of at least this magnitude.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the average number of earthquakes in a year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>Y</mi><mo>></mo><mn>100</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em><strong>A</strong></em><sup>2</sup> = 2<em><strong>A</strong></em> + <em><strong>I</strong></em> where <em><strong>A</strong></em> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em><strong>A</strong></em><sup>4</sup> = 12<em><strong>A</strong></em> + 5<em><strong>I</strong></em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <em><strong>B</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
<p>Given that <em><strong>B</strong></em><sup>2</sup> – <em><strong>B</strong></em> – 4<em><strong>I</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} k&0 \\ 0&k \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The matrix <strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&0 \\ { - 3}&1&{ - 1} \\ 2&{ - 2}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> has inverse <strong><em>A</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1}&{ - 2}&{ - 2} \\ 3&1&1 \\ a&6&b \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the simultaneous equations</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x + 2y = 7">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>=</mo>
<mn>7</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext=" - 3x + y - z = 10">
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="2x - 2y + z = - 12">
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>12</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write these equations as a matrix equation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Solve the matrix equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong><em> = </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&{ - 1}&2 \\ 3&{ - 3}&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em>, <strong>D</strong> =</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 4}&{13}&{ - 7} \\ { - 2}&7&{ - 4} \\ 3&{ - 9}&5 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>9</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em>, </em>and<em> <strong>C</strong> =</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5 \\ 7 \\ {10} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em>. </em></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">Given matrices <strong><em>A</em></strong><em>, <strong>B</strong>, <strong>C </strong></em>for which <strong><em>AB</em></strong><em> = <strong>C</strong> </em>and det <strong><em>A</em></strong> ≠ 0, express <strong><em>B</em></strong> in terms of <strong><em>A</em></strong> and <strong><em>C</em></strong><em>.</em></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">Find the matrix<em> <strong>DA</strong>.</em></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">Find <strong><em>B </em></strong>if <strong><em>AB</em></strong> = <strong><em>C</em></strong>.</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 class="indent1" style="margin-top:12pt;text-align: left;">Find the coordinates of the point of intersection of the planes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y + 3z = 5">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y + 2z = 7">
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>7</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 3y + 2z = 10">
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">The square matrix <strong><em>X</em></strong> is such that <strong><em>X</em></strong><sup>3</sup> = 0. Show that the inverse of the matrix (<strong><em>I</em></strong> <em>– <strong>X</strong></em>) is <strong><em>I</em></strong> <em>+</em> <strong><em>X</em></strong> <em>+</em> <strong><em>X</em></strong><sup>2</sup><em>.</em></p>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ 3&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong><strong><em> </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&0 \\ { - 2}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>A</em></strong> + <strong><em>B</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find −3<strong><em>A</em></strong>. </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>AB</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a relationship between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> if the matrices <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M = \left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)">
<mi>M</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N = \left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)">
<mi>N</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> commute under matrix multiplication.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> if the determinant of matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M">
<mi>M</mi>
</math></span> is −1.</p>
<p> </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>Write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^{ - 1}}">
<mrow>
<msup>
<mi>M</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> for this value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>If <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x&4 \\ 4&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&y \\ 8&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find 2 values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>, given that <strong><em>AB</em></strong> = <strong><em>BA</em></strong>.</p>
</div>
<br><hr><br><div class="specification">
<p>The equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> can be expressed in vector form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi>λ</mi><mi mathvariant="bold-italic">b</mi></math>.</p>
</div>
<div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn><mo> </mo><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>≠</mo><mo>−</mo><mn>2</mn></math>) is transformed into a new line using the transformation described by matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det </mtext><mi mathvariant="bold-italic">M</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the equation of the resulting line does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are 2 × 2 matrices, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left[ {\begin{array}{*{20}{c}} 5&2 \\ 2&0 \end{array}} \right]">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="BA = \left[ {\begin{array}{*{20}{c}} {11}&2 \\ {44}&8 \end{array}} \right]">
<mi>B</mi>
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>44</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span></p>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 3&1&2 \\ 2&0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {18} \\ {23} \\ {13} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>23</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, and <strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the equation <strong><em>AX</em></strong> = <strong><em>B</em></strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the inverse matrix <strong><em>A</em></strong><sup>−1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Express <strong><em>X</em></strong> in terms of <strong><em>A</em></strong><sup>−1</sup> and <strong><em>B</em></strong>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2"><strong>Hence</strong>, solve for <strong><em>X</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>The sum of an infinite geometric sequence is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math>.</p>
<p>The first term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> more than the second term.</p>
<p>Find the third term. Justify your answer.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of equations 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>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + 4y - z = 10">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y + az = 5">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5x + 12y = 2a">
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> for which the system of equations does not have a unique solution.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the solution of the system of equations when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2">
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An arithmetic sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{\text{, }}{u_2}{\text{, }}{u_3} \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>…<!-- … --></mo>
</math></span> has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> and common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2}{\text{, }}{u_3}">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</math></span> are the first three terms of a geometric sequence</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_N} = - 15">
<mrow>
<msub>
<mi>u</mi>
<mi>N</mi>
</msub>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>15</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^N {{u_r}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>r</mi>
</msub>
</mrow>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">Find the determinant of <strong><em>A</em></strong>, where <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1&2 \\ 9&5&8 \\ 7&4&6 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r,\,x \in {\mathbb{R}^ + }">
<mi>r</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p{x^q}">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>q</mi>
</msup>
</mrow>
</math></span>, where <em>p</em> , <em>q</em> are constants.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region <em>R</em>, is bounded by the graph of the function found in part (b), the <em>x</em>-axis, and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \alpha ">
<mi>x</mi>
<mo>=</mo>
<mi>α</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha > 1">
<mi>α</mi>
<mo>></mo>
<mn>1</mn>
</math></span>. The area of <em>R</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 2 ">
<msqrt>
<mn>2</mn>
</msqrt>
</math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the system of equations <strong><em>A</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2 \\ { - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <strong><em>A</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {k + 1}&{ - k} \\ 2&{k - 1} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <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>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find det <strong><em>A</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> for which the system has a unique solution.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The matrix <em><strong>A</strong></em> is given by <em><strong>A </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The matrix <em><strong>B</strong></em> is given by <em><strong>B</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 2&3 \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the eigenvalues of <em><strong>A</strong></em> are real if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a - d} \right)^2} + 4bc \geqslant 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>c</mi>
<mo>⩾</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce that the eigenvalues are real if <em><strong>A</strong></em> is symmetric.</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>Determine the eigenvalues of <em><strong>B</strong></em>.</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>Determine the corresponding eigenvectors.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Matrices <strong><em>A</em></strong>, <strong><em>B</em> </strong>and <strong><em>C</em> </strong>are defined as</p>
<p style="text-align: center;"><strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&5&1 \\ 3&{ - 1}&3 \\ { - 9}&3&7 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>9</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&{ - 1} \\ 3&{ - 1}&0 \\ 0&3&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 8 \\ 0 \\ { - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Given that <strong><em>AB</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&0&0 \\ 0&a&0 \\ 0&0&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Hence, or otherwise, find <strong><em>A</em></strong><sup>–1</sup>.</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 class="indent1" style="margin-top:12.0pt;">Find the matrix <strong><em>X</em></strong>, such that <strong><em>AX</em> </strong>= <strong><em>C</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ { - b}&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> are non-zero real numbers.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Show that <strong><em>M</em> </strong>is non-singular.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"> Calculate <strong><em>M</em></strong><sup>2</sup>.</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 class="indent1" style="margin-top:12.0pt;"> Show that det(<strong><em>M</em></strong><sup>2</sup>) is positive.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12pt;text-align: left;">Given the matrix <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&2 \\ { - 1}&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> find the values of the real number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{det}}\left( {A - kI} \right) = 0">
<mrow>
<mtext>det</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>−</mo>
<mi>k</mi>
<mi>I</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)">
<mi>I</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p class="indent1" style="margin-top:12pt;text-align: left;"> </p>
</div>
<br><hr><br><div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>A switch has two states, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. Each second it either remains in the same state or moves according to the following rule: If it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math> and if it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>7</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an eigenvector corresponding to the eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>, 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>.</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>Using your answer to (a), or otherwise, find the long-term probability of the switch being in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>c</mi><mi>d</mi></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i, </mtext>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \frac{{{z_1}}}{{{z_2}}}">
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the modulus of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>;</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>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest positive integer value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^n}">
<mrow>
<msup>
<mi>w</mi>
<mi>n</mi>
</msup>
</mrow>
</math></span> is a real number.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrix <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 5&{ - 2} \\ 7&1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p><strong><em>B</em></strong>, <strong><em>C</em></strong> and <strong><em>X</em></strong> are also 2 × 2 matrices.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the inverse, <strong><em>A</em></strong><sup>–1</sup>.</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>Given that <strong><em>XA</em></strong> + <strong><em>B</em></strong> = <strong><em>C</em></strong>, express <strong><em>X</em></strong> in terms of <strong><em>A</em></strong><sup>–1</sup>, <strong><em>B</em></strong> and <strong><em>C</em></strong><em>.</em></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>Given that <em><strong>B</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 6&7 \\ 5&{ - 2} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, and <em><strong>C</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 5}&0 \\ { - 8}&7 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find <em><strong>X</strong></em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> given that the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left( {\begin{array}{*{20}{c}} a&{ - 4}&{ - 6} \\ { - 8}&5&7 \\ { - 5}&3&4 \end{array}} \right)">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is the inverse of the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = \left( {\begin{array}{*{20}{c}} 1&2&{ - 2} \\ 3&b&1 \\ { - 1}&1&{ - 3} \end{array}} \right)">
<mi>B</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">For the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> found in part (a), solve the system of linear equations</p>
<p class="indent1" style="margin-top:12pt;text-align: left;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{c}} {x + 2y - 2z = 5} \\ {3x + by + z = 0} \\ { - x + y - 3z = a - 1.} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>−</mo>
<mn>1.</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <em>f</em> : <em><strong>M</strong></em> → <em><strong>M</strong></em> where <em><strong>M</strong></em> is the set of 2 × 2 matrices, is given by <em>f</em>(<em><strong>X</strong></em>) = <em><strong>AX</strong></em> where <em><strong>A</strong></em> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em><strong>A</strong></em> is non-singular, prove that<em> f</em> is a bijection.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is now given that <em><strong>A</strong></em> is singular.</p>
<p>By considering appropriate determinants, prove that <em>f</em> is not a bijection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi ">
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>θ<!-- θ --></mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ ">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msup>
<mn>0</mn>
<mo>∘</mo>
</msup>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<msup>
<mn>180</mn>
<mo>∘</mo>
</msup>
</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus and argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>. Express each answer in its simplest form.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cube roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in modulus-argument form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Find the solution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}x - {\log _2}5 = 2 + {\log _2}3">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>5</mn>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Write down the inverse of the matrix</p>
<p class="indent1" style="margin-top:12pt;text-align: center;"><strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{ - 3}&1 \\ 2&2&{ - 1} \\ 1&{ - 5}&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong>Hence</strong>, find the point of intersection of the three planes.</p>
<p class="indent1" style="margin-top:12pt;text-align: center;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{c}} {x - 3y + z = 1} \\ {2x + 2y - z = 2} \\ {x - 5y + 3z = 3} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">A fourth plane with equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = d">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mi>d</mi>
</math></span> passes through the point of intersection. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span><em>.</em></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&2 \\ k&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&2 \\ 1&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">2<strong><em>A </em></strong>− <strong><em>B</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">det (2<strong><em>A </em></strong>− <strong><em>B</em></strong>).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The square matrix <strong><em>X</em></strong> is such that <strong><em>X</em></strong><sup>3</sup> = 0. Show that the inverse of the matrix (<strong><em>I</em> </strong>–<strong> <em>X</em></strong>) is <strong><em>I</em></strong> + <strong><em>X</em></strong> + <strong><em>X</em></strong><sup>2</sup>.</p>
</div>
<br><hr><br><div class="specification">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mtext> cis</mtext><mfenced><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mtext> cis</mtext><mfenced><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi></mrow><mn>16</mn></mfrac></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>In parts (a)(i) and (a)(ii), give your answers in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>−</mo><mi>π</mi><mo><</mo><mi>θ</mi><mo>≤</mo><mi>π</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>3</mn></msup></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mfrac></mfenced><mn>4</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that<strong><em> A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ { - 3}&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> for which (<strong><em>A</em> </strong>– <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><em><strong>I</strong></em>) is a singular matrix.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse of the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&1 \\ 1&1&2 \\ 2&1&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Hence</strong> solve the system of equations</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x + 2y + z = 0">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x + y + 2z = 7">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>7</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="2x + y + z = 17">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>17</mn>
</math></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&3 \\ 1&{ - 2} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find <strong><em>X</em></strong> if <strong><em>BX</em></strong> = <strong><em>A</em></strong> <em>–</em> <strong><em>AB</em></strong>.</p>
<p> </p>
</div>
<br><hr><br><div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the determinant of a relevant matrix, show that the eigenvalues, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math>, of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> satisfy the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mi>α</mi><mi>λ</mi><mo>+</mo><mi>β</mi><mo>=</mo><mn>0</mn></math>,</p>
<p style="text-align: left;">where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> are functions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi></math> to be determined.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mn>2</mn></msup><mo>-</mo><mi>α</mi><mi mathvariant="bold-italic">A</mi><mo>+</mo><mi>β</mi><mi mathvariant="bold-italic">I</mi><mo>=</mo></math> <em><strong>0</strong></em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is non-singular, use the result in part (b)(i) to show that</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>β</mi></mfrac><mfenced><mrow><mi>α</mi><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Roger buys a new laptop for himself at a cost of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>£</mo><mn>495</mn></math>. At the same time, he buys his daughter Chloe a higher specification laptop at a cost of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>£</mo><mn>2200</mn></math>.</p>
<p>It is anticipated that Roger’s laptop will depreciate at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> per year, whereas Chloe’s laptop will depreciate at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo>%</mo></math> per year.</p>
</div>
<div class="specification">
<p>Roger and Chloe’s laptops will have the same value <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> years after they were purchased.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Estimate the value of Roger’s laptop after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment on the validity of your answer to part (b).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&x&{ - 1} \\ 3&1&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ x \\ 2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Find <strong><em>AB</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">The matrix <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {20} \\ {28} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>28</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and 2<strong><em>AB</em></strong> = <strong><em>C</em></strong>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A meteorologist models the height of a hot air balloon launched from the ground. The model assumes the balloon travels vertically upwards and travels <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>450</mn></math> metres in the first minute.</p>
<p>Due to the decrease in temperature as the balloon rises, the balloon will continually slow down. The model suggests that each minute the balloon will travel only <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>82</mn><mo>%</mo></math> of the distance travelled in the previous minute.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how high the balloon will travel in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> minutes after it is launched.</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>The balloon is required to reach a height of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2520</mn></math> metres.<br><br>Determine whether it will reach this height.</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>Suggest a limitation of the given model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrix <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&2 \\ a&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find the matrix <strong><em>A</em></strong><sup>2</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">If det <strong><em>A</em></strong><sup>2</sup> = 16, determine the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span><em>.</em></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">Consider the matrix <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {{{\text{e}}^x}}&{{{\text{e}}^{ - x}}} \\ {2 + {{\text{e}}^x}}&1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p class="question" style="margin-top:12.0pt;">Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which <strong><em>A</em> </strong>is singular.</p>
</div>
<br><hr><br><div class="question">
<p>The matrices <strong><em>A</em></strong>, <strong><em>B</em></strong>, <strong><em>C</em></strong> and <strong><em>X</em></strong> are all non-singular 3 × 3 matrices.</p>
<p>Given that <strong><em>A</em></strong><sup><em>–</em>1</sup><strong><em>XB</em></strong> = <strong><em>C</em></strong>, express <strong><em>X</em></strong> in terms of the other matrices.</p>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">If <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ k&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>A</em></strong><sup>2</sup> is a matrix whose entries are all 0, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">Given that <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 3}&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and that <strong><em>M</em></strong><sup>2</sup> <em>–</em> 6<strong><em>M</em></strong> <em>+</em> <em>k<strong>I</strong></em> = 0 find <em>k</em>.</p>
</div>
<br><hr><br><div class="question">
<p>In the following Argand diagram the point A represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 + 4{\text{i}}">
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mn>4</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and the point B represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 0{\text{i}}">
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mn>0</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_06.11.20.png" alt="M17/5/MATHL/HP1/ENG/TZ2/05"></p>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12pt;text-align: left;">Find the values of the real number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> for which the determinant of the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {k - 4}&3 \\ { - 2}&{k + 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is equal to zero.</p>
</div>
<br><hr><br><div class="question">
<p>If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left( {\begin{array}{*{20}{c}} {2p}&3 \\ { - 4p}&p \end{array}} \right)">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>p</mi>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
<mi>p</mi>
</mrow>
</mtd>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 14">
<mi>A</mi>
<mo>=</mo>
<mn>14</mn>
</math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^x} + {2^{x + 2}} = 3">
<mrow>
<msup>
<mn>4</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}(x + 3) + {\log _2}(x - 3) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>The rate, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, of a chemical reaction at a fixed temperature is related to the concentration of two compounds, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>, by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = k{B^x}{C^y}">
<mi>A</mi>
<mo>=</mo>
<mi>k</mi>
<mrow>
<msup>
<mi>B</mi>
<mi>x</mi>
</msup>
</mrow>
<mrow>
<msup>
<mi>C</mi>
<mi>y</mi>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>A scientist measures the three variables three times during the reaction and obtains the following values.</p>
<p><img style="display: block;margin-left:auto;margin-right:auto;" 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"></p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</div>
<br><hr><br>