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<h2>HL Paper 1</h2><div class="question">
<p>Three planes have equations:</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y + z = 5"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>5</mn> </math></span></p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 3y - z = 4"> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>4</mn> </math></span> , where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{, }}b \in \mathbb{R}"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 5y + az = b"> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mi>a</mi> <mi>z</mi> <mo>=</mo> <mi>b</mi> </math></span></p>
<p>Find the set of 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> such that the three planes have no points of intersection.</p>
</div>
<br><hr><br><div class="specification">
<p>Two distinct lines, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, intersect at a point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>. In addition to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>, four distinct points are marked out on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and three distinct points on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>. A mathematician decides to join some of these eight points to form polygons.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> has vector equation <em><strong>r</strong></em><sub>1</sub> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 1 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ<!-- λ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda \in \mathbb{R}">
<mi>λ<!-- λ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> has vector equation <em><strong>r</strong></em><sub>2</sub> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 2 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 5 \\ 6 \\ 2 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>μ<!-- μ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu \in \mathbb{R}">
<mi>μ<!-- μ --></mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> has coordinates (4, 6, 4).</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> has coordinates (3, 4, 3) and lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> has coordinates (−1, 0, 2) and lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how many sets of four points can be selected which can form the vertices of a quadrilateral.</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 how many sets of three points can be selected which can form the vertices of a triangle.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> is the point of intersection of the two lines.</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>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> corresponding to the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span>.</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>Write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PA}}} ">
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PB}}} ">
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span> be the point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> with coordinates (1, 0, 1) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> be the point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> with parameter <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = - 2">
<mi>μ</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span>.</p>
<p>Find the area of the quadrilateral <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{CDBA}}">
<mrow>
<mtext>CDBA</mtext>
</mrow>
</math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>S</em> be the sum of the roots found in part (a).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^{24}} = 1"> <mrow> <msup> <mi>z</mi> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> </math></span> which satisfy the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{arg}}\left( z \right) < \frac{\pi }{2}"> <mn>0</mn> <mo><</mo> <mrow> <mtext>arg</mtext> </mrow> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, expressing your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{e^{{\text{i}}\theta }}"> <mi>r</mi> <mrow> <msup> <mi>e</mi> <mrow> <mrow> <mtext>i</mtext> </mrow> <mi>θ</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta \in {\mathbb{R}^ + }"> <mi>θ</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Re <em>S</em> = Im <em>S</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>, find the value of cos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{{12}}"> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sqrt a + \sqrt b }}{c}"> <mfrac> <mrow> <msqrt> <mi>a</mi> </msqrt> <mo>+</mo> <msqrt> <mi>b</mi> </msqrt> </mrow> <mi>c</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <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> are integers to be determined.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, show that <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left( {1 + \sqrt 2 } \right)\left( {1 + \sqrt 3 } \right)\left( {1 + {\text{i}}} \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.iii.</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="specification">
<p>Consider the three planes</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle><mo>:</mo><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>4</mn></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle><mo>:</mo><mo> </mo><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></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle><mo>:</mo><mo>-</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>32</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the three planes do not intersect.</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 the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> lies on both <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math>.</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>Find a vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>, the line of intersection of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder></mstyle></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>In the following diagram, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} ">
<mover>
<mrow>
<mtext>OA</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span> = <strong><em>a</em></strong>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OB}}} ">
<mover>
<mrow>
<mtext>OB</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span> = <strong><em>b</em></strong>. C is the midpoint of [OA] and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OF}}} = \frac{1}{6}\overrightarrow {{\text{FB}}} ">
<mover>
<mrow>
<mtext>OF</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mover>
<mrow>
<mtext>FB</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.26.10.png" alt="N17/5/MATHL/HP1/ENG/TZ0/09"></p>
</div>
<div class="specification">
<p>It is given also that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AD}}} = \lambda \overrightarrow {{\text{AF}}} ">
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mover>
<mrow>
<mtext>AF</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CD}}} = \mu \overrightarrow {{\text{CB}}} ">
<mover>
<mrow>
<mtext>CD</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mi>μ<!-- μ --></mi>
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ,{\text{ }}\mu \in \mathbb{R}">
<mi>λ<!-- λ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>μ<!-- μ --></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, in terms of <strong><em>a </em></strong>and <strong><em>b </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OF}}} ">
<mover>
<mrow>
<mtext>OF</mtext>
</mrow>
<mo>→</mo>
</mover>
</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>Find, in terms of <strong><em>a </em></strong>and <strong><em>b </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AF}}} ">
<mover>
<mrow>
<mtext>AF</mtext>
</mrow>
<mo>→</mo>
</mover>
</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>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OD}}} ">
<mover>
<mrow>
<mtext>OD</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> in terms of <strong><em>a</em></strong>, <strong><em>b </em></strong>and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OD}}} ">
<mover>
<mrow>
<mtext>OD</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> in terms of <strong><em>a</em></strong>, <strong><em>b </em></strong>and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = \frac{1}{{13}}">
<mi>μ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
</math></span>, and find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CD}}} ">
<mover>
<mrow>
<mtext>CD</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> in terms of <strong><em>a </em></strong>and <strong><em>b </em></strong>only.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})">
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>OAB</mtext>
</mrow>
<mo>=</mo>
<mi>k</mi>
<mo stretchy="false">(</mo>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>CAD</mtext>
</mrow>
<mo stretchy="false">)</mo>
</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">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><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="{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = 1 + {\text{sin}}\,2x"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sec}}\,2x + {\text{tan}}\,2x = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}"> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{\pi }{6}} {\left( {{\text{sec}}\,2x + {\text{tan}}\,2x} \right)} {\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {a + \sqrt b } \right)"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <msqrt> <mi>b</mi> </msqrt> </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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{Z}"> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following figure shows a square based pyramid with vertices at O(0, 0, 0), A(1, 0, 0), B(1, 1, 0), C(0, 1, 0) and D(0, 0, 1).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The Cartesian equation of the plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _2}">
<mrow>
<msub>
<mi mathvariant="normal">Π<!-- Π --></mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, passing through the points B , C and D , is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + z = 1">
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _3}">
<mrow>
<msub>
<mi mathvariant="normal">Π<!-- Π --></mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> passes through O and is normal to the line BD.</p>
</div>
<div class="specification">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _3}">
<mrow>
<msub>
<mi mathvariant="normal">Π<!-- Π --></mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> cuts AD and BD at the points P and Q respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of the plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _1}">
<mrow>
<msub>
<mi mathvariant="normal">Π</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>, passing through the points A , B and D.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle between the faces ABD and BCD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _3}">
<mrow>
<msub>
<mi mathvariant="normal">Π</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that P is the midpoint of AD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the triangle OPQ.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" 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="\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}"> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi "> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>≡</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>k</mi> <mi>π</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </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>Use the principle of mathematical induction to prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi "> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>n</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>k</mi> <mi>π</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> in the interval <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \pi "> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mi>π</mi> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the binomial theorem to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">i</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> are expressed in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</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>Use de Moivre’s theorem and the result from part (a) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><msup><mi>cot</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cot</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cot</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math>.</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>Use the identity from part (b) to show that the quadratic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce a quadratic equation with integer coefficients, having roots <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>In the following diagram, the points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> are on the circumference of a circle with centre <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{O}}">
<mrow>
<mtext>O</mtext>
</mrow>
</math></span> and radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{AC}}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mrow>
<mtext>AC</mtext>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> is a diameter of the circle. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = r">
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>=</mo>
<mi>r</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AD}} = {\text{CD}}">
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>=</mo>
<mrow>
<mtext>CD</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\mathop {\text{B}}\limits^ \wedge {\text{C}} = {\text{A}}\mathop {\text{D}}\limits^ \wedge {\text{C}} = 90^\circ ">
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>B</mtext>
</mrow>
<mo>∧<!-- ∧ --></mo>
</mover>
<mo><!-- --></mo>
<mrow>
<mtext>C</mtext>
</mrow>
<mo>=</mo>
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>D</mtext>
</mrow>
<mo>∧<!-- ∧ --></mo>
</mover>
<mo><!-- --></mo>
<mrow>
<mtext>C</mtext>
</mrow>
<mo>=</mo>
<msup>
<mn>90</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
</math></span>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,75^\circ = q"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>75</mn> <mo>∘</mo> </msup> <mo>=</mo> <mi>q</mi> </math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,105^\circ = - q"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>105</mn> <mo>∘</mo> </msup> <mo>=</mo> <mo>−</mo> <mi>q</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 75^\circ "> <mrow> <mtext>B</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>D</mtext> </mrow> <mo>=</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </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>By considering triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABD}}"> <mrow> <mtext>ABD</mtext> </mrow> </math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}{{\text{D}}^2} = 5{r^2} - 2{r^2}q\sqrt 6 "> <mrow> <mtext>B</mtext> </mrow> <mrow> <msup> <mrow> <mtext>D</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>5</mn> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mi>q</mi> <msqrt> <mn>6</mn> </msqrt> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{CBD}}"> <mrow> <mtext>CBD</mtext> </mrow> </math></span>, find another expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}{{\text{D}}^2}"> <mrow> <mtext>B</mtext> </mrow> <mrow> <msup> <mrow> <mtext>D</mtext> </mrow> <mn>2</mn> </msup> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> and <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">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answers to part (c) to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,75^\circ = \frac{1}{{\sqrt 6 + \sqrt 2 }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>75</mn> <mo>∘</mo> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Given any two non-zero 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>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mn>2</mn></msup><msup><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi mathvariant="bold-italic">a</mi><mo>·</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced><mn>2</mn></msup></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, with asymptotes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe a sequence of transformations that transforms the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan </mtext><mi>x</mi></math> to the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>p</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi><mo>≡</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>p</mi><mi>q</mi></mrow></mfrac></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo><</mo><mn>1</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan </mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mtext>arctan </mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi mathvariant="normal">+</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using mathematical induction and the result from part (b), prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>n</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> defined by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}:">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>:</mo>
</math></span> <em><strong>r</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ a \end{array}} \right) + \beta \left( {\begin{array}{*{20}{c}} 1 \\ 4 \\ 2 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>β<!-- β --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}:\frac{{6 - x}}{3} = \frac{{y - 2}}{4} = 1 - z">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>:</mo>
<mfrac>
<mrow>
<mn>6</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>z</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is a constant.</p>
<p>Given that the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_1}">
<mrow>
<msub>
<mi>l</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l_2}">
<mrow>
<msub>
<mi>l</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> intersect at a point P,</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>;</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 coordinates of the point of intersection P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a triangle OAB such that O has coordinates (0, 0, 0), A has coordinates (0, 1, 2) and B has coordinates (2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, 0, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> − 1) where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> < 0.</p>
</div>
<div class="specification">
<p>Let M be the midpoint of the line segment [OB].</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, a Cartesian equation of the plane <em>Π</em> containing this triangle.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, the equation of the line <em>L</em> which passes through M and is perpendicular to the plane <em>П</em>.</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>Show that<em> L</em> does not intersect the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis for any negative value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The lengths of two of the sides in a triangle are 4 cm and 5 cm. Let <em>θ</em> be the angle between the two given sides. The triangle has an area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5\sqrt {15} }}{2}">
<mfrac>
<mrow>
<mn>5</mn>
<msqrt>
<mn>15</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> cm<sup>2</sup>.</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{sin}}\,\theta = \frac{{\sqrt {15} }}{4}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>15</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</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 two possible values for the length of the third side.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The points A and B are given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}(0,{\text{ }}3,{\text{ }} - 6)">
<mrow>
<mtext>A</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}(6,{\text{ }} - 5,{\text{ }}11)">
<mrow>
<mtext>B</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>11</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p>The plane <em>Π</em> is defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 3y + 2z = 20">
<mn>4</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>20</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation of the line <em>L </em>passing through the points A and B.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point of intersection of the line <em>L </em>with the plane <em>Π</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^x}\,{\text{cos}}{\,^2}x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<msup>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</math></span>, where 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5. The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is shown on the following graph which has local maximum points at A and C and touches the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at B and D.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use integration by parts to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x{\text{d}}x = } \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x + c"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <msup> <mspace width="thinmathspace"></mspace> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinates of A and of C , giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{arctan}}\,b"> <mi>a</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{R}"> <mi>b</mi> <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>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area enclosed by the curve and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis between B and D, as shaded on the diagram.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The points A, B, C and D have position vectors <em><strong>a</strong></em>, <em><strong>b</strong></em>, <em><strong>c</strong></em> and <em><strong>d</strong></em>, relative to the origin O.</p>
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to = \mathop {{\text{DC}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
<mo>=</mo>
<mover>
<mrow>
<mrow>
<mtext>DC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</math></span>.</p>
</div>
<div class="specification">
<p>The position vectors <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OA}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>OA</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OB}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OC}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>OC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OD}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>OD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</math></span> are given by</p>
<p style="padding-left: 150px;"><em><strong>a</strong></em> = <em><strong>i</strong></em> + 2<em><strong>j</strong></em> − 3<em><strong>k</strong></em></p>
<p style="padding-left: 150px;"><em><strong>b</strong></em> = 3<em><strong>i</strong></em> − <em><strong>j</strong></em> + <em>p<strong>k</strong></em></p>
<p style="padding-left: 150px;"><em><strong>c</strong></em> = <em>q<strong>i</strong></em> + <em><strong>j</strong></em> + 2<em><strong>k</strong></em></p>
<p style="padding-left: 150px;"><em><strong>d</strong></em> = −<em><strong>i</strong></em> + <em>r<strong>j</strong></em> − 2<em><strong>k</strong></em></p>
<p>where <em>p</em> , <em>q</em> and <em>r</em> are constants.</p>
</div>
<div class="specification">
<p>The point where the diagonals of ABCD intersect is denoted by M.</p>
</div>
<div class="specification">
<p>The plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Pi ">
<mi mathvariant="normal">Π<!-- Π --></mi>
</math></span> cuts the <em>x</em>, <em>y</em> and <em>z</em> axes at X , Y and Z respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why ABCD is a parallelogram.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using vector algebra, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AD}}}\limits^ \to = \mathop {{\text{BC}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>AD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mover>
<mrow>
<mrow>
<mtext>BC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>p</em> = 1, <em>q</em> = 1 and <em>r</em> = 4.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the parallelogram ABCD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector equation of the straight line passing through M and normal to the plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Pi ">
<mi mathvariant="normal">Π</mi>
</math></span> containing ABCD.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Pi ">
<mi mathvariant="normal">Π</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of X, Y and Z.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find YZ.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The inverse of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfrac><mi>x</mi><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mn>2</mn></mfrac><msqrt><mi>r</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>By using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi></math> or otherwise, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></munderover><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is a non-zero real number.</p>
</div>
<br><hr><br><div class="question">
<p>A straight line, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_\theta }"> <mrow> <msub> <mi>L</mi> <mi>θ</mi> </msub> </mrow> </math></span>, has vector equation <em><strong>r</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 0 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right){\text{, }}\lambda {\text{, }}\theta \in \mathbb{R}"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>λ</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mi>λ</mi> <mrow> <mtext>, </mtext> </mrow> <mi>θ</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<p>The plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _p}"><msub><mi>Π</mi><mi>p</mi></msub></math></span>, has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = p{\text{, }}p \in \mathbb{R}"> <mi>x</mi> <mo>=</mo> <mi>p</mi> <mrow> <mtext>, </mtext> </mrow> <mi>p</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<p>Show that the angle between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_\theta }"> <mrow> <msub> <mi>L</mi> <mi>θ</mi> </msub> </mrow> </math></span> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _p}"><msub><mi>Π</mi><mi>p</mi></msub></math> is independent of both <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> have the following vector equations where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>,</mo><mo> </mo><mi>μ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub><mo> </mo><mo>:</mo><mo> </mo><msub><mi mathvariant="bold-italic">r</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable></mfenced><mo> </mo><msub><mi>l</mi><mn>2</mn></msub><mo> </mo><mo>:</mo><mo> </mo><msub><mi mathvariant="bold-italic">r</mi><mn>2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>m</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>m</mi></mtd></mtr></mtable></mfenced></math></p>
</div>
<div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> has Cartesian equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>=</mo><mi>p</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p> </p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> have no points in common, find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> are never perpendicular to each other.</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 value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</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>the condition on the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> defined by the Cartesian equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mi>y</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>z</mi></math>.</p>
</div>
<div class="specification">
<p>Consider a second line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> defined by the vector equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></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>Find a vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></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 possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> when the acute angle between <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is given that the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math> have a unique point of intersection, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mi>k</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, and find the coordinates of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span>(0 , 0 , 10) , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span>(0 , 10 , 0) , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span>(10 , 0 , 0) , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{V}}">
<mrow>
<mtext>V</mtext>
</mrow>
</math></span>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>) form the vertices of a tetrahedron.</p>
</div>
<div class="specification">
<p>Consider the case where the faces <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABV}}">
<mrow>
<mtext>ABV</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ACV}}">
<mrow>
<mtext>ACV</mtext>
</mrow>
</math></span> are perpendicular.</p>
</div>
<div class="specification">
<p>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> against <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>. The maximum point is shown by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{X}}">
<mrow>
<mtext>X</mtext>
</mrow>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} = - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>×</mo> <mover> <mrow> <mtext>AV</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mo>−</mo> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> and find a similar expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}} "> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> <mo>×</mo> <mover> <mrow> <mtext>AV</mtext> </mrow> <mo>→</mo> </mover> </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>Hence, show that, if the angle between the faces <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABV}}"> <mrow> <mtext>ABV</mtext> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ACV}}"> <mrow> <mtext>ACV</mtext> </mrow> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span>, then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>p</mi> <mo>−</mo> <mn>20</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>6</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>40</mn> <mi>p</mi> <mo>+</mo> <mn>100</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the two possible coordinates of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{V}}"> <mrow> <mtext>V</mtext> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment on the positions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{V}}"> <mrow> <mtext>V</mtext> </mrow> </math></span> in relation to the plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABC}}"> <mrow> <mtext>ABC</mtext> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{X}}"> <mrow> <mtext>X</mtext> </mrow> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the horizontal asymptote of the graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Find the coordinates of the point of intersection of the planes defined by the equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = 3,{\text{ }}x - y + z = 5"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>5</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + 2z = 6"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>6</mn> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 4\,{\text{cos}}\,x + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \leqslant x \leqslant \frac{\pi }{2}">
<mi>a</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a < \frac{\pi }{2}">
<mi>a</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - \frac{\pi }{2}"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>. Indicate clearly the maximum and minimum values of the function.</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>Write down the least value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> has an inverse.</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>For the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> found in part (b), write down the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}"> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> found in part (b), find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}\left( x \right)"> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the vectors <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{ }}3">
<mo>−<!-- − --></mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
</math></span><strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{ }}2">
<mo>−<!-- − --></mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
</math></span><strong><em>k</em></strong>, <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ }}3">
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
</math></span><strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + {\text{ }}2">
<mo>+</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
</math></span><strong><em>k</em></strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <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>Hence find the Cartesian equation of the plane containing the vectors <strong><em>a </em></strong>and <strong><em>b</em></strong>, and passing through the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}0,{\text{ }} - 1)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</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 acute angles such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,A = \frac{2}{3}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,B = \frac{1}{3}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </math></span>.</p>
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {2A + B} \right) = - \frac{{2\sqrt 2 }}{{27}} - \frac{{4\sqrt 5 }}{{27}}"> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mn>4</mn> <msqrt> <mn>5</mn> </msqrt> </mrow> <mrow> <mn>27</mn> </mrow> </mfrac> </math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>The acute angle between the vectors 3<em><strong>i</strong></em> − 4<em><strong>j</strong></em> − 5<em><strong>k</strong></em> and 5<em><strong>i</strong></em> − 4<em><strong>j</strong></em> + 3<em><strong>k</strong></em> is denoted by <em>θ</em>.</p>
<p>Find cos <em>θ</em>.</p>
</div>
<br><hr><br><div class="specification">
<p>ABCD is a parallelogram, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} ">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span> = –<strong><em>i</em></strong> + 2<strong><em>j</em></strong> + 3<strong><em>k</em></strong> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AD}}} ">
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
</math></span> = 4<strong><em>i</em></strong> – <strong><em>j</em></strong> – 2<strong><em>k</em></strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the parallelogram ABCD.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable scalar product of two vectors, determine whether <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{A\hat BC}}">
<mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mrow>
<mover>
<mi mathvariant="normal">B</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
</math></span> is acute or obtuse.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cosec</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo><</mo><mi>θ</mi><mo><</mo><mfrac><mstyle displaystyle="true"><mn>3</mn><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle></mfrac></math>. Find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi></math>.</p>
</div>
<br><hr><br><div class="question">
<p>Let <em><strong>a</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ k \\ { - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <em><strong>b</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</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>Given that <em><strong>a</strong></em> and <em><strong>b</strong></em> are perpendicular, find the possible values of <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="specification">
<p>Three points in three-dimensional space have coordinates A(0, 0, 2), B(0, 2, 0) and C(3, 1, 0).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} ">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
</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>Find the vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AC}}} ">
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→</mo>
</mover>
</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>Hence or otherwise, find the area of the triangle ABC.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi "> <mrow> <msup> <mi>sec</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>2</mn> <mi>π</mi> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> have the following vector equations where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>,</mo><mo> </mo><mi>μ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub><mo>:</mo><msub><mi mathvariant="bold-italic">r</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub><mo>:</mo><msub><mi mathvariant="bold-italic">r</mi><mn>2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> do not intersect.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the minimum distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = {\text{sin}}\,b,\,\,0 < b < \frac{\pi }{2}">
<mi>a</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo><</mo>
<mi>b</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<p>Find, in terms of <em>b</em>, the solutions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,2x = - a,\,\,0 \leqslant x \leqslant \pi ">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mi>a</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mi>π</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> defined on the domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < 2\pi ">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
</math></span> by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 3\,{\text{cos}}\,2x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 4 - 11\,{\text{cos}}\,x">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
<p>The following diagram shows the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span></p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinates of the points of intersection of the two graphs.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact area of the shaded region, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\pi + q\sqrt 3 "> <mi>p</mi> <mi>π</mi> <mo>+</mo> <mi>q</mi> <msqrt> <mn>3</mn> </msqrt> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}"> <mi>q</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the points A and B on the diagram, the gradients of the two graphs are equal.</p>
<p>Determine the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-coordinate of A on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Consider quadrilateral <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PQRS</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mi>PQ</mi></mfenced></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mi>SR</mi></mfenced></math>.</p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<p style="text-align:left;">In <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PQRS</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PQ</mi><mo>=</mo><mi>x</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>SR</mi><mo>=</mo><mi>y</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mover><mi mathvariant="normal">S</mi><mo>^</mo></mover><mi mathvariant="normal">P</mi><mo>=</mo><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Q</mi><mover><mi mathvariant="normal">R</mi><mo>^</mo></mover><mi mathvariant="normal">S</mi><mo>=</mo><mi>β</mi></math>.</p>
<p style="text-align:left;">Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PS</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mi>b</mi><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mi>b</mi><mtext>i</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <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></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arg</mtext><mfenced><mrow><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mrow></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The vectors <strong><em>a</em></strong> and <em><strong>b</strong></em> are defined by <strong><em>a </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ t \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>t</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>b</em><em> </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ { - t} \\ {4t} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>4</mn>
<mi>t</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \in \mathbb{R}">
<mi>t</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 and simplify an expression for <em><strong>a</strong></em> • <em><strong>b</strong></em> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> for which the angle between<em><strong> a</strong></em> and <em><strong>b</strong></em> is obtuse .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mrow></mfrac></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>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>n</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>n</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>,</mo><mo> </mo><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>×</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>It is given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> term in the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>+</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn><mi mathvariant="normal">π</mi></math>.</p>
</div>
<br><hr><br><div class="question">
<p>A sector of a circle with radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> cm , where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> > 0, is shown on the following diagram.<br>The sector has an angle of 1 radian at the centre.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Let the area of the sector be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> cm<sup>2</sup> and the perimeter be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> cm. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = P">
<mi>A</mi>
<mo>=</mo>
<mi>P</mi>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>The plane <em>П</em> has the Cartesian equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + y + 2z = 3"> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>3</mn> </math></span></p>
<p>The line <em>L</em> has the vector equation <strong><em>r</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 3 \\ { - 5} \\ 1 \end{array}} \right) + \mu \left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \\ p \end{array}} \right){\text{,}}\,\,\mu {\text{,}}\,p \in \mathbb{R}"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>μ</mi> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. The acute angle between the line <em>L</em> and the plane <em>П</em> is 30°.</p>
<p>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>
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