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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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to eliminate a variable (or attempt to find det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A"> <mi>A</mi> </math></span>) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 1&3&{ - 1} \\ 3&{ - 5}&a \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 4 \\ b \end{array}} \right) \to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ - 14}&{a + 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 12} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </mtd> <mtd> <mrow> <mi>a</mi> <mo>+</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>b</mi> <mo>−</mo> <mn>12</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> (or det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 14\left( {a - 3} \right)"> <mi>A</mi> <mo>=</mo> <mn>14</mn> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </math></span>)</p>
<p>(or two correct equations in two variables) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \to \left( {\left. {\begin{array}{*{20}{c}} 2&{ - 1}&1 \\ 0&7&{ - 3} \\ 0&{ 0}&{a - 3} \end{array}\,} \right|\begin{array}{*{20}{c}} 5 \\ 3 \\ {b - 6} \end{array}} \right)"> <mo stretchy="false">→</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo stretchy="true" symmetric="true" fence="true"></mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mn>0</mn> </mrow> </mtd> <mtd> <mrow> <mi>a</mi> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> <mspace width="thinmathspace"></mspace> </mrow> <mo>|</mo> </mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>b</mi> <mo>−</mo> <mn>6</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> (or solving det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 0"> <mi>A</mi> <mo>=</mo> <mn>0</mn> </math></span>)</p>
<p>(or attempting to reduce to one variable, e.g. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {a - 3} \right)z = b - 6"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mi>z</mi> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mn>6</mn> </math></span>) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 3{\text{, }}b \ne 6"> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>≠</mo> <mn>6</mn> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>appreciation that two points distinct from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}"> <mrow> <mtext>P</mtext> </mrow> </math></span> need to be chosen from each line <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{}^4{C_2} \times {}^3{C_2}"> <msup> <mrow> </mrow> <mn>4</mn> </msup> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> <mo>×</mo> <msup> <mrow> </mrow> <mn>3</mn> </msup> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </math></span></p>
<p>=18 <em><strong>A</strong><strong>1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>consider cases for triangles including <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> <strong>or</strong> triangles not including <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}">
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 \times 4 + 4 \times {}^3{C_2} + 3 \times {}^4{C_2}">
<mn>3</mn>
<mo>×</mo>
<mn>4</mn>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<msup>
<mrow>
</mrow>
<mn>3</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<msup>
<mrow>
</mrow>
<mn>4</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> <em><strong>(A</strong><strong>1)(</strong></em><em><strong>A</strong><strong>1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for 1st term, <em><strong>A1</strong></em> for 2nd & 3rd term.</p>
<p><strong>OR</strong></p>
<p>consider total number of ways to select 3 points and subtract those with 3 points on the same line <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{}^8{C_3} - {}^5{C_3} - {}^4{C_3}">
<msup>
<mrow>
</mrow>
<mn>8</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<msup>
<mrow>
</mrow>
<mn>5</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<msup>
<mrow>
</mrow>
<mn>4</mn>
</msup>
<mrow>
<msub>
<mi>C</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> <em><strong>(A</strong><strong>1)(</strong></em><em><strong>A</strong><strong>1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for 1st term, <em><strong>A1</strong></em> for 2nd & 3rd term.</p>
<p>56−10−4</p>
<p><strong>THEN</strong></p>
<p>= 42 <em><strong>A</strong><strong>1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitution of (4, 6, 4) into both equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 3">
<mi>λ</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 1">
<mi>μ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>(4, 6, 4) <em><strong>AG</strong></em></p>
<p><strong>METHOD 2</strong></p>
<p>attempting to solve two of the three parametric equations <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 3">
<mi>λ</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = 1">
<mi>μ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>check both of the above give (4, 6, 4) <em><strong>M1</strong></em><em><strong>AG</strong></em></p>
<p><strong>Note:</strong> If they have shown the curve intersects for all three coordinates they only need to check (4,6,4) with one of "<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>" or "<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span>".</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 2">
<mi>λ</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PA}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ { - 2} \\ { - 1} \end{array}} \right)">
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PB}}} = \left( {\begin{array}{*{20}{c}} { - 5} \\ { - 6} \\ { - 2} \end{array}} \right)">
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if both are given as coordinates.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABP}} = \frac{1}{2}\left| {\overrightarrow {{\text{PB}}} \times \overrightarrow {{\text{PA}}} } \right|">
<mrow>
<mtext>ABP</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>PB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>PA</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}} { - 5} \\ { - 6} \\ { - 2} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} { - 1} \\ { - 2} \\ { - 1} \end{array}} \right)} \right|} \right) = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}} 2 \\ { - 3} \\ 4 \end{array}} \right)} \right|">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt {29} }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PC}}} = 3\overrightarrow {\,{\text{PA}}} ">
<mover>
<mrow>
<mtext>PC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mn>3</mn>
<mover>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>PA</mtext>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PD}}} = 3\overrightarrow {\,{\text{PB}}} ">
<mover>
<mrow>
<mtext>PD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mn>3</mn>
<mover>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>PB</mtext>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> <em><strong>(M1)</strong></em></p>
<p>area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PCD}} = 9 \times ">
<mrow>
<mtext>PCD</mtext>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo>×</mo>
</math></span> area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABP}}">
<mrow>
<mtext>ABP</mtext>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{9\sqrt {29} }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> has coordinates (−11, −12, −2) <em><strong>A1</strong></em></p>
<p>area triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PCD}} = \frac{1}{2}\left| {\overrightarrow {{\text{PD}}} \times \overrightarrow {{\text{PC}}} } \right| = \frac{1}{2}\left| {\left( {\begin{array}{*{20}{c}} { - 15} \\ { - 18} \\ { - 6} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 6} \\ { - 3} \end{array}} \right)} \right|">
<mrow>
<mtext>PCD</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>PD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>PC</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>15</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note: <em>A1</em></strong> is for the correct vectors in the correct formula.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{9\sqrt {29} }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{CDBA}} = \frac{{9\sqrt {29} }}{2} - \frac{{\sqrt {29} }}{2}">
<mrow>
<mtext>CDBA</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>29</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt {29} ">
<mo>=</mo>
<mn>4</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{D}}">
<mrow>
<mtext>D</mtext>
</mrow>
</math></span> has coordinates (−11, −12, −2) <em><strong>A1</strong></em></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left| {\overrightarrow {{\text{CB}}} \times \overrightarrow {{\text{CA}}} } \right| + \frac{1}{2}\left| {\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{BD}}} } \right|">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>BD</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for use of correct formula on appropriate non-overlapping triangles.</p>
<p><strong>Note:</strong> Different triangles or vectors could be used.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CB}}} = \left( {\begin{array}{*{20}{c}} { - 2} \\ 0 \\ 1 \end{array}} \right)">
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</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="\overrightarrow {{\text{CA}}} = \left( {\begin{array}{*{20}{c}} 2 \\ 4 \\ 2 \end{array}} \right)">
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CB}}} \times \overrightarrow {{\text{CA}}} = \left( {\begin{array}{*{20}{c}} { - 4} \\ 6 \\ { - 8} \end{array}} \right)">
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BC}}} = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)">
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BD}}} = \left( {\begin{array}{*{20}{c}} { - 10} \\ { - 12} \\ { - 4} \end{array}} \right)">
<mover>
<mrow>
<mtext>BD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BC}}} \times \overrightarrow {{\text{BD}}} = \left( {\begin{array}{*{20}{c}} { - 12} \\ {18} \\ { - 24} \end{array}} \right)">
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>BD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>24</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Other vectors which might be used are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DA}}} = \left( {\begin{array}{*{20}{c}} {14} \\ {16} \\ {5} \end{array}} \right)">
<mover>
<mrow>
<mtext>DA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>16</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BA}}} = \left( {\begin{array}{*{20}{c}} {4} \\ {4} \\ {1} \end{array}} \right)">
<mover>
<mrow>
<mtext>BA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DC}}} = \left( {\begin{array}{*{20}{c}} {12} \\ {12} \\ {3} \end{array}} \right)">
<mover>
<mrow>
<mtext>DC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><strong>Note:</strong> Previous <em><strong>A1A1A1A1</strong></em> are all dependent on the first <em><strong>M1</strong></em>.</p>
<p>valid attempt to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left| {a \times b} \right|">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>independent of triangle chosen.</p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times 2 \times \sqrt {29} + \frac{1}{2} \times 6 \times \sqrt {29} ">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<msqrt>
<mn>29</mn>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<msqrt>
<mn>29</mn>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt {29} ">
<mo>=</mo>
<mn>4</mn>
<msqrt>
<mn>29</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \sqrt {116} + \frac{1}{2}\sqrt {1044} ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msqrt>
<mn>116</mn>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msqrt>
<mn>1044</mn>
</msqrt>
</math></span> or equivalent.</p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {r\left( {{\text{cos}}\,\theta + {\text{i}}\,{\text{sin}}\,\theta } \right)} \right)^{24}} = 1\left( {{\text{cos}}\,0 + {\text{i}}\,{\text{sin}}\,0} \right)"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>use of De Moivre’s theorem <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^{24}} = 1 \Rightarrow r = 1"> <mrow> <msup> <mi>r</mi> <mrow> <mn>24</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mo stretchy="false">⇒</mo> <mi>r</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\theta = 2\pi n \Rightarrow \theta = \frac{{\pi n}}{{12}}"> <mn>24</mn> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>n</mi> <mo stretchy="false">⇒</mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mi>π</mi> <mi>n</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {n \in \mathbb{Z}} \right)"> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{arg}}\left( z \right) < \frac{\pi }{2} \Rightarrow n = "> <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> <mo stretchy="false">⇒</mo> <mi>n</mi> <mo>=</mo> </math></span> 1, 2, 3, 4, 5</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = {\text{e}}\frac{{\pi {\text{i}}}}{{12}}"> <mi>z</mi> <mo>=</mo> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{2\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{3\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{4\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{e}}\frac{{5\pi {\text{i}}}}{{12}}"> <mrow> <mtext>e</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> if additional roots are given or if three correct roots are given with no incorrect (or additional) roots.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Re <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{{12}} + {\text{cos}}\frac{{2\pi }}{{12}} + {\text{cos}}\frac{{3\pi }}{{12}} + {\text{cos}}\frac{{4\pi }}{{12}} + {\text{cos}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span></p>
<p>Im <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{\pi }{{12}} + {\text{sin}}\frac{{2\pi }}{{12}} + {\text{sin}}\frac{{3\pi }}{{12}} + {\text{sin}}\frac{{4\pi }}{{12}} + {\text{sin}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for both parts correct.</p>
<p>but <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{5\pi }}{{12}} = {\text{cos}}\frac{\pi }{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{4\pi }}{{12}} = {\text{cos}}\frac{{2\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{3\pi }}{{12}} = {\text{cos}}\frac{{3\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{{2\pi }}{{12}} = {\text{cos}}\frac{{4\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\frac{\pi }{{12}} = {\text{cos}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p>⇒ Re <em>S</em> = Im <em>S <strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept a geometrical method.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{{12}} = {\text{cos}}\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right) = {\text{cos}}\frac{\pi }{4}{\text{cos}}\frac{\pi }{6} + {\text{sin}}\frac{\pi }{4}{\text{sin}}\frac{\pi }{6}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <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> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 2 }}{2}\frac{{\sqrt 3 }}{2} + \frac{{\sqrt 2 }}{2}\frac{1}{2}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 6 + \sqrt 2 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span><em> <strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{12}} = {\text{cos}}\left( {\frac{\pi }{6} + \frac{\pi }{4}} \right) = {\text{cos}}\frac{\pi }{6}{\text{cos}}\frac{\pi }{4} - {\text{sin}}\frac{\pi }{6}{\text{sin}}\frac{\pi }{4}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>+</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Allow alternative methods <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{{5\pi }}{{12}} = {\text{sin}}\frac{\pi }{{12}} = {\text{sin}}\left( {\frac{\pi }{4} - \frac{\pi }{6}} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <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>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 3 }}{2}\frac{{\sqrt 2 }}{2} - \frac{1}{2}\frac{{\sqrt 2 }}{2} = \frac{{\sqrt 6 - \sqrt 2 }}{4}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p>Re <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{{12}} + {\text{cos}}\frac{{2\pi }}{{12}} + {\text{cos}}\frac{{3\pi }}{{12}} + {\text{cos}}\frac{{4\pi }}{{12}} + {\text{cos}}\frac{{5\pi }}{{12}}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span></p>
<p>Re <em>S</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sqrt 2 + \sqrt 6 }}{4} + \frac{{\sqrt 3 }}{2} + \frac{{\sqrt 2 }}{2} + \frac{1}{2} + \frac{{\sqrt 6 - \sqrt 2 }}{4}"> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <msqrt> <mn>6</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {\sqrt 6 + 1 + \sqrt 2 + \sqrt 3 } \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><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)"> <mo>=</mo> <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> </math></span></p>
<p><em>S</em> = Re(<em>S</em>)(1 + i) since Re <em>S</em> = Im <em>S</em>, <em><strong>R1</strong></em></p>
<p><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> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\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>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(\sin x\cos 60^\circ + \cos x\sin 60^\circ ) = \cos x\cos 30^\circ - \sin x\sin 30^\circ ">
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin x \times \frac{1}{2} + 2\cos x \times \frac{{\sqrt 3 }}{2} = \cos x \times \frac{{\sqrt 3 }}{2} - \sin x \times \frac{1}{2}">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{3}{2}\sin x = - \frac{{\sqrt 3 }}{2}\cos x">
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \tan x = - \frac{1}{{\sqrt 3 }}">
<mo stretchy="false">⇒</mo>
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 150^\circ ">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<msup>
<mn>150</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>choosing two appropriate angles, for example 60° and 45° <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ ">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
</math></span> and</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 105^\circ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ ">
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \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>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{\sqrt 2 }}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>OR</strong></p>
<p>attempt to square the expression <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\sin 105^\circ + \cos 105^\circ )^2} = {\sin ^2}105^\circ + 2\sin 105^\circ \cos 105^\circ + {\cos ^2}105^\circ ">
<mrow>
<mo stretchy="false">(</mo>
<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>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\sin 105^\circ + \cos 105^\circ )^2} = 1 + \sin 210^\circ ">
<mrow>
<mo stretchy="false">(</mo>
<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>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>210</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><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> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta ">
<mi>z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {{{(1 - \cos 2\theta )}^2} + {{(\sin 2\theta )}^2}} ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {1 - 2\cos 2\theta + {{\cos }^2}2\theta + {{\sin }^2}2\theta } ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mn>2</mn>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mn>2</mn>
<mi>θ</mi>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 2 \sqrt {(1 - \cos 2\theta )} ">
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<msqrt>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {2(2{{\sin }^2}\theta )} ">
<mo>=</mo>
<msqrt>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta ">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \alpha ">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \alpha = - \frac{{\sin 2\theta }}{{1 - \cos 2\theta }}">
<mi>tan</mi>
<mo></mo>
<mi>α</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2\sin \theta \cos \theta }}{{2{{\sin }^2}\theta }}">
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \cot \theta ">
<mo>=</mo>
<mo>−</mo>
<mi>cot</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \alpha = - \arctan \left( {\tan \left( {\frac{\pi }{2} - \theta } \right)} \right)">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α</mi>
<mo>=</mo>
<mo>−</mo>
<mi>arctan</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>tan</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \theta - \frac{\pi }{2}">
<mo>=</mo>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta ">
<mi>z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2{\sin ^2}\theta - 2{\text{i}}\sin \theta \cos \theta ">
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta (\sin \theta - {\text{i}}\cos \theta )">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2{\text{i}}\sin \theta (\cos \theta + {\text{i}}\sin \theta )">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta \left( {\cos \left( {\theta - \frac{\pi }{2}} \right) + {\text{i}}\sin \left( {\theta - \frac{\pi }{2}} \right)} \right)">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = 2\sin \theta ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \theta - \frac{\pi }{2}">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to apply De Moivre’s theorem <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(1 - \cos 2\theta - {\text{i}}\sin 2\theta )^{\frac{1}{3}}} = {2^{\frac{1}{3}}}{(\sin \theta )^{\frac{1}{3}}}\left[ {\cos \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right)} \right]">
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>for modulus, <strong><em>A1 </em></strong>for dividing argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> by 3 and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\pi ">
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</math></span>.</p>
<p> </p>
<p>Hence cube roots are the above expression when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = - 1,{\text{ }}0,{\text{ }}1">
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</math></span>. Equivalent forms are acceptable. <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to eliminate a variable <em><strong>M1</strong></em></p>
<p style="text-align:left;">obtain a pair of equations in two variables</p>
<p style="text-align:left;"><br><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> and <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>44</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>7</mn></math> and <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>40</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></math> and <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mo>-</mo><mfrac><mn>79</mn><mn>5</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;">the two lines are parallel (<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>≠</mo><mn>44</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>7</mn><mo>≠</mo><mn>40</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>≠</mo><mo>-</mo><mfrac><mn>79</mn><mn>5</mn></mfrac></math>) <em><strong>R1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> There are other possible pairs of equations in two variables.<br>To obtain the final <em><strong>R1</strong></em>, at least the initial <em><strong>M1</strong> </em>must have been awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">hence the three planes do not intersect <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">vector product of the two normals <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</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>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>A0</strong></em> if “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo></math>” is missing. Subsequent marks may still be awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">Attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>+</mo><mi>λ</mi><mo>,</mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>5</mn><mi>λ</mi><mo>,</mo><mn>3</mn><mi>λ</mi></mrow></mfenced></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mn>5</mn><mi>λ</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>λ</mi></mrow></mfenced><mo>=</mo><mn>32</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>15</mn><mo>=</mo><mn>32</mn></math>, a contradiction <em><strong>R1</strong></em></p>
<p style="text-align:left;">hence the three planes do not intersect <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 3</strong></p>
<p style="text-align:left;">attempt to eliminate a variable <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>100</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>94</mn></math>, a contradiction <em><strong>R1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Accept other equivalent alternatives. Accept other valid methods.<br>To obtain the final <em><strong>R1</strong></em>, at least the initial <em><strong>M1</strong> </em>must have been awarded.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">hence the three planes do not intersect <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>1</mn></munder><mo>:</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>4</mn></mstyle></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>2</mn></munder><mo>:</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>=</mo><mn>5</mn></mstyle></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to find the vector product of the two normals <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</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>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>” is missing.<br>Accept any multiple of the direction vector.<br>Working for (b)(ii) may be seen in part (a) Method 2. In this case penalize lack of “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>” only once.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">attempt to eliminate a variable from <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> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>y</mi><mo>-</mo><mn>5</mn><mi>z</mi><mo>=</mo><mo>-</mo><mn>6</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>7</mn></math></p>
<p style="text-align:left;">Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math></p>
<p style="text-align:left;">substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>t</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></math> to obtain</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>+</mo><mn>3</mn><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><mi>t</mi><mo>-</mo><mn>7</mn></math> (for all three variables in parametric form) <em><strong>A1</strong></em></p>
<p style="text-align:left;"><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><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>” is missing.<br>Accept any multiple of the direction vector. Accept other position vectors which satisfy both the planes <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>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">the line connecting <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> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math></p>
<p style="text-align:left;">attempt to substitute position and direction vector to form <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">s</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mn>9</mn><mi>t</mi><mo>,</mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>t</mi><mo>,</mo><mo>-</mo><mn>2</mn><mi>t</mi></mrow></mfenced></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>9</mn><mi>t</mi></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mn>3</mn><mi>t</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mn>32</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>94</mn><mi>t</mi><mo>=</mo><mn>47</mn><mo>⇒</mo><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">attempt to find distance between <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and their point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="|" close="|"><mrow><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><msup><mfenced><mrow><mo>-</mo><mn>9</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msqrt><mn>94</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">unit normal vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced></mrow><msqrt><mn>81</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>4</mn></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><msqrt><mn>94</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">let <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>4</mn></munder></mstyle></math> be the plane parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>3</mn></munder></mstyle></math> and passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, <br>then the normal vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>4</mn></munder></mstyle></math> is given by</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mo>-</mo><mn>15</mn></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">unit normal vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="false"><munder><mo>∏</mo><mn>4</mn></munder></mstyle></math> is given by</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mtable><mtr><mtd><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced></mrow><msqrt><mn>81</mn><mo>+</mo><mn>9</mn><mo>+</mo><mn>4</mn></msqrt></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>15</mn></mrow><msqrt><mn>94</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">distance between the planes is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><msqrt><mn>94</mn></msqrt></mfrac><mo>-</mo><mfrac><mrow><mo>-</mo><mn>15</mn></mrow><msqrt><mn>94</mn></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>47</mn><msqrt><mn>94</mn></msqrt></mfrac><mfenced><mrow><mo>=</mo><mfrac><msqrt><mn>94</mn></msqrt><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was well attempted using a variety of approaches. Most candidates were able to gain marks for part (a) through attempts to eliminate a variable with many subsequently making algebraic errors. Part (b)(i) was well done. For part (b)(ii) few successful attempts were noted, many candidates failed to use an appropriate notation "<em>r</em> =" while giving the vector equation of a line. Part (c) proved to be challenging for most candidates with very few correct answers seen. Many candidates did not attempt part (c).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OF}}} = \frac{1}{7}">
<mover>
<mrow>
<mtext>OF</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>7</mn>
</mfrac>
</math></span><strong><em>b</em></strong> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AF}}} = \overrightarrow {{\text{OF}}} - \overrightarrow {{\text{OA}}} ">
<mover>
<mrow>
<mtext>AF</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mover>
<mrow>
<mtext>OF</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>−</mo>
<mover>
<mrow>
<mtext>OA</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{7}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>7</mn>
</mfrac>
</math></span><strong><em>b</em></strong> – <strong><em>a </em></strong><strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><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>
<mo>=</mo>
</math></span> <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + \lambda \left( {\frac{1}{7}b -a} \right){\text{ }}\left( { = (1 - \lambda )a + \frac{\lambda }{7}b} \right)">
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>7</mn>
</mfrac>
<mi>b</mi>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
<mo stretchy="false">)</mo>
<mi>a</mi>
<mo>+</mo>
<mfrac>
<mi>λ</mi>
<mn>7</mn>
</mfrac>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OD}}} = \frac{1}{2}">
<mover>
<mrow>
<mtext>OD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + \mu \left( { - \frac{1}{2}a + b} \right){\text{ }}\left( { = \left( {\frac{1}{2} - \frac{\mu }{2}} \right)a + \mu b} \right)">
<mo>+</mo>
<mi>μ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mi>μ</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>a</mi>
<mo>+</mo>
<mi>μ</mi>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>equating coefficients: <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\lambda }{7} = \mu ,{\text{ }}1 - \lambda = \frac{{1 - \mu }}{2}">
<mfrac>
<mi>λ</mi>
<mn>7</mn>
</mfrac>
<mo>=</mo>
<mi>μ</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>μ</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p>solving simultaneously: <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = \frac{7}{{13}},{\text{ }}\mu = \frac{1}{{13}}">
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mn>7</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>μ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CD}}} = \frac{1}{{13}}\overrightarrow {{\text{CB}}} ">
<mover>
<mrow>
<mtext>CD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mover>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{13}}\left( {b - \frac{1}{2}a} \right){\text{ }}\left( { = - \frac{1}{{26}}a + \frac{1}{{13}}b} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>26</mn>
</mrow>
</mfrac>
<mi>a</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area }}\Delta {\text{ACD}} = \frac{1}{2}{\text{CD}} \times {\text{AC}} \times \sin {\rm{A\hat CB}}">
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>ACD</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>CD</mtext>
</mrow>
<mo>×</mo>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>×</mo>
<mi>sin</mi>
<mo></mo>
<mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mrow>
<mover>
<mi mathvariant="normal">C</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area }}\Delta {\text{ACB}} = \frac{1}{2}{\text{CB}} \times {\text{AC}} \times \sin {\rm{A\hat CB}}">
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>ACB</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>CB</mtext>
</mrow>
<mo>×</mo>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>×</mo>
<mi>sin</mi>
<mo></mo>
<mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mrow>
<mover>
<mi mathvariant="normal">C</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ratio }}\frac{{{\text{area }}\Delta {\text{ACD}}}}{{{\text{area }}\Delta {\text{ACB}}}} = \frac{{{\text{CD}}}}{{{\text{CB}}}} = \frac{1}{{13}}">
<mrow>
<mtext>ratio </mtext>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>ACD</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>ACB</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>CD</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>CB</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{{{\text{area }}\Delta {\text{OAB}}}}{{{\text{area }}\Delta {\text{CAD}}}} = \frac{{13}}{{{\text{area }}\Delta {\text{CAB}}}} \times {\text{area }}\Delta {\text{OAB}}">
<mi>k</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>OAB</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>CAD</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>CAB</mtext>
</mrow>
</mrow>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>OAB</mtext>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 13 \times 2 = 26">
<mo>=</mo>
<mn>13</mn>
<mo>×</mo>
<mn>2</mn>
<mo>=</mo>
<mn>26</mn>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area }}\Delta {\text{OAB}} = \frac{1}{2}\left| {a \times b} \right|">
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>OAB</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area }}\Delta {\text{CAD}} = \frac{1}{2}\left| {\overrightarrow {{\text{CA}}} \times \overrightarrow {{\text{CD}}} } \right|">
<mrow>
<mtext>area </mtext>
</mrow>
<mi mathvariant="normal">Δ</mi>
<mrow>
<mtext>CAD</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>CD</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left| {\overrightarrow {{\text{CA}}} \times \overrightarrow {{\text{AD}}} } \right|">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>CA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left| {\frac{1}{2}a \times \left( { - \frac{1}{{26}}a + \frac{1}{{13}}b} \right)} \right|">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>26</mn>
</mrow>
</mfrac>
<mi>a</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left| {\frac{1}{2}a \times \left( { - \frac{1}{{26}}a} \right) + \frac{1}{2}a \times \frac{1}{{13}}b} \right|">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>26</mn>
</mrow>
</mfrac>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{{13}}\left| {a \times b} \right|{\text{ }}\left( { = \frac{1}{{52}}\left| {a \times b} \right|} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>52</mn>
</mrow>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area }}\Delta {\text{OAB}} = k({\text{area }}\Delta {\text{CAD}})">
<mrow>
<mtext>area </mtext>
</mrow>
<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></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left| {a \times b} \right| = k\frac{1}{{52}}\left| {a \times b} \right|">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mi>k</mi>
<mfrac>
<mn>1</mn>
<mrow>
<mn>52</mn>
</mrow>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
</mrow>
<mo>|</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 26">
<mi>k</mi>
<mo>=</mo>
<mn>26</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = {\text{si}}{{\text{n}}^2}\,x + 2{\text{sin}}\,x\,{\text{cos}}\,x + {\text{co}}{{\text{s}}^2}\,x"> <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> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Do not award the <em><strong>M1</strong></em> for just <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^2}\,x + {\text{co}}{{\text{s}}^2}\,x"> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<p><strong><span style="background-color: #ffffff;">Note: </span></strong><span style="background-color: #ffffff;">Do not award <em><strong>A1</strong> </em>if correct expression is followed by incorrect working.</span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1 + {\text{sin}}\,2x"> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sec}}\,2x + {\text{tan}}\,2x = \frac{1}{{{\text{cos}}\,2x}} + \frac{{{\text{sin}}\,2x}}{{{\text{cos}}\,2x}}"> <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> <mn>1</mn> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong></em> is for an attempt to change both terms into sine and cosine forms (with the same argument) or both terms into functions of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,x"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{1}} + {\text{sin}}\,2x}}{{{\text{cos}}\,2x}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>1</mtext> </mrow> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)}^2}}}{{{\text{co}}{{\text{s}}^2}\,x - {\text{si}}{{\text{n}}^2}\,x}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <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> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for numerator, <em><strong>A1</strong></em> for denominator.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)}^2}}}{{\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <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> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <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> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <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> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}"> <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> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Apply MS in reverse if candidates have worked from RHS to LHS.</p>
<p><strong>Note:</strong> Alternative method using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2x"> <mrow> <mtext>tan</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="{\text{sec}}\,2x"> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,x"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{\pi }{6}} {\left( {\frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}} \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> <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> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct expression with or without limits.</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - {\text{ln}}\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)} \right]_0^{\frac{\pi }{6}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <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> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{ln}}\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)} \right]_{\frac{\pi }{6}}^0"> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <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> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mn>0</mn> </msubsup> </math></span> <em><strong>(M1)</strong><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for integration by inspection or substitution, <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{ln}}\left( {{\text{cos}}\,x - {\text{sin}}\,x} \right)}"> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <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> <mo>)</mo> </mrow> </mrow> </math></span>, <em><strong>A1</strong></em> for completely correct expression including limits.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ln}}\left( {{\text{cos}}\,\frac{\pi }{6} - {\text{sin}}\,\frac{\pi }{6}} \right) + {\text{ln}}\left( {{\text{cos}}\,0 - {\text{sin}}\,0} \right)"> <mo>=</mo> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for substitution of limits into their integral and subtraction.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ln}}\left( {\frac{{\sqrt 3 }}{2} - \frac{1}{2}} \right)"> <mo>=</mo> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{cos}}\,x - {\text{sin}}\,x"> <mi>u</mi> <mo>=</mo> <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> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = - {\text{sin}}\,x - {\text{cos}}\,x = - \left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <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> <mo>=</mo> <mo>−</mo> <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> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \int_1^{\frac{{\sqrt 3 }}{2} - \frac{1}{2}} {\left( {\frac{1}{u}} \right)} {\text{d}}u"> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>1</mn> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>u</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>u</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct limits even if seen later, <em><strong>A1</strong></em> for integral.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - {\text{ln}}\,u} \right]_1^{\frac{{\sqrt 3 }}{2} - \frac{1}{2}}"> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <mn>1</mn> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {{\text{ln}}\,u} \right]_{\frac{{\sqrt 3 }}{2} - \frac{1}{2}}^1"> <msubsup> <mrow> <mo>[</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>u</mi> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mn>1</mn> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {\text{ln}}\left( {\frac{{\sqrt 3 }}{2} - \frac{1}{2}} \right)\left( {{\text{ + ln}}\,1} \right)"> <mo>=</mo> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext> + ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\left( {\frac{2}{{\sqrt 3 - 1}}} \right)"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for both putting the expression over a common denominator and for correct use of law of logarithms.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\left( {1 + \sqrt 3 } \right)"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\frac{1}{2}{\text{ln}}\left( {{\text{tan}}\,2x + {\text{sec}}\,2x} \right) - \frac{1}{2}{\text{ln}}\left( {{\text{cos}}\,2x} \right)} \right]_0^{\frac{\pi }{6}}"> <msubsup> <mrow> <mo>[</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{ln}}\left( {\sqrt 3 + 2} \right) - \frac{1}{2}{\text{ln}}\left( {\frac{1}{2}} \right) - 0"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em><em><strong>A1(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{ln}}\left( {4 + 2\sqrt 3 } \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>+</mo> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{ln}}\left( {{{\left( {1 + \sqrt 3 } \right)}^2}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ln}}\left( {1 + \sqrt 3 } \right)"> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>recognising normal to plane or attempting to find cross product of two vectors lying in the plane <em><strong>(M1)</strong></em></p>
<p>for example, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to \,\, \times \mathop {{\text{AD}}}\limits^ \to = \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 0 \hfill \\ \end{gathered} \right) \times \left( \begin{gathered} - 1 \hfill \\ \,0 \hfill \\ \,1 \hfill \\ \end{gathered} \right) = \left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>×</mo>
<mover>
<mrow>
<mrow>
<mtext>AD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _1}\,{\text{:}}\,\,x + z = 1">
<mrow>
<msub>
<mi mathvariant="normal">Π</mi>
<mn>1</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) \bullet \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{gathered} \right) = 1 = \sqrt 2 \sqrt 2 \,{\text{cos}}\,\theta ">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>∙</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<msqrt>
<mn>2</mn>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong> M1A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\left( \begin{gathered} 1 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) \times \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 1 \hfill \\ \end{gathered} \right)} \right| = \sqrt 3 = \sqrt 2 \sqrt 2 \,{\text{sin}}\,\theta ">
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<msqrt>
<mn>2</mn>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>M1</strong> </em>is for an attempt to find the scalar or vector product of the two normal vectors.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \theta = 60^\circ \left( { = \frac{\pi }{3}} \right)">
<mo stretchy="false">⇒</mo>
<mi>θ</mi>
<mo>=</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>angle between faces is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20^\circ \left( { = \frac{{2\pi }}{3}} \right)">
<msup>
<mn>20</mn>
<mo>∘</mo>
</msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{DB}}}\limits^ \to = \left( \begin{gathered} \,1 \hfill \\ \,1 \hfill \\ - 1 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>DB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{BD}}}\limits^ \to = \left( \begin{gathered} - 1 \hfill \\ - 1 \hfill \\ \,1 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>BD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _3}\,{\text{:}}\,\,x + y - z = k">
<mrow>
<msub>
<mi mathvariant="normal">Π</mi>
<mn>3</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mi>k</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _3}\,{\text{:}}\,\,x + y - z = 0">
<mrow>
<msub>
<mi mathvariant="normal">Π</mi>
<mn>3</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>line AD : (<strong>r</strong> =)<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 0 \hfill \\ 0 \hfill \\ 1 \hfill \\ \end{gathered} \right) + \lambda \left( \begin{gathered} \,1 \hfill \\ \,0 \hfill \\ - 1 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>intersects <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> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda - \left( {1 - \lambda } \right) = 0">
<mi>λ</mi>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = \frac{1}{2}">
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>hence P is the midpoint of AD <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>midpoint of AD is (0.5, 0, 0.5) <em><strong>(M1)A1</strong></em></p>
<p>substitute into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y - z = 0">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong> M1</strong></em></p>
<p>0.5 + 0.5 − 0.5 = 0 <em><strong>A1</strong></em></p>
<p>hence P is the midpoint of AD <em><strong> AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OP}} = \frac{1}{{\sqrt 2 }},\,\,{\text{O}}\mathop {\text{P}}\limits^ \wedge {\text{Q}} = 90^\circ ,\,\,{\text{O}}\mathop {\text{Q}}\limits^ \wedge {\text{P}} = 60^\circ ">
<mrow>
<mtext>OP</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>O</mtext>
</mrow>
<mover>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>Q</mtext>
</mrow>
<mo>=</mo>
<msup>
<mn>90</mn>
<mo>∘</mo>
</msup>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>O</mtext>
</mrow>
<mover>
<mrow>
<mtext>Q</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>=</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
</math></span> <em><strong>A1A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PQ}} = \frac{1}{{\sqrt 6 }}">
<mrow>
<mtext>PQ</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>6</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong> A1</strong></em></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{2\sqrt {12} }} = \frac{1}{{4\sqrt 3 }} = \frac{{\sqrt 3 }}{{12}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>12</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>4</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>line BD : (<strong>r </strong> =)<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 1 \hfill \\ 1 \hfill \\ 0 \hfill \\ \end{gathered} \right) + \lambda \left( \begin{gathered} - 1 \hfill \\ - 1 \hfill \\ \,1 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \lambda = \frac{2}{3}">
<mo stretchy="false">⇒</mo>
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OQ}}}\limits^ \to = \left( \begin{gathered} \frac{1}{3} \hfill \\ \frac{1}{3} \hfill \\ \frac{2}{3} \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>OQ</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\left| {\mathop {{\text{OP}}}\limits^ \to \, \times \mathop {{\text{OQ}}}\limits^ \to } \right|">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OP</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mo>×</mo>
<mover>
<mrow>
<mrow>
<mtext>OQ</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OP}}}\limits^ \to = \left( \begin{gathered} \frac{1}{2} \hfill \\ 0 \hfill \\ \frac{1}{2} \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>OP</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: This <em><strong>A1</strong> </em>is dependent on <em><strong>M1</strong></em>.</p>
<p>area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sqrt 3 }}{{12}}">
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><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} = \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} = \frac{{\sqrt 2 }}{2}"> <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> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for 5 equal terms with \) + \) or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - "> <mo>−</mo> </math></span> signs.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \frac{{1 - (1 - 2{{\sin }^2}x)}}{{2\sin x}}"> <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> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{2{{\sin }^2}x}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \sin x"> <mo>≡</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(n):\sin x + \sin 3x + \ldots + \sin (2n - 1)x \equiv \frac{{1 - \cos 2nx}}{{2\sin x}}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>:</mo> <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> </math></span></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(1):\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <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> </math></span> which is true (as proved in part (b)) <strong><em>R1</em></strong></p>
<p>assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(k)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </math></span> true, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x + \ldots + \sin (2k - 1)x \equiv \frac{{1 - \cos 2kx}}{{2\sin x}}"> <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>k</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>k</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Only award <strong><em>M1 </em></strong>if the words “assume” and “true” appear. Do not award <strong><em>M1 </em></strong>for “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span><em>” </em>only. Subsequent marks are independent of this <strong><em>M1</em></strong><em>.</em></p>
<p> </p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(k + 1)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span>:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(k + 1):\sin x + \sin 3x + \ldots + \sin (2k - 1)x + \sin (2k + 1)x \equiv \frac{{1 - \cos 2(k + 1)x}}{{2\sin x}}"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>:</mo> <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>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</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> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="LHS = \sin x + \sin 3x + \ldots + \sin (2k - 1)x + \sin (2k + 1)x"> <mi>L</mi> <mi>H</mi> <mi>S</mi> <mo>=</mo> <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>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2kx}}{{2\sin x}} + \sin (2k + 1)x"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <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> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2kx + 2\sin x\sin (2k + 1)x}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>sin</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2kx + 2\sin x\cos x\sin 2kx + 2{{\sin }^2}x\cos 2kx}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \left( {(1 - 2{{\sin }^2}x)\cos 2kx - \sin 2x\sin 2kx} \right)}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - (\cos 2x\cos 2kx - \sin 2x\sin 2kx)}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos (2kx + 2x)}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \equiv \frac{{1 - \cos 2(k + 1)x}}{{2\sin x}}"> <mo>≡</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span></p>
<p>so if true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k"> <mi>n</mi> <mo>=</mo> <mi>k</mi> </math></span> , then also true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1"> <mi>n</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p>as true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 1"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </math></span> then true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }"> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept answers using transformation formula for product of sines if steps are shown clearly.</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>R1 </em></strong>only if candidate is awarded at least 5 marks in the previous steps.</p>
<p> </p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x \Rightarrow \frac{{1 - \cos 4x}}{{2\sin x}} = \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> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 - \cos 4x = 2\sin x\cos x,{\text{ }}(\sin x \ne 0)"> <mo stretchy="false">⇒</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo></mo> <mn>4</mn> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 1 - (1 - 2{\sin ^2}2x) = \sin 2x"> <mo stretchy="false">⇒</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> </mrow> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2x(2\sin 2x - 1) = 0"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2x = 0"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = \frac{1}{2}"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \pi ,{\text{ }}2x = \frac{\pi }{6}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \frac{{5\pi }}{6}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x + \sin 3x = \cos x \Rightarrow 2\sin 2x\cos x = \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> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow (2\sin 2x - 1)\cos x = 0,{\text{ }}(\sin x \ne 0)"> <mo stretchy="false">⇒</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \sin 2x = \frac{1}{2}"> <mo stretchy="false">⇒</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos x = 0"> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \frac{\pi }{6},{\text{ }}2x = \frac{{5\pi }}{6}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore x = \frac{\pi }{2},{\text{ }}x = \frac{\pi }{{12}}"> <mo>∴</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{5\pi }}{{12}}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Do not award the final <strong><em>A1 </em></strong>if extra solutions are seen.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p style="text-align:left;">uses the binomial theorem on <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> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mmultiscripts><mi>C</mi><mn>0</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mfenced><mrow><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>3</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>3</mn></msup><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>4</mn><mprescripts></mprescripts><mn>4</mn></mmultiscripts><mfenced><mrow><msup><mi mathvariant="normal">i</mi><mn>4</mn></msup><mo> </mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mfenced><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">(using de Moivre’s theorem with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math> gives) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> <strong>(A1)</strong></p>
<p style="text-align:left;">equates both the real and imaginary parts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>+</mo><mi mathvariant="normal">i</mi><mo> </mo><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced><mo>+</mo><mi mathvariant="normal">i</mi><mfenced><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow></mfenced></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo>-</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi></math></p>
<p style="text-align:left;">recognizes 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><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow></mfrac></math> <strong>(A1)</strong></p>
<p style="text-align:left;">substitutes for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>cos</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow><mrow><mi>sin</mi><mo> </mo><mn>4</mn><mi>θ</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p style="text-align:left;"><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>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow></mfrac></math></p>
<p style="text-align:left;">divides the numerator and denominator by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></math> to obtain</p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mrow><msup><mi>cos</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mo>-</mo><mn>6</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mi>θ</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi><mo mathvariant="italic">-</mo><mn>4</mn><mo mathvariant="italic"> </mo><mi>cos</mi><mo mathvariant="italic"> </mo><mi>θ</mi><mo mathvariant="italic"> </mo><msup><mi>sin</mi><mn>3</mn></msup><mo mathvariant="italic"> </mo><mi>θ</mi></mrow><mrow><msup><mi>sin</mi><mn>4</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac></mstyle></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"><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> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn></math> and putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> in the numerator of <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> gives <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> <strong>M1</strong></p>
<p style="text-align:left;">attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>…</mo><mo> </mo><mfenced><mrow><mn>4</mn><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi mathvariant="normal">n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mo>…</mo></mrow></mfenced></math> <strong>(A1)</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Do not award the final <strong>A1</strong> if solutions other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> are listed.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">finding the roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cot</mi><mo> </mo><mn>4</mn><mi>θ</mi><mo>=</mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>8</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></mrow></mfenced></math> corresponds to finding the roots of <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> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <strong>R1</strong></p>
<p style="text-align:left;">so the 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> as 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> <strong>AG</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">attempts to solve <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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><mo>±</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;">since <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><mo>></mo><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac><mo>,</mo><mo> </mo><msup><mtext>cot</mtext><mn>2</mn></msup><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>8</mn></mfrac></math> has the smaller value of the two roots <strong>R1</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>R1</strong> for an alternative convincing valid reason.</p>
<p style="text-align:left;"> </p>
<p style="text-align:left;">so <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><mo>=</mo><mn>3</mn><mo>-2</mo><msqrt><mn>2</mn></msqrt></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math></p>
<p style="text-align:left;">uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><msup><mi>cosec</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mi>cot</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <strong>(M1)</strong></p>
<p style="text-align:left;"><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><mo>⇒</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>6</mn><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <strong>M1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>y</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></math> <strong>A1</strong></p>
<p style="text-align:left;"> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,105^\circ = {\text{cos}}\left( {180^\circ - 75^\circ } \right) = - {\text{cos}}\,75^\circ "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>105</mn> <mo>∘</mo> </msup> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mn>180</mn> <mo>∘</mo> </msup> <mo>−</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>75</mn> <mo>∘</mo> </msup> </math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - q"> <mo>=</mo> <mo>−</mo> <mi>q</mi> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept arguments using the unit circle or graphical/diagrammatical considerations.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AD}} = {\text{CD}} \Rightarrow {\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 45^\circ "> <mrow> <mtext>AD</mtext> </mrow> <mo>=</mo> <mrow> <mtext>CD</mtext> </mrow> <mo stretchy="false">⇒</mo> <mrow> <mtext>C</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>D</mtext> </mrow> <mo>=</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> </math></span> <em><strong>A1</strong></em></p>
<p>valid method to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}}"> <mrow> <mtext>B</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>C</mtext> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>for example: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = r \Rightarrow {\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{A}} = 60^\circ "> <mrow> <mtext>BC</mtext> </mrow> <mo>=</mo> <mi>r</mi> <mo stretchy="false">⇒</mo> <mrow> <mtext>B</mtext> </mrow> <mover> <mrow> <mtext>C</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{C}} = 30^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>B</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> </math></span> <em><strong>A1</strong></em></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D}} = 45^\circ + 30^\circ = 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>45</mn> <mo>∘</mo> </msup> <mo>+</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo>=</mo> <msup> <mn>75</mn> <mo>∘</mo> </msup> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AB}} = r\sqrt 3 "> <mrow> <mtext>AB</mtext> </mrow> <mo>=</mo> <mi>r</mi> <msqrt> <mn>3</mn> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AD}} = \left( {{\text{CD}}} \right) = r\sqrt 2 "> <mrow> <mtext>AD</mtext> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>CD</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>r</mi> <msqrt> <mn>2</mn> </msqrt> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p>applying cosine rule <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}{{\text{D}}^2} = {\left( {r\sqrt 3 } \right)^2} + {\left( {r\sqrt 2 } \right)^2} - 2\left( {r\sqrt 3 } \right)\left( {r\sqrt 2 } \right){\text{cos}}\,75^\circ "> <mrow> <mtext>B</mtext> </mrow> <mrow> <msup> <mrow> <mtext>D</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>75</mn> <mo>∘</mo> </msup> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3{r^2} + 2{r^2} - 2{r^2}\sqrt 6 \,{\text{cos}}\,75^\circ "> <mo>=</mo> <mn>3</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> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <msqrt> <mn>6</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>75</mn> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5{r^2} - 2{r^2}q\sqrt 6 "><mo>=</mo><mn>5</mn><msup><mi>r</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>r</mi><mn>2</mn></msup><mi>q</mi><msqrt><mn>6</mn></msqrt></math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\mathop {\text{C}}\limits^ \wedge {\text{D}} = 105^\circ "> <mrow> <mtext>B</mtext> </mrow> <mover> <mrow> <mtext>C</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>D</mtext> </mrow> <mo>=</mo> <msup> <mn>105</mn> <mo>∘</mo> </msup> </math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to use cosine rule on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Delta {\text{BCD}}"> <mi mathvariant="normal">Δ</mi> <mrow> <mtext>BCD</mtext> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}{{\text{D}}^2} = {r^2} + {\left( {r\sqrt 2 } \right)^2} - 2r\left( {r\sqrt 2 } \right){\text{cos}}\,105^\circ "> <mrow> <mtext>B</mtext> </mrow> <mrow> <msup> <mrow> <mtext>D</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>r</mi> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <msup> <mn>105</mn> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3{r^2} + 2{r^2}q\sqrt 2 "> <mo>=</mo> <mn>3</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>2</mn> </msqrt> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5{r^2} - 2{r^2}q\sqrt 6 = 3{r^2} + 2{r^2}q\sqrt 2 "> <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> <mo>=</mo> <mn>3</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>2</mn> </msqrt> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{r^2} = 2{r^2}q\left( {\sqrt 6 + \sqrt 2 } \right)"> <mn>2</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> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for any correct intermediate step seen using only two terms.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = \frac{1}{{\sqrt 6 + \sqrt 2 }}"> <mi>q</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>6</mn> </msqrt> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> if follow through is being applied.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>×</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced><mo>=</mo><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mi>sin</mi><mo> </mo><mi>θ</mi></math> on the LHS <em><strong>(M1)</strong></em></p>
<p><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><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><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><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><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><mo>-</mo><msup><mfenced><mrow><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>·</mo><mi mathvariant="bold-italic">b</mi><mo>=</mo><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mi>cos</mi><mo> </mo><mi>θ</mi></math> on the RHS <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><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><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><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></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> If candidates attempt this question using cartesian vectors, e.g</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><msub><mi>a</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced></math> <strong>and </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi><mo>=</mo><mfenced><mtable><mtr><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced></math>,</p>
<p style="padding-left:30px;">award full marks if fully developed solutions are seen.<br>Otherwise award no marks.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong><br>horizontal stretch/scaling with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><br><strong>Note:</strong> Do not allow ‘shrink’ or ‘compression’</p>
<p><br>followed by a horizontal translation/shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> units to the left <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Do not allow ‘move’</p>
<p><br><em><strong>OR</strong></em></p>
<p>horizontal translation/shift <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> unit to the left</p>
<p>followed by horizontal stretch/scaling with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A2</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>vertical translation/shift up by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> (or translation through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><br>(may be seen anywhere)</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mi>p</mi></math></strong> and <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mi>q</mi></math> <em>M1</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mtext>tan</mtext><mo> </mo><mi>α</mi></math> </strong>and <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mtext>tan</mtext><mo> </mo><mi>β</mi></math> <em>(A1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mo>=</mo><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></math> <em>A1</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</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> <em>A1</em></strong></p>
<p>so <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>. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mfenced><mn>1</mn></mfenced></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p>Consider <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><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfenced><mrow><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></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mn>2</mn><mi>x</mi><mo>+1</mo><mo>-</mo><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mstyle displaystyle="true"><mn>1</mn><mo>+</mo><mfrac><mrow><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+1</mo></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>x</mi></mstyle><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mstyle></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan 1</mtext></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>tan </mtext><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mi mathvariant="normal">=</mi><mi>tan</mi><mo> </mo><mfenced><mrow><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></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> (or equivalent)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mrow></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>n</mi></mfenced></math> be the proposition 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>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mn>1</mn></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><mn>1</mn><mn>2</mn></mfrac></mfenced><mo>=</mo><mtext>RHS</mtext></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math> is true <strong><em>R1</em></strong></p>
<p>assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> is true, ie. <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</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>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mi> </mi><mfenced><mrow><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>”.<br><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong></em> are independent of this mark and can be awarded.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></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><munderover><mtext>Σ</mtext><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow><mi>k</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><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mstyle><mrow><mn>1</mn><mo>-</mo><mfenced><mstyle displaystyle="true"><mfrac><mi>k</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfenced><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></mstyle></mfenced></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mi>k</mi></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct numerator, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> factored. Denominator does not need to be simplified</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mi>k</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for denominator correctly expanded. Numerator does not need to be simplified. These two <em><strong>A</strong></em> marks may be awarded in any order</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mfenced></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> The word ‘arctan’ must be present to be able to award the last three A marks</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> is true whenever <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> is true and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mn>1</mn></mfenced></math> is true, so</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>n</mi></mfenced></math> is true for 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> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the final <em><strong>R1</strong></em> mark provided at least four of the previous marks have been awarded.<br><strong>Note:</strong> To award the final <em><strong>R1</strong></em>, the truth of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> must be mentioned. ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>k</mi></mfenced></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>’ is insufficient to award the mark.</p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></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><strong><em>r</em></strong> <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) \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = - 3 + \beta } \\ {y = - 2 + 4\beta } \\ {z = a + 2\beta } \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</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>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mi>β</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>4</mn>
<mi>β</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>β</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 - ( - 3 + \beta )}}{3} = \frac{{( - 2 + 4\beta ) - 2}}{4} \Rightarrow 4 = \frac{{4\beta }}{3} \Rightarrow \beta = 3">
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mi>β</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>4</mn>
<mi>β</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>4</mn>
</mfrac>
<mo stretchy="false">⇒</mo>
<mn>4</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>β</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mo stretchy="false">⇒</mo>
<mi>β</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{6 - ( - 3 + \beta )}}{3} = 1 - (a + 2\beta ) \Rightarrow 2 = - 5 - a \Rightarrow a = - 7">
<mfrac>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mi>β</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>β</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
<mo>−</mo>
<mi>a</mi>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} { - 3 + \beta = 6 - 3\lambda } \\ { - 2 + 4\beta = 4\lambda + 2} \\ {a + 2\beta = 1 - \lambda } \end{array}} \right.">
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mi>β</mi>
<mo>=</mo>
<mn>6</mn>
<mo>−</mo>
<mn>3</mn>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>4</mn>
<mi>β</mi>
<mo>=</mo>
<mn>4</mn>
<mi>λ</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>β</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p>attempt to solve <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 2,{\text{ }}\beta = 3">
<mi>λ</mi>
<mo>=</mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>β</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 1 - \lambda - 2\beta = - 7">
<mi>a</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mi>λ</mi>
<mo>−</mo>
<mn>2</mn>
<mi>β</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OP}}} = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ { - 7} \end{array}} \right) + 3 \bullet \left( {\begin{array}{*{20}{c}} 1 \\ 4 \\ 2 \end{array}} \right)">
<mover>
<mrow>
<mtext>OP</mtext>
</mrow>
<mo>→</mo>
</mover>
<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>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mo>∙</mo>
<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> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 0 \\ {10} \\ { - 1} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore {\text{P}}(0,{\text{ 10, }} - 1)">
<mo>∴</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> 10, </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = \left( {\begin{array}{*{20}{c}} 0 \\ 1 \\ 2 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} {2b} \\ 0 \\ {b - 1} \end{array}} \right)">
<mi>n</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {b - 1} \\ {4b} \\ { - 2b} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>4</mn>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p>(0, 0, 0) on <em>Π</em> so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {b - 1} \right)x + 4by - 2bz = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using equation of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="px + qy + rz = 0">
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
<mi>y</mi>
<mo>+</mo>
<mi>r</mi>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>(0, 1, 2) on <em>Π</em> ⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q + 2r = 0">
<mi>q</mi>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>(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) on <em>Π</em> ⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2bp + r\left( {b - 1} \right) = 0">
<mn>2</mn>
<mi>b</mi>
<mi>p</mi>
<mo>+</mo>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)A1</strong></em> for both equations seen.</p>
<p>solve for <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">
<mi>q</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {b - 1} \right)x + 4by - 2bz = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>M has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {b,\,\,0,\,\,\frac{{b - 1}}{2}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em><strong>r</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} b \\ 0 \\ {\frac{{b - 1}}{2}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {b - 1} \\ {4b} \\ { - 2b} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>4</mn>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> if <em><strong>r</strong></em> = (or equivalent) is not seen.</p>
<p><strong>Note:</strong> Allow equivalent forms such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{x - b}}{{b - 1}} = \frac{y}{{4b}} = \frac{{2z - b + 1}}{{ - 4b}}">
<mfrac>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>y</mi>
<mrow>
<mn>4</mn>
<mi>b</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>z</mi>
<mo>−</mo>
<mi>b</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>4</mn>
<mi>b</mi>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = z = 0">
<mi>x</mi>
<mo>=</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or both.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b + \lambda \left( {b - 1} \right) = 0">
<mi>b</mi>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{b - 1}}{2} - 2\lambda b = 0">
<mfrac>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mn>2</mn>
<mi>λ</mi>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to eliminate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow - \frac{b}{{b - 1}} = \frac{{b - 1}}{{4b}}">
<mo stretchy="false">⇒</mo>
<mo>−</mo>
<mfrac>
<mi>b</mi>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>4</mn>
<mi>b</mi>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4{b^2} = {\left( {b - 1} \right)^2}">
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p>consideration of the signs of LHS and RHS <em><strong>(M1)</strong></em></p>
<p>the LHS is negative and the RHS must be positive (or equivalent statement) <em><strong>R1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4{b^2} = {b^2} - 2b + 1">
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 5{b^2} - 2b + 1 = 0">
<mo stretchy="false">⇒</mo>
<mn>5</mn>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Delta = {\left( { - 2} \right)^2} - 4 \times 5 \times 1 = - 16\,\left( { < 0} \right)">
<mi mathvariant="normal">Δ</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo>×</mo>
<mn>5</mn>
<mo>×</mo>
<mn>1</mn>
<mo>=</mo>
<mo>−</mo>
<mn>16</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo><</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore ">
<mo>∴</mo>
</math></span> no real solutions <em><strong>R1</strong></em></p>
<p><strong>THEN</strong></p>
<p>so no point of intersection <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = z = 0">
<mi>x</mi>
<mo>=</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> or both.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b + \lambda \left( {b - 1} \right) = 0">
<mi>b</mi>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{b - 1}}{2} - 2\lambda b = 0">
<mfrac>
<mrow>
<mi>b</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mn>2</mn>
<mi>λ</mi>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to eliminate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{\lambda }{{1 + \lambda }} = \frac{1}{{1 - 4\lambda }}">
<mo stretchy="false">⇒</mo>
<mfrac>
<mi>λ</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>λ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>4</mn>
<mi>λ</mi>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4{\lambda ^2} = 1\left( { \Rightarrow {\lambda ^2} = - \frac{1}{4}} \right)">
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>consideration of the signs of LHS and RHS <em><strong>(M1)</strong></em></p>
<p>there are no real solutions (or equivalent statement) <em><strong>R1</strong></em></p>
<p>so no point of intersection <em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5\sqrt {15} }}{2} = \frac{1}{2} \times 4 \times 5\,{\text{sin}}\,\theta ">
<mfrac>
<mrow>
<mn>5</mn>
<msqrt>
<mn>15</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>4</mn>
<mo>×</mo>
<mn>5</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p>height of triangle is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5\sqrt {15} }}{4}">
<mfrac>
<mrow>
<mn>5</mn>
<msqrt>
<mn>15</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> if using 4 as the base or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sqrt {15} }">
<mrow>
<msqrt>
<mn>15</mn>
</msqrt>
</mrow>
</math></span> if using 5 as the base <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><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> <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let the third side be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = {4^2} + {5^2} - 2 \times 4 \times 5 \times {\text{cos}}\,\theta ">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>4</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>×</mo>
<mn>4</mn>
<mo>×</mo>
<mn>5</mn>
<mo>×</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p>valid attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta ">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Do not accept writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {{\text{arcsin}}\left( {\frac{{\sqrt {15} }}{4}} \right)} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>15</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> as a valid method.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \pm \sqrt {1 - \frac{{15}}{{16}}} ">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mo>±</mo>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>15</mn>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{4}{\text{,}}\,\, - \frac{1}{4}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 16 + 25 - 2 \times 4 \times 5 \times \pm \frac{1}{4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>16</mn>
<mo>+</mo>
<mn>25</mn>
<mo>−</mo>
<mn>2</mn>
<mo>×</mo>
<mn>4</mn>
<mo>×</mo>
<mn>5</mn>
<mo>×</mo>
<mo>±</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt {31} ">
<mi>x</mi>
<mo>=</mo>
<msqrt>
<mn>31</mn>
</msqrt>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {51} ">
<msqrt>
<mn>51</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 6 \\ { - 8} \\ {17} \end{array}} \right)">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong><em>r </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ 3 \\ { - 6} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 6 \\ { - 8} \\ {17} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <strong><em>r</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ { - 5} \\ {11} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} 6 \\ { - 8} \\ {17} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1A0 </em></strong>if <strong><em>r </em></strong>= is not seen (or equivalent).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitute line <em>L </em>in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Pi :4(6\lambda ) - 3(3 - 8\lambda ) + 2( - 6 + 17\lambda ) = 20">
<mi mathvariant="normal">Π</mi>
<mo>:</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mn>6</mn>
<mi>λ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>−</mo>
<mn>8</mn>
<mi>λ</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>6</mn>
<mo>+</mo>
<mn>17</mn>
<mi>λ</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>20</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="82\lambda = 41">
<mn>82</mn>
<mi>λ</mi>
<mo>=</mo>
<mn>41</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = \frac{1}{2}">
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong><em>r</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ 3 \\ { - 6} \end{array}} \right) + \frac{1}{2}\left( {\begin{array}{*{20}{c}} 6 \\ { - 8} \\ {17} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 3 \\ { - 1} \\ {\frac{5}{2}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>so coordinate is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3,{\text{ }} - 1,{\text{ }}\frac{5}{2}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept coordinate expressed as position vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ { - 1} \\ {\frac{5}{2}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>attempt at integration by parts with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {{\text{e}}^x}"> <mi>u</mi> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = {\text{cos}}\,2x"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x = } \frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x\,{\text{d}}x - \int {\frac{{{{\text{e}}^x}}}{2}} {\text{sin}}\,2x\,{\text{d}}x"> <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> <mspace width="thinmathspace"></mspace> <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>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x - \frac{1}{2}\left( { - \frac{{{{\text{e}}^x}}}{2}{\text{cos}}\,2x + \int {\frac{{{{\text{e}}^x}}}{2}} {\text{cos}}\,2x} \right)"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{4}{\text{cos}}\,2x - \frac{1}{4}\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} "> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</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>4</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore \frac{5}{4}\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = \frac{{{{\text{e}}^x}}}{2}{\text{sin}}\,2x\, + \frac{{{{\text{e}}^x}}}{4}{\text{cos}}\,2x"> <mo>∴</mo> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><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\left( { + c} \right)"> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <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> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt at integration by parts with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{cos}}\,2x"> <mi>u</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = {{\text{e}}^x}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x = } {{\text{e}}^x}\,{\text{cos}}\,2x + 2\int {{{\text{e}}^x}\,{\text{sin}}\,2x\,{\text{d}}x} "> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <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> <mo>+</mo> <mn>2</mn> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^x}\,{\text{cos}}\,2x + 2\left( {{{\text{e}}^x}\,{\text{sin}}\,2x - 2\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} } \right)"> <mo>=</mo> <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> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x - 4\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} "> <mo>=</mo> <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> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore 5\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x"> <mo>∴</mo> <mn>5</mn> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <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> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><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\left( { + c} \right)"> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <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> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>attempt at use of table <em><strong>M1</strong></em></p>
<p><em>eg</em></p>
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong> </em></p>
<p><strong>Note:</strong> <em><strong>A1</strong> </em>for first 2 lines correct, <em><strong>A1</strong> </em>for third line correct.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x = \,} {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x - 4\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} "> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mspace width="thinmathspace"></mspace> </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> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore 5\int {{{\text{e}}^x}\,{\text{cos}}\,2x\,{\text{d}}x} = {{\text{e}}^x}\,{\text{cos}}\,2x + 2{{\text{e}}^x}\,{\text{sin}}\,2x"> <mo>∴</mo> <mn>5</mn> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <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> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><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\left( { + c} \right)"> <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> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <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> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{co}}{{\text{s}}^2}\,x{\text{d}}x = } \int {\frac{{{{\text{e}}^x}}}{2}} \left( {{\text{cos}}\,2x + 1} \right){\text{d}}x"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mo>∫</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {\frac{{{\text{2}}{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x} \right) + \frac{{{{\text{e}}^x}}}{2}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>2</mtext> </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> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2}\left( { + c} \right)"> <mo>=</mo> <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> <mrow> <mo>(</mo> <mrow> <mo>+</mo> <mi>c</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Do not accept solutions where the RHS is differentiated.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {{\text{e}}^x}\,{\text{co}}{{\text{s}}^{\text{2}}}\,x - 2{{\text{e}}^x}\,{\text{sin}}\,x\,{\text{cos}}\,x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <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>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mtext>2</mtext> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for an attempt at both the product rule and the chain rule.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,{\text{cos}}\,x\left( {{\text{cos}}\,x - 2\,{\text{sin}}\,x} \right) = 0"> <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> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to factorise <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{cos}}\,x}"> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </math></span> or divide by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x\left( {{\text{cos}}\,x \ne 0} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>discount <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = 0"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> (as this would also be a zero of the function)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{cos}}\,x - 2\,{\text{sin}}\,x = 0"> <mo stretchy="false">⇒</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{tan}}\,x = \frac{1}{2}"> <mo stretchy="false">⇒</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = {\text{arctan}}\left( {\frac{1}{2}} \right)"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (at A) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi + {\text{arctan}}\left( {\frac{1}{2}} \right)"> <mi>x</mi> <mo>=</mo> <mi>π</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (at C) <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct answer. If extra values are seen award <em><strong>A1A0</strong></em>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = 0 \Rightarrow x = \frac{\pi }{2}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\pi }}{2}"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> The <em><strong>A1</strong></em>may be awarded for work seen in part (c).</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {\left( {{{\text{e}}^x}\,{\text{co}}{{\text{s}}^{\text{2}}}\,x} \right)} \,{\text{d}}x = \left[ {\frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2}} \right]_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}}"> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mtext>2</mtext> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <msubsup> <mrow> <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> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( { - \frac{{{{\text{e}}^{\frac{{3\pi }}{2}}}}}{{10}} + \frac{{{{\text{e}}^{\frac{{3\pi }}{2}}}}}{2}} \right) - \left( { - \frac{{{{\text{e}}^{\frac{\pi }{2}}}}}{{10}} + \frac{{{{\text{e}}^{\frac{\pi }{2}}}}}{2}} \right)\left( { = \frac{{{\text{2}}{{\text{e}}^{\frac{{3\pi }}{2}}}}}{5} - \frac{{{\text{2}}{{\text{e}}^{\frac{\pi }{2}}}}}{5}} \right)"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>2</mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mo>−</mo> <mfrac> <mrow> <mrow> <mtext>2</mtext> </mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1(A1</strong><strong>)</strong><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for substitution of the end points and subtracting, <em><strong>(A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,3\pi = {\text{sin}}\,\pi = 0"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mi>π</mi> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,3\pi = {\text{cos}}\,\pi = - 1"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mi>π</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>π</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> and <em><strong>A1</strong></em> for a completely correct answer.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>a pair of opposite sides have equal length and are parallel <em><strong>R1</strong></em></p>
<p>hence ABCD is a parallelogram <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rewrite the given information in vector form <em><strong>M1</strong></em></p>
<p><em><strong>b</strong></em> − <em><strong>a</strong></em> = <em><strong>c</strong></em> − <em><strong>d</strong></em> <em><strong>A1</strong></em></p>
<p>rearranging <em><strong>d</strong></em> − <em><strong>a</strong></em> = <em><strong>c</strong></em> − <em><strong>b </strong> <strong>M1</strong></em></p>
<p>hence <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> <em><strong>AG</strong></em></p>
<p><strong>Note</strong>: Candidates may correctly answer part i) by answering part ii) correctly and then deducing there<br>are two pairs of parallel sides.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>use of <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> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 2 \hfill \\ - 3 \hfill \\ p + 3 \hfill \\ \end{gathered} \right) = \left( \begin{gathered} q + 1 \hfill \\ 1 - r \hfill \\ 4 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>p</mi>
<mo>+</mo>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mi>q</mi>
<mo>+</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>OR</strong></p>
<p>use of <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> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} - 2 \hfill \\ r - 2 \hfill \\ 1 \hfill \\ \end{gathered} \right) = \left( \begin{gathered} q - 3 \hfill \\ 2 \hfill \\ 2 - p \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mi>q</mi>
<mo>−</mo>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
<mo>−</mo>
<mi>p</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>attempt to compare coefficients of <em><strong>i</strong></em>, <em><strong>j</strong></em>, and <em><strong>k</strong></em> in their equation or statement to that effect <em><strong>M1</strong></em></p>
<p>clear demonstration that the given values satisfy their equation <em><strong>A1</strong></em><br><em>p</em> = 1, <em>q</em> = 1, <em>r</em> = 4 <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at computing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to \, \times \mathop {{\text{AD}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mo>×</mo>
<mover>
<mrow>
<mrow>
<mtext>AD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span> (or equivalent) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} - 11 \hfill \\ - 10 \hfill \\ - 2 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>11</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>10</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left| {\mathop {{\text{AB}}}\limits^ \to \, \times \mathop {{\text{AD}}}\limits^ \to } \right|\left( { = \sqrt {225} } \right)">
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mo>×</mo>
<mover>
<mrow>
<mrow>
<mtext>AD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msqrt>
<mn>225</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 15 <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OM}}}\limits^ \to = \left( {\frac{1}{2}\left( {a + c} \right)} \right)">
<mover>
<mrow>
<mrow>
<mtext>OM</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 1 \hfill \\ \frac{3}{2} \hfill \\ - \frac{1}{2} \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>the equation is</p>
<p><em><strong>r</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 1 \hfill \\ \frac{3}{2} \hfill \\ - \frac{1}{2} \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} 11 \hfill \\ 10 \hfill \\ 2 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>11</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>10</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> or equivalent <em><strong>M1A1</strong></em></p>
<p><strong>Note</strong>: Award maximum <em><strong>M1A0</strong></em> if '<em><strong>r</strong></em> = …' (or equivalent) is not seen.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to obtain the equation of the plane in the form <em>ax</em> + <em>by</em> + <em>cz</em> = <em>d</em> <em><strong>M1</strong></em></p>
<p>11<em>x</em> + 10<em>y</em> + 2<em>z</em> = 25 <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong> </em>for right hand side, <em><strong>A1</strong></em> for left hand side.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>putting two coordinates equal to zero <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{X}}\left( {\frac{{25}}{{11}},\,0,\,0} \right),\,\,{\text{Y}}\left( {0,\,\frac{5}{2},\,0} \right),\,\,{\text{Z}}\left( {0,\,0,\,\frac{{25}}{2}} \right)">
<mrow>
<mtext>X</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mrow>
<mn>11</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>Y</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>Z</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{YZ}} = \sqrt {{{\left( {\frac{5}{2}} \right)}^2} + {{\left( {\frac{{25}}{2}} \right)}^2}} ">
<mrow>
<mtext>YZ</mtext>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>25</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {\frac{{325}}{2}} \left( { = \frac{{5\sqrt {104} }}{4} = \frac{{5\sqrt {26} }}{2}} \right)">
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mn>325</mn>
</mrow>
<mn>2</mn>
</mfrac>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<msqrt>
<mn>104</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<msqrt>
<mn>26</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><img 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hraUU/faOY6thPVyPm5MZJJDWya6J2MLvYOrLF5bB+hR/TBEX1aWOhNZZg1a1ZoUU9jvHpwcCeffHJN7O3oTGpg0yNGjMgOOeSQ8JmonTW2P2DAgDAELpkBR/PpYaE3lWH27NnBiQ0ZMqShkBPp4/SIcuzsTGqolIrudNi3hB7I/FKtpUjfpIWF3lQGHBmti6mPrC+a13ecH8eoCNOYVMCuWbB1RfKAzdNAlRnsaLBq0sPezFQGBgTBuanfcD2XX355bcAcR/QmNYjWp06dmt122221iF52Xi/0jUq8THmx0JvKgNDjwBgZrx62/+lPfwoizzj4dnQmNbDpadOmZQ899FCI7LVN7wSZXMTemdz0sNCbykAdPSy11FINnRkDh1CsqcWYlFD7E9YqvhcIPVVbEnqLfVpY6E1lYLAcGiJpEo8YnB7FlmzHEcZO0JhUIJJndMhYzFnT2wTbj7vYmXTwEzWVgaJ7HFqjxnYSdortcXwWepMaalGPbfMOSOj5Th09fezpYmfbTw8LvakMRPSKXOqdGU5v4403rkXz9RG/MWUHcafaaoMNNgj2HpdcUXTPfo13b9LCQm8qAw2R6C/cqGgSxzd69OhsscUWs8ibJMGuV1hhhWzHHXds3fL+NmyfDAClWYx3b9LDQm+SRo6MYst58+aFQXGIXNqDjAC/MSYliN71LghF9WSAEfopU6a07jEpYaE3lYAGSAg9kYscXgzbaHXMmsxBR5kBY8oIdo99I/ASfLZRR09DVerogffApIOF3iSNHJa6DiH0OLhGjuzUU0/N5syZM58TNCYlJk6cmF1wwQVhvAi9A6wR+2HDhtWmqq3PCJtyY6E3lYApallodNTIieHcXn755dq+RvX4xpQZbJyMLItKrGTv7GNiG401YaFPC3szkyw4K9XRE8HQT57GdtoXw3HqWmdMilBKJRtX9RQCr4V6emZ4ZB+LSQcLvUkWOTAgiqH4fskll6w5uBi+4wCJ5MkQ1O83puzIthF77LtezIcOHRrekTjiN2lgoTdJI/GmDz3Oi1HxtL0eIh4a7bnY3qQKg+Iwpr3eixjq6GnHwsBSJi3s0UyySMyJXGhkhJNbeumlw7ZGEf0xxxwTWh8rujcmNVZbbbXsoIMOahjRf/zjHw89Uyi+t/2nhYXeJA1RC04Noaf7EA2OoN6R8X3VVVcNUX+jon1jyg4lVowMSRE99l4f0S+77LKhwSoDS7lUKy38NE2yKDJnQegRcLW6rxdyvlNsD+w3JjWom6dRKvAu6LOg/QqlXrwrfgfSwkJvkiUW9JkzZ7Yr9EAfY4m9Wx2b1MDuaYw3adKk8Jn3IYbSLkaOtNCnh4XeJA2CjdOigRFd62iMp0i/ntNPPz00VDImRXgXnnjiiWzcuHEN7Z/3g4ieTLFJCwu9SRZF7awRek1o01b9Y5wBcB2lSY04gm9UokUmmMaor7zyiiP6xLA3M8mCs0KwWTPiF0LPZzsxYz4IGQF6pTBCpDO6aeGnaZJFos5ClyGEnpbHfG8U0RhTZXgnPvaxj4WJbfx+pIWF3iSLiuIRd+reEfr6BkiC4yi21PHGpAh18DS4o76+vsEpto/QM7gU/elNOljoTfLgtP7973+Hkb8Qf2UAYvg+ZsyYIPZkBtzq3qQGNj1ixIjswAMPrL0HMXxnLAmGwX3ppZdat5oUsNCbZEG8cV6vv/566DOM0BOtt+XkFlxwwVrRvjGpgdDTl56oXu9GPSuuuGLoYjp58uTWLSYFLPQmeYjoKbofPHhw65b3MwExcnr1a2NSAZFH7LHtthrbkRnm3WB0PJMOFnqTLDgzInSKIono43Hs60cFwwFefvnlYQhQHWNMSmDjdJ27/fbbg33zPQa7p9X9oosumk2ZMiVsqz/GlBMLvUkaxB6hJ5pZaqmlwjacXH2jPI67+eabawPmuHuRSQ3s/sUXX8weeOCBhu8A22iox3sioec9YDuLKS/2ZiZpcGY0xMOBaYpaIheWemLHZ8dmUgObV/E9At4oWmfipyFDhoTGeI7m08FCb5IFsVbRPQ5Mw9/iwOqFnmNppOQIxqSKbJ9Gp1RdNcrsso0GeUT0mveBbY2ONeXBQm+Shiid4W9xbtQ9gsS8HurnlQmwYzMpgl0zsQ3vRaN3gG3LLLNMmNhG1Vi8E874lhsLvUkaHBRCzxSciL1o5Lj233//WmbAmNTA5onWP//5z4fvjTKzHLPyyiuHUjDq88GZ3vJjoTfJg9B//OMfDw4LR8a63nmxfeuttw5C7wjGpAglWUzTvNFGGwUbbwTvxeqrrx6OfeaZZ8J70NaxpjxY6E2y4KRYGDAn7kMP9UKvTIAG1MHRGZMS2DZF9hTdY+P17wDwDiy33HJh33PPPVfL9LKY8mJvZpIGR8XY3bQk7kjAcYDAMThFY1KCBngIvcaQaCTevCNE/dTTP/vss+FdYGmUKTDlwUJvkkVOipnrGPGrI4477riQKZBDNCYlaKMyYcKE7He/+11D4SZTzHaO431R0T1Y6MuNhd4kz5w5c8KsXO2BI+M4uhQh8q6XNKmBaGPf2Hlb0TzbKc2iQR7T1dIoz5QfC71JGhXd19fRN4JIngFFcHaOYIqBBKmIGS+uqZFgFpm4DUq9jXMvWtZYY41s9uzZ2fTp08M+Z3zLjYXeJA8j4zGGtykfCBML4oPYqH65v5HwIZZlas+BwOt6G103+ynRGj58eNhPgzzS3hnfcmOhN8mCMyaax1ENGjSodWtjFJ3JqTmCKQaUsMSNx4rWdoJrKguyaQk8oh4jMec4+tszbDT19Gy30JcbC71JFpwTxY8Mfbvwwgu3bm0Mx9IYb8CAAcHR2bEVA+qUx48fHzJsPJOiCCuzwDEefFFKGJqBTBJF8t/4xjdqGdsY0pdMAMetsMIKoV3LPffc07rXlBkLvUmal19+OUQmHQk9To8ohggSh1cf7Zi+A3FHiF544YVs7733Ds9QExL1ptDznxJAlrfeeqsW/db/L1VB55xzTnbLLbcEW2E/3TN17UWFAaHoOsf11mdm+Y7Is6bl/VprrZX985//LFVmxjTG3swkC87sX//6V5h2s95R1yOnh6OO6zFN38OzePLJJ7PddtstO+aYY7Iddtih1kiyo+fYHfhfFgk+Ysda4yvEYCPHHntsdt5552WXXnppuC6JZJHhfrBtZU7a45Of/GRojKcGeaa8WOhNsuDMJk6cGIS+IwfMfgYIAZxh0eqCqwSR9F577ZX98pe/zNZee+3wHCXACH5vwflZ+C9E8MEHH8yef/75sK/efrgmZkQ8//zzQ2R/11131YSzNzMj3YHrIm2nTp0arr8jSHsmeuJ4U24s9CZZcGxPPfVUqGvsSOjh9NNPD3XBOEEcvukffvvb32YHHnhg9qlPfaom8oDI9+ZzkfgxZPLhhx+ebbPNNtmjjz4a/re+hEeZDhp5MgDNIYccEvqnF9l2SEfeh8suu6wW1bfHKquski222GLZ5MmT/T6UHAu9SQZFUnJK1C3SapgJbZqJsnCEOD+ObSZjYHoG0lsLz+7+++/Pvv71r9eeB6UrKhbvSJy6A/bC+Zmidb311gvRrK6pvoRH14XYr7nmmtl2220XonvgN0VENs31ce0diTdjT1Cfr1INU14s9CYZcGQ4L4k102wy/C2N7DqC4+WgLfJ9C+nOcyPKJGNGCQyNJ/v6OajEAHvZeOONa3akdXt897vfzX7zm9+EzAHHFxXStNl05Tga5NG7wO9EubHQm6SInTINuqiTxHF3FL3E8PuOHLvpOYikFTETFW+11VbhMw0j+1JglEkUapjZjC3QCn+zzTbLbrzxxsKKouya9G6m6J79W2yxRWh5b8qNhd4kA446dma33nprtuSSS4Y+wR2Bc6aVMQ2s9N30DaoD57ndcMMN2aabbhrSXxF2XyFRZx0//2bEnowKvQSuu+661i3FhIap6667blP3xH6EnpIxDTylxZQLC71JBjln1nSJuvPOO0ODLhoU1dexNuKggw4Kx4KFvm/h+dBvnilSiY4ltqz7Cv0faz1/iVpH18H+z3zmM9ljjz2WzZs3r3Vr8aB0a+eddw7315Fg80yYm55n8vDDD9dEvr5hoik+FnqTDDghHBhR4COPPBJaC9NyWvvaQ8W2Wnd0vOk5KH2Bxx9/PDSCKysMzETf+6LO+KYMjEq+lJlpC45bYIEFsnXWWSf74x//WNtGBsDvR7mw0JukkAO7/vrrQzHl5ptvHrZ35NQQdwSH4+zE+haEAwGZNGlStvzyy7duLRdcP7az+uqrh0Gaigh2rfeDhWtuD/Zz/IgRI7K77747TA6lZ2XKhYXeJANiTZE9LZ+pn6fbE8IhB9ceOK8rrrjC82/3Azw3hOe1114L3bliJCqsJTzx0tPE55UgQkfipt8R1TNkbxHhXqZMmZLddttt4X4Q7fZgP8X0VH+RCVPxPb9VuphyYKE3yYATogEXI5pR3/uFL3whFD3ilDqqV+QYHCCZBNP3kP4ISP2cBGznuUpYJLgSnJ5EJTrA+Vmwm2aEDbsTRbUh7ufVV1/NHnroodr3juC+iOgZI18NDXmnmvmtKQ4WepMMOB8cMiN/UWy/44471up/iRpN+ZDQS1h4jkSll1xySYfi21k4tzKEjHKHyNGHvKczFP0J99JsYzrSl+NpjEcV2E033RQyCvzeQl8u7P1MUjB9KMX2W265ZRh4RcWTFvpyI4G68sorQz97qlkaTTbTHRA2FhpxMgbDz3/+89C47vbbb289otzEGaNm3of4+K9+9ash83PttdfW0smUB3s/U2qILOKonXHHaTTEWOWdjToQEjc2KiY8F54nLcCpM+Z7XFzeU2BDDJnMWPujR4/OvvWtb4WeG6kJW2ffjZEjR4aGhoz+x1wApIfeE71/prhY6E2pwdngnInuGMELR7TnnnuGoTsVebC/GUfNjGkqAbDYFwc9P6LrNdZYI/SzZ9Q6PaueQjYiu+H8vZWh6A/IyNIv/rOf/WwQ+mZsnPsn/Vl///vfDzPZXXDBBbWMAudkf2czDqZvsdCbUoNDxsmwPvroo4NDOvHEE1v3dg4iNxodAc7LmJTg3WCkyE984hOdtm/eLzIIDLZDlcZzzz0XInm9eyymuNibmdKD07rqqquyv//976GolS5OnYXoRg7L0YlJEUXfLKqmaha9E9/+9rfD+pRTTpnvffE7U2ws9KbUINCzZ8/OfvzjH4fhPQ844IBal7rOFL/j/DQtaSpFtanCs20LnjnVOAhZRxCRslANwPESrEZLvJ//6IxtFQVsnLTjXiTQzcKxvBfDhw/P9t133+zmm2+uNVIsY1pUDQu9KTU4L4rqGaRk7NixYVIatuF82hOEenBkJ510UmhZrCJJU06oy2cSlk9/+tNhApe2lr333jscj/DtscceYaz6thYao2277bZhkpc77rgj/K6M0I7lnHPOCe9IZxrRKePM8p3vfCcMbMT7Qjor42CKy4dyh2aPZkrLvffeGxrf0ZCOqD4G59OsA8LpHXLIIdkZZ5xRm9gGZ1h0eH3pZbDrrrtmf/nLX0pxzfVwDzynE044IVtttdVCxNge9Kh49tlnG04Ji+gok0cf+PYgQ0DDPoqwGWe/PeFTRM9vaNBGXXd9WrP/sMMOC33O99lnn9atxYF7GD9+fHbLLbcEsea+m30/FLVLLv70pz9lX/va18KzOvPMM8O2Zs9l+h5H9KbU/OpXvwrD3B5zzDGtW953Rp2N6CmWVDEu2GmVE547AoyIEXVSndPWMnTo0CDcPGt6aWy44YZtLkxhzEJJwKBBg0qZoeKaSR/snM+dsXGO53f8hvfr85//fLbffvuF8Qyuueaa1qNMUbHQm1Kh+lQcFg3wmGzj7LPPDg3wcEYsOPnO1rPjvOQI5cxM/6NnrTXPhefDZy1vvvlmKIlhREQ9OxaJeFuLbITnTruO9uB42RZLGZGNA/fTWRsnPfk96Ua6f/e7381WWGGF7LjjjgtVZ5yP58RiioWF3pQKHBSNrYjkaQFMlzoGUZFIdxXOS6SGE1fEY/ofnoOEm+d7//33Z0899VQYAfGGG27IZs2aFUZCpETn0EMPDdOp6jdmfkgXFoa0he6kEecZMmRI9j//8z9hfeSRR4YJodqr/jD9h72ZKRVE9Mcee2xoCETR4de//vVadNKdSAIR+dGPfhScIGLfnUyD6Tl4tnq+iAtF7AzDet9994V+3Twv5jR4/vnns2984xshuuc480F4d9Zff/2QTiqG7yrKJFA9Mm7cuDA1L+8iPVe6c17TO/iNMIUDkZUjYc3CNiJ5oniiCMbePvnkk2tRCnRU/NoRKpJksbMqBrIDFbMvvvjioQiZ3hUMbsSzR1xmzJgRBnH50pe+1GURU+ZO/0nGkYwDDc9Ylx3ShLQhI8vn7tg46a4qDOaUoESF3gi8n/RcIe1IR95ZpafpPyz0plCo2FyCy0IkMnPmzNAqnqkyaWFPVzocCMfisLpbbyrHp2jezqnviNO6Xnz4rmfMWoueF89q4sSJoQU+wx8zzS2ZAh3TGXQ818OC3Z166qkh80BVQdkhXbgvZZq6C+mlTANTQlN1QsO8I444Ips3b14QezLfLs7vfyz0plDghOQYcEo4EWYTo1iQ+tdvfvOb2WmnnRacCM68J6H4kfNyDZ0VCdN1JPRdTXPGv6dhGNF+d+D/uRbsijXtAWjoCWQoyg73wLtFw7metm/O94Mf/CBE9ExnS4mbprTt6ffUdB4/AVM4FJ3jPB599NHQR5z1T3/60+x73/te2I/zIOLqKXDsY8aMyV577bXgEHvaEZq2kRCQ7p0V1FiYu1uqwzkERfW0BRk1alQyQoVNa+InPvd05oUM8g9/+MMwnsUjjzwSxsV/5plnevQ9NV3DQm/6HTlY1nI+OCIGw9l///1DcSz9dYkScLrsY009bU/BOTkf/4/DIhIxfYOeP8+A+l1gWyy8baHhjlWE3F147ix01aPbGAPjEAU32/6DOumiQnqSRkqnnkivGM7Pe8lAOueee24Ympr+9rSpYR/pGmfm9I4185xN97DQm36Fl54XPX75iapp3ENdPAObUGTPHORyJDgorXsKzoVDR+xx1t2NDk3zkPY822HDhoVpULEBtml7e3AMtgBadxUiT87H86cEaeuttw7nZJEotQX7udYpU6ZkI0aMaN1aLLgP7lGZoo7StjMonfQ86BHx5z//OVt77bXDKHy0qeH9YuF/SS9dj95903tY6E2/IofDi08/3Ouvvz7bbrvtst/97ndhbPGLLroo9NOV8+9NFMmz7kknaNqHtGZh1LlJkyYFAeiP9JcAMvjOTjvtVMuENiOKsk36+DNSYxFBZOmxgI1LaHsL0pJM+iWXXBLa1fziF78IjfUYlpj/leCTbs5U9z4WetOv8LIzbjnOFYH//ve/H8YKJxq44IILsgEDBoTjetMpAddBoy6KaPnckWM3PQuiuvrqq2cPPfRQrU63LzJ3MfwfRfavv/56sEVsQaLYkRhhn0888UQQt+42CuwtuAcGhWI+AehNG0fIeaYDBw4MVSC///3vs2nTpmXbbLNNaFjLzHdk7Hm+juh7H09qY3ocvbhy0piYXmgcImu2kbunLzxdo4jaDzrooFC/J3HvS7genBPRPNfH9fZ25qIn4LpTmNRGbL/99qFLG9E9zwBx6qt7ok4ZodbIcYDok770Fac4+vzzz/+AbfAd2/nJT34Siv2pdmq2Tr+v4Tq53njpC/TO33PPPdnpp58e2t9Q8sH7zgRAtIWQf4ivS7YRX2dfXXNKWOhNj4NJ8TKyVjSkNbl4xqe/8MILs7/+9a/hBad/PI3uEHiiOYoX+xqulQVnA1xrGRwK11x2oQelPRE1fbAPPPDA8L0vbYG2IX/4wx+yt956Kwg2Yk3JEnZKSROT2tC4TLYtUQLse7PNNst+/etfh3pp7Kdo8G5xTyo278vMiISeNCN9EXoyRsymxzwV9G444IADQoYfOJYFyHyTzqRxnOameSz0plfQi8rCC/rwww+HwW6I3mlwxQu92267hSieYTR5gXGuOPb+eJHliOTEWZdBNLnWsgu9HDjQ6p6pXmmbgVjyXPpKNJUZZaQ9xAUoXaAhGd3SVlpppVq1gjKCshci1eOPPz5kDNim3xeJ2MZFX9kL/8v/C/4XwX/ggQeyiy++OEydu8gii4RSEwbf2XTTTUMViK6P3/I5vnbTPOXzCqbw4DBZaLlMX+Qtt9wyNG6iYQ5Foz//+c9Djp66u2WXXTa8wDgCRJ51f4AjYVQvMhtgh9J38PyxF8C5L7bYYtnTTz/9AVHqbRBvroNIl//FJljrGpQRqBd5tlNcT9RfZLhWBrG5+eab+zxtY6Em/YB0xjecd9552dVXXx0+M9ww3Wg//elPhzXj6NPAkTTuL9+QAo7oTS2nLcEFOTGIt+lYfZaz4Bi6FiHuRDd33XVXaGSH495ggw2C0NPAiS5U/E/R4PopLqY4kQZLoHsrMlx3CkX3MVTtMGkRJUAqXkYcZId9cY/YNv/HNVx++eXZOeecE6qZBPt0DPtoiHfWWWfVemxIzIrGP/7xj1DqwCh2RbJv0pLnylDXBAGU/DFxEaP4kaaUAG6yySahy+Oaa66ZLbfccqGqT2lNBr0+A8Y5ZTfaBvEx7cEx/YGuV/5V90L6aA2deQ8s9BVFj13GLGPSd70ksYHppSF3TfEmk4g8+OCDwTE/9thjYTxwijapc0PcER+6yC211FLhnPF5igb3ZaEvBtQhn3DCCaFol1HWuEfuS8+jL+5R7wPwf7oGwTUC3UG5RsSTkggdW0QbhyIKPenMtajEhPQDvpOBwscg+qyZvIjjyXQh/quuumoQfsYuoNcG1YBkDHgO/B7iahSdm/9j0XeIP0Nf2FkjuA7ukf/nGrE17oG10gji++oIC31F0WPXWi+bjFvbEfQXXngh1Ksj7EwHysvHuPDTp08PxwwdOjS8cOS4EXimwqRbDecDzoWBYph6wYoG12ihLwbYCnZHVI9dnnjiibVRC/uqcZ5sVs4WYntgH6JJ1dRvf/vbIDBcH05Ytl5EihrRy9+A0p0oXenIfr7TVoKFoXUff/zxMO6CJhzifsgAEO2z0KaCEkQWek3Qyp93W/8V37/8n7bpWfYHun9A3OPMDw1VNSJoZ0YGbVromzys39GD4nrjB1mP9nf2vtr6Tfxfjc6t/fW/1bEi/gzsZ+HB67fxuRodz7Z4DXzGePnOON50JaKYjDUNoFiov5s1a1ZYeHlYyEFjaBTBI+jUqS+99NLhRSInvfLKK4eIHaPTf9a/NPE1xOsiwTUffPDBoeGVMilFvM56SNvUhJ5nIRuitIg62j333DN0e+N+++Ie9a5wDXzmHYjtgT7hCCbXhRhxTTpWn4sIjWKpBz/66KMLYyvyDzFsi5+1olodq32IP/aP72LyK4ISng2+be7cuSFAoTcFx9Djh9JGfBnvOPZExoA11QAsZATYpuOYCpmSGj4zFDf/qWvgGeuZs9Z2jpEdxMfyWdfdHvyWrse33nprduaZZ4YuyJSMMugQbRlGjx4djqMhc7M0LfQktG6Mpa9QwiiROvpvHdPouEa3quPq95HYMR09nBjO1dY163uca4uP4TPoO2uuBaMFIh2EmgUjx3jp60uROfswavaR86N+S6LOds7BZ4we+D2/w7hZEPCPf/zjtWX48OFByBF3roMXTUat60sF0p3ogPvmuZTl/rju1IQ+hvuL34miPJf4/YUy2AvpiI8gI0+Uy/fU7EV+m+eh+5N24f94x8kUKJAhLcgY4CsJdhQA4RspPVJUT1sRgpnBgwcHP4n4k0HAb7ImQ0DmgQyBMglkDNjHOahmwH/KTtpLd9k8xzJJEBMEkaFkjAnOyQL0UmiWpoUeQeBCMXDdfF+gB6d1ewnEMVwb14hB199a/DJyLHWAHMM5+R6fW8IqtA/DwAjicyG2CKrAsDQ5B/AdMeYYfsfvMSjSlOvFUSPM/AfXxMI+ommuj4Vt+i3Xyn7Oy8Jn4PecD4NkwRDYhrERdWOUGCOROYJGVE5xFjlZjDquY9QalPYQ708J7on71HNlHT/josJ1py70PAfW+lyE5xLbS1GuqRnwjSA7Kct1N0vsq7g3vsfvhL7LnrSP7fhSfDnvE/6Zqkn8OD6fzIG2kUEgcMJnozP4X/wyaSv/rFIeRBmfzGiA9MqI97UFx3DtXN8dd9yR7bvvvqG7Kb0S1CME2jtHPU0LPcWa1I3EUV1fwH8hcjwEPaD24LpIeES1PiHia+Z8eihK1PjcPPAYHadzxMfyIEkXER/Hmv2suR7dAwbAOflODpAcIYaC4PKZB0qukO8cg0jzH4g3ws1vOAfHKPfIwnZ+y+8QcD7z//wX5+cc/L+uUfcRb2uLOA3aO66MkDakE7ZDmik9ig7XWZWIXu9OEZ6LrkPXVgZbwcaVhix8Tu091nPBV8X3yHfQM1M6xNtAx8lnsi8+H5/1W3yFAjPS9oYbbghTaTOeAn6abWQQyAwwiNLIkSNrvoVztAX70SeugaoHxhRgICYieq6D8wI+vlmaFnpmIKJvKyLEjbV3oT0Jl0ei8b9K9LZgHwmE4CGGJGpMfKvsk6DGD1SopbjgtyyIan0jCCJhFsF5OI4HxW/4zn/xYPgv/pPrYzvflZb6rO26Hs6hY+J70H7Q9XFc/f3oGvSi6zPXB/XHg36j87Lo+FThhVSGTelUdHiORBYIPYOOyE5SQXaoz1CE56Lriq+vDGDj+CGumyXFd7qtZ1K/ne+C7bHPBX1WWvE93h/DbzmGKXlp0Hv44YeHoX1JX37DPj7rP9p7T8lAKKCmVweDNu2yyy6hRIDfdeUd71TRPQKFkMbRa1/ADQOX2uxN8pv2juVc8YPTf4i2HqhJE+yBF4mBT8ikdeVl6g+wW4oRcSo33XRTaa7b9D1kCp988skwpC8zyaWece9LpE2kMQ0eGQwMGBxslVVWCfuBY/jcXtpzDhZ6FjAWCRM9UU9PQzyC7fXWW6/1yOZpWuipq1A9cV86E10e/9vkpdYEvBmx5pzxf4j63+qYRsea8sNzpdRKxW5EPWV4xqqC+spXvhIGdrHzNm2BjSNCDGjF1LHYt/1Yz4E4k55kvklr5m1gMKVvfetboY6eUl1AP9vTUGb64/cU2/+///f/QsbsG9/4RnbkkUdmn/jEJ7Ktttqq9cjmaVroadJP32lFyvURcG/B5VGCQCJCR4apRO7oOO1v6/brf6/jOjqvKS+0yKW/LdUwiGcZnjXCTgYcp0ADSwu9aQt8KP6RxmX0+4+rqkz3IG0JDvAbKnZHJ+nyR3ozqBKN6tiHlrT3nv7xj3/MJkyYEDJjtLOiNJ2xJKijZ6wSqqY7S9NCz4XHjq+vnSCX2dn/bObW4vM2e/4mk8yUDKbMpK8qOe8yOUB6dOy+++5hDHMLvWkPioHp/69+9H3tx1NGDeiA6jSm473tttuyI444Itt7771DZkCRfHsRvYJojpHWaN3e79qj6V/JKLT0NV35z/h621ri+2qW+Pde0liAaJ4XSi9To+OKtug6dd31+7140YKNEA0SJZbJxsuwkLaIPK3sib4pqqfrMu1m9tprr3AMwQPprrRvi/gYnb+Z37VH0xG9MSlDbpuFl5UcNS8lL1jR4fVNuXud6TkUKRJ5qreRS4B6DhrM0dqeWff222+/Wu8GiXV/Yq9gTI5yzcBaTtGYlJC4s3amsOdA0Gkrc8opp4RoXj3U2F4EX+InbUyOCraU83ak03vI+bHGGao0Rc/A9A7KzCrClK2b7kNaMoc+o9cB6YzY40eK4Ess9Mbk8GLSZ1UCZNHpXUhvxF0RJo6Sz7QEN70HI4ZOmTKllt6mGljojclB2H/2s5+FrjAu0uxdFMkjNkQ9Eny+s5jegXRnJkCm1eWz7bw6+EkbkyOnp5axpvegX/C1116b7b///tlOO+0UBgO56667atG96R3IRJGpYghv0prMlqkG9mhtQI6XRfCC8GLgpFjzoigSMf2LnhXPgwErWHfl2fBsoUzPlPuN71+fuYeO7oPjQMfFv9W5tF3ni7c1u+i3+h0jhh122GFhvm0GBqFfN6J/1llnhUwWx+hYLfqua4g/N9pWv6gtAJ91TFUhHUjnomSqeB7KdHBtLNg17yNr030s9O1QX4zI7H1MWkA3JuoS5UQwTNM/4CQEz4vviMff/va3TjlzfsfvJTRlEQKNtAVcO/ehNOmoGJzfcSxrOdr4vvkcf4f4N1r0f4LvOOl44R2hfph5v88444wgMly7/pc1M38xLWejc7IN9Iz5zud4Hb+HHKPjIG4HUHVIJ6VLUeB6sA/6nWMfF154YTZx4sQObdg0h/vRd4CS53e/+12YhvA3v/lNdtVVVwWxHzduXHAcLu7tP0h/OXomgaD+8ZJLLgljv48dO7bp58J5cDJbb711qDcuUz/6F198MfTdxT6Zpph74b47un7ENRZA1ojyzJkzw7zbTNXMCF/Mw80MeYzAxzJ37twQabGP/+c49gsywThtzglcC8dzbfyW+byVKam/xqFDh2YrrrjifOOCM8CLZoxku2aL5DPPikFgmIiIbazZzm8GDhz4gZkmY3sow/PtSXjePFfq6Wkd3oyN9AXYHDCuO7OGLrnkktkVV1yRvfDCC+Gd3GCDDcJ+03VKJfRcKoYZr+MF5yzn0ciA49/V79e22Pmw5uV46aWXss985jNhsoGNN9449JdEEA488MBs1KhRHzhXVVC6c//xOkaOlShCzr2+yDAWBJ2nWfR/rPkPntdGG20UBpAZM2ZM0+fiGljk/Difrr1IvJffn+7ppZcmZyedeGJ25513BnEdtsyw7OT8nj/3uc+F/cB9kN6sub9p06YFoWUEL8bh5jtj/LMN4Sbi5lykpRbSlP/UqGoIKudEuIFJgNgmEFf2IcL8J7/VsyXTQJG90pnzcy6cPdtWWmml4OzJPAD/zfWw5ve8exzLJFu6Lv0H5+Ecsi+ul2tD8LnuQYMGhWXppZcOGQrmBuA7/8c2BIZzQfzs+V8W/oNF24Dv2ge6Fn0vGro/0o4BXTqC40kLjuczaat7bpRGbKvfB0oXaJQ2ZD6wyc022yz8F41isWPGd2faV9M9Sif0OAaMiJcdg+Eza7bru9ALL3RMI0MTcXJwHAZ62mmnZZdeemloMIQz4Lw0ICLHSf2irqFKyGFw37yYQLrIYZPpAjkK0lWf47RiO9u0T+uuwvmY4WmHHXbIfvSjH7Vu7Rg9d12nrr9ovPe/uQ3nawRz1113yZ7/1/Pv78j58EfyyDl/FpQ6EbGRQWUiKqa2pNoJZ4qgI6KMpocgL7HEEkEMBw8enA0ZMiRbbrnlwtCdiB8CycIxRMqIN+KgEiyEXO9I/MyUlmzTZ45hoUSAiTlY6/3EZsgocE9M6IGz53fsZ81+vbuUDHBeBF/no/RAa5U8ELlSMoFgcF6tOYb7J6PAebkf0kGlAAyDTBqMGDEiW3311UPpAmnB9amHAHAtXIfuX/CZ645tvCjItvWs+B4/t3piHwvyvXomGvmNc5I28fnr71//xVqfBedlG79h4dzY2Oc///kwRjxzUJjuUTqhF3zGWeHIeGk333zzsJ2+0Gzbdtttg/HF6IXEmDCu2Jnr3PGLIHbcccfgQG655ZbabxCRc889N3vwwQeDgyyqMPQWpB+i8Y9//CMIK0JBut5+++0hOlp//fVrL67SlDRmkYMHpTtrjutuOnKergg9cE+yC1170fjfd97NSLGLLrwwO/bYY7MFovR6L0+/BRdeqBapqasgkS0ZVObFXmeddbJVV101iPmwYcOCkEvA9SyAe+d7fRrE2+N1TP13nqueLccfddRRoXSMtEZgySgi3F/4whfCO4Xwxueov4b4f+s/c79859wifp/5H8Se0gsyAyrZ0EIJBzMBkimgJAGI+nnHhw8fHtJv3XXXDWm47LLLBltmke1w7axjGy8KPAOtlSZx2tSjDDylPOPHjw9F6GR68Kv4QjKAn/zkJ2v3T7pzPj7Xv8dKH2UI4v/Vs2MBzkP6//CHPwwTw5ABM92jdEKvBZG//vrrs4suuqgmOBgcLXcp6nnssceCc4shE8CLjpHJOcTw+9VWWy18Zj+Q+99iiy1CMd8111xTM1DmGWbyAq6BEZGqBumIqJMGG264YXb22WeHl5KSD5wgjRbl7EhLoksiLaVrDA4Ap4IDJX1jJ9BZeKZdEXqcH20vyNQR4TZyVoXgP3mGKL/HfffZJ7vn7rvnt+E8bd99733nTNUForTGGmuEqBRRwsmSttxr/Bz4zP3qs45h4RnqndNz4Ri+s9a5WISOZ9H3WAwQWTIpN954Y9hGRoPIjXm7iZzJmOi3oP+L0f/F2/nMouuM75Ptuhd9B/az6P75LZ9p9/Dcc8+FoOH5558PC5/JDHBebIOME+8+ArjeeutlK6+8cvgPzqdrKBJcN6L9xBNPZCNHjvxAIFSPWrzznp9wwgnh/SQj9t///d+h1IgSD94ZpTlr0omMlNJdsI//h7XXXvsD6aPSAyCzhV/ZY489sm222aaY72HJKGUdffyZxhqHHHJIyGGS48Y4v//972fHHHNMqIeLj//BD34QcusYDi9zbGwYIVHOcccdV3tZWYjkqfPFaV5++eU1Z/XrX/86O+mkk8K2rbbaqvUs1YH0JM0QdMZ3ZnpXGlrhQIiESHuQA6CR3N25MNXDOUhvoIqEkgE5ZNJZa8H3GPbpmei/ulp0T3sLWvwS5UL8v30BaaH7wUa5pvgzJVe333pbEEhap7+ZZ0LZx+/CvefneK/lfYG+//77g8DXn4MFlFb8n7bpfuPj2BYfo9+IeF9MfC4+cw08VzJ0fOf/EQScOnOjq0SBhWvVb/V70BrYDmzTZ6HfkA7Af+k7a50nPreoPxf72caCGFFKglDSs4O2EUS6BB1kDon6qTLZbrvtQuZXpVz6H51b30X9NfQmlEDSkBj/2NH/cr9kBkg3ukTynp933nnZjBkzggATBPGek77AM8Yv/v3vf294bs5DJo73PA7ClC78/r777gtBA2vOy2dKenRMX6ZVSpRK6Bvx+OOPB6OjxTEvF9E9UeV3vvOdYBQyDG5TLzrE+4B9LBiXnBL7eYmpFiAyQtR0jp///OdBSKhTrGJETzoAkToZnQMOOCA4D5CjFqQnx0ts5BiA7/GzYR/f+cy26667LtQvKzNQDw4Dgda5+V1XhZ7GlZQGUVQLnKsvIH34L64BlLa6H0qnrrzyymDjr748LVt8wIBslTx6nJBvz4/KPvLhj4T6+ffe+0/Wkl8ypU9kfKlXN72DnhGZL0oK77333tCGhwwAxc5USSD41C9jj9g1Noodqw6a5872tmy7N6Dkk3ZFBD0d2Tf2p/eSYXNpO0HbJAIprjl+z5UeZOTIHNSfW+fR/hjtA/leMqr0JKHYnq6yZKSgo2s2jSle+VInwBBxyhgPuUy+Ezkye5AML4b9Wtgff2fBwNiuF49tvLBEHLy8KspiO8WP1MetsMIKDf8rdUgD7htRQWwpDeElZTvrenBoSieO0QL6HWgNvNQUh9J4i7rAT33qU6HXQ7xQZBo7nDJCunCv8YI906WT3h5UJ5B55X6vvuaa7NFHH81+k9v5MsOGhWNJs7feejukMVUq3/zmN4PdxmlpehbSnedG6Q92yQBABAJEs1dffXXo3slnqlAoEaTom5IA4DnhS4put9gP1wq05+BdRHDxj9ouSA9lXPQ+x4tgP/46ht+C/ANpQ5qSqeA9oMoP6s9lmqfUET2XjuDyIlHnhyFSXEbdEcajRcf+8pe/DI1vtD2+db7z29GjR4ftGDPbMErq43/1q19l99xzT63IaZ999gmGSZFWR3VdKaKXksgRx0YPhJtvvjlsV9oJ0pPIHMenxkoxevmPPPLIEIUqw8V2/Y/OFz8zYLscixxn2SJ64B4oHfnTn/4UioSJaKhXp7QK+0bkg+39J4/y83Thykjzyy67NHvwgQezxx5/LNtiyy2zUXkmlxIoPYO+vIcqgV1iMyyyU9bANtKd0sCHH344LESllD5SvP/Zz342dB2jBBLfwbPqKzob0Wvh+vF1tFug6J93Lf690oM6+yeffDJ8rkf3ecQRR3ygtAn753wsqq//61//Ghrk4XdVShD/p+kE+QMpLblxtbzxxhstebTXctRRR7XkgtOSG0nYp3V3yMUnrOfMmdOy1VZbtVx88cXhvP/6179a1lhjjZbHHnssfM+NNByXMqQ198madMlz2i0TJ05sufHGG1vuu+++ljyyb3n66adb8he9JXdwrb/qW7gunkce5bfkjixsi6+7I1588cWWPCIOxzZzfFfhOll0bfPmzWsZO3ZsSx41tSy33HIteUTe8uCDD9bupz04x9y5c1vyDEEl7LCM8AxZsK+zzz67ZZNNNmnJBb8lz5C13HXXXeG5vf322/PZXfy5p5CtTZ48ualz55F1eM/zDGXL9ddf35KLfcsyyyzT8swzzwRfi1+U/+vOtfLbSZMmteSZ22Dzb775ZngP99hjj+BfTPcpvdDzgnz6059u2WWXXVpef/31YJzQE06Pc+g8Dz30UMvuu+/eMm7cuGCAeYQajByjrAISJtYI+vDhw1vGjBkTXkiEZq211mrZcccdW/LopUfSvitMmzat5corrwximUdNwYliE/FzbAvuKxbW7jiujuBa+B/+L4+OWrbeeuuWPHpvOfzww4NT1fVyDR1dN8dY6IsNz1rvDs8IgcSPbLbZZiGDfNppp7XMnj077I/9Cb/pSTuUPWndkb08//zzLSuttFLLCSecEH7z2muvtay//vrhPf/73/8ejmG77q+rcN8XXnhhy8CBA1u++MUvthx77LEto0aNCpkJ/IvpPqUuus+NKywUudIQjKJ7yA0nFBOpKK0rcN76omPqiij6oo84dVUUJ+k/Ui9SUpoqzUkHutHRWIZ7p0ESXaMYcET9uPsaWvtTNUP1AHXVQKtgnhNm3l4RKftzxxeO4XN3bKcjSD9ab9Nrg0Z2FMkzzjuNt0hL/l/Xy7EdXTdFxNQFU6Tam9dtugbvDs+FZwVa0+vg5JNPzi6++OJszTXXzE499dRs0003DfvkTzi2J32L3mPo6Ly5gGd5gBMaImvAIN5zWs7TNof9KlKHrl4nNs67y4BODHLE+0DXRYr3uV7+w3SP0tfRn3/++WE4WgwvNjqMpzuNXfg955fjjJOJz/wPS/w5ZUgP4D75zEuOw+A7a15IhBKRZX974tRb8P/xc4ifIdvbE0F+q2ep4+Jz9SS0Zfjud78bBmyhtwIDyODc+D9dg65dadwWHGOhLzY8oxieLc+Ud4Y143Mg+LQ3whZoTNnRc+8Kej+U8ejoP7hOUAaYtc5BZp79erdkt12B3+JP8Nf6T87LdbLNNt19StePHmiwRIMpGiMRxTOQgxqH2ChMV8CBMYYCjX+IXqA7jksOS06Q7zguBhWirzGtmGn4R7/r7mRIObeFvtxgFwzEQ7c1GqDRkJQSHkahk5jynCW2rLsCv2VGOAbAodExttlVGzflonRegZwfLwGtRhnFju5VGD4vgo3WdBVsh1HD6NqDLXUHfo9z5pxy0hTTEr0zshitrhFlBheyMBtsgIwfsy4ygxtCTPc8+q5LjDkG0e+qyAPnoN8/1VumWpTKy8gBM+wsxZ9rrbVWKK7nBWCx0JuuIkcKKp7sKvyWc3BOVXHQ/ZPR7L797W+H0cXof409s98YbAW7Ofroo8PIcdSF77nnnqE4H7tU5rG7YItqQyN7N+lTKqHXy0BjEARedTp8luAb01WwIYhFvyvwW9km0dOhhx4a+sYzsh0RG/+DHbPoP/sCxEJLkYivq2jX1ldgL4gwtscYHfRZ5zvj/1MaJHvsjl1ib8A5ZH+mGpSu3BDHyMvAS6CXQ0bbEzleU13oMSCB7o4TxCZpRcy8CgyBytj0jHLH5Cc0VlSGlXVf26yEtDuC0dNI4Lub7mWl3h5IC0aGYyAqRtMjygcEv7v2gn0zuyT/0de2Z/oPP2ljcnB8DFOqIT67I4Sci6J7iusZDY15EWhLgmNle3+h6oSiCSrXw6JrM+830KMxHnN2MNocET4lmd1JH6qJmJyLMeQ5D3ZqqoGF3phWEBpEXoLYVXCgY8eODQ3uGN9hl112qW3vb4Hl3riG/sxw1KNo1iL/Pgiyuq/tvvvuodEx9fa0zO+O/VAaynNnjZ2b6mChNyZHIoxzpXizsyiKB2Yxo6h+2223zQ466KCakEF3HHVXUGM/7uvaa68N4+lDX15HffSoz6xZ2M/kJUwRzIAp2k56sq9qIMQ8N2yGEibmf6fYfsyYMSFNlH6dTRt+JzusfyYmbSz0xuQgfC+++GL43BUHiGMmSkKwDj/88FAPinCRaaAYVmLflwKrjMe8efNCH+25c+eGgaW4Hu3rCxAV/k8ixXcW0kLiQy8ESj722muvMNUr1whVFCPsSG2RuH9G4qRrJr2NmA4XW9Pz62z60HZEdt6Xtmj6Fwu9MTk4TEYnY/CZroggThlx+tnPfhaGt/3xj39cGx64vxyqMhbMFsbIe/TNRkTYxr6+gv/S/0rYgc8SKrattNJKYTpehoJFzJRBqjLcP5kiem7QWJQZGUlLZZQ6yz//+c/s8ssv79OMnul/LPTG5Ej8cIBqed8Z+D1TeI4bNy7MIc/CNgbgwTH3FwwuhWgy5C5wPX0N6aoInTkSGD8dgZdQSfD5TgO0L37xi2GUQn6nY6oMNklGbY899ghTPZOG2FSchs2iEianbbWw0BuTg9OLxb2z4szvzzzzzLCmtb0i2K5kGroLwoAzp2sW9fK6Lq6J6+nrxlhcz2OPPRYGgNlqq62CWHE9EioWrklrBO2BBx4Ic6DrOJYqRqHcN8+MdKBEhiqhc845p3Vv54ruOYcyXP1hl6b/sNAbk4PTw2kigAhKZ50g3egYEIfuUMy8hVPlXCxET32B7oH/5h6OPPLI0E6A+m/QNelzX8H90zaAngigNJFI6ZoUZdLi/KyzzgqjCFKVAtwbx1VNnPQsSZtll102dI1jhEUyS3rWzcLxshFXi1QLC70xrXz5y18OA9p0xQEy3ewyyywTWtn3F4gBjpzlz3/+c/bRj34023jjjWv7+gvSk8zGcsstV2tk1h4IGxNVMScA479zP/yGKoiqgSiTwWHNcsghh4Q1bRk6a6f8boUVVsi22GKLSmaaqoyF3pgcnObnPve5MChJV4Se1tD77bdfKBLFofYHOG6unYWMx2677RYEEuFEJLtyXz0B14XAS6ibERiudd999w1VD0Caci/9dQ/9BffMvXPffKaBJ0LNLHdvvPFGp8SacwwZMiTbbLPNwu84n6kGftLG5MiRUqQpx9oeOEoWRJS+6c8//3y24447hn39JfS6Zu6BOu5PfvKTYRuZD01k0h9IrGLRag+OI22ZtIqpqOkeyO9YOiNsqUD0TZqxkDY777xzNnv27PCMO4PsMj6fqQYWemNy1EhJxaQIeHvgJCU+t956a7b88suH7mH93coeIWAENVh11VXDumxIhMigMH0r07VyXyxVFyfsbZtttgklJExnS5p0Fmwd+65ipqmqWOiNyUGc//CHP4QBRRCTjhwojpJjcLyMRc4kJKChS/sLrodob/jw4eFzGeG6eR6k74YbbphNnDixlqZVF3run/YOG220USi+74ytkZ5kmphJsSsZBFNe/LSNaYWx6an3xAl2JCgIEWLPfOFPPvlktvbaa/e7CPH/OP5JkyaFYu+yiiJCrxIWBolB6JWpKmvmpafg+ZIWm266aajWeOWVV1r3dAz2On369NCepBkbN+lgoTcmBwdKUTHOsFlBwVnSP5wImolH9Bu29wf8PxkQ2gsw8ExHjpx7ZqG6gbWKdCnV4HtPovPpmvTfjf6H9FNEP2jQoOzll18Ov7M4/V/6bbnlluF5M1ZCW+lYD7+lQSRp28zxJh0s9MbkICIInVrNdyQoEnWK7alHHjFiRPjOecriRJWpYcIURvXj/rlvDV7TU0iI+C/WiA3/TVopHU3zkHaMf08bDA0+BB2lJfs5VovTvjpY6I3JwfEhQM0KNcczSQwz1VGMSp91OU/OUQa4zrPPPju04v79738fGnlRTN7Tw+SSJlRvnHLKKaEfN2l23333BcEynQMb47mxbL/99kHooRmbU3UTdm6RrxYWemNaYc7vWLDbg8zA008/nU2bNi0M6wpFd55cX5yJYVx+7pm+6kcffXSY4e7444/vlQaFtBlgQpbx48dnl112WejLjfCwdATXbWF6H6UDgs1gSLTHIGPWTKaJ31ISwAQ5fC5LhtR0Hz9pY3IQNoZpxWE24wA5hvpR1uuss05tW9GdJ5kY7vX1118Pos4EMgzCguCOHDkyRIjUiffkfXBuqgP0WUsz6cX1gsRe36sMz48qptVXXz2bM2dOmBqZdFQmoC2wbTJxtN8w1cJCb0wOjhIHiiNsRkw45sEHHwxOk4Z4ZYJrZ7AVokEiPAnoYostFkZO++Mf/9h6pCkaPCcySazpkUBjRRqENpMJ4neIPXbOZ1MdLPTG5OAo1eq8o8gIKC5lbm+KpBdZZJHWrcWG+9K90cUKh/+JT3yiJhCsGeuf+zLFRM+QBbFeeeWVs7vvvnu+Z9sW2HYs9iymGljojWmFxmI0sEPwOypSprj0xRdfDMX2HTnYoiBB53pnzZoVvsddCmmkxfdnnnnmA9EhoqD75PhG6BgJSFvHcUy86Hh9N23Dc9GzIX3XWGONWsv7+mdWD8+WRpEXXHBBh/Zt0sJP25gcnCSjhrHGIXYEDfHeeuut2njyZQEHz/UiqKypqiDKYzv16BLaRoKLIKtFPp/jRdtZc14yDcA+lZS0tfBf9N3X8aZ9SF8W7JShlxk4hwxqM2CzjPug52+qgYXemC7A/POII/3nyyJOOHctiDrXLdHQPSC4TCdbD/fKQnc8+m8z53680DCMYXdJDxr6cf6DDz44W3rppcP0vcyl3tZCo0CqDNztq/OsuOKKoXSJ9hZOO9MWFnpjugCN2agfHTp0aKmiUIn71ltvHSJ5eg4I9jEEMF3f6kWDyJv9hx12WDZhwoQPLDQII/PzyCOPhEZ9RPFnnnlm9uyzz4aqgKeeeqq2UBoSL8cee2w4XiULpnnIXNFGhHEJjGkLv1XG5CBsRJ4ITRzhCvZrYSQ5Gqwx4YqKvssA9yUh/cxnPhOi8JtvvjmILPfLuP2Mhf6lL30pHEMRL8Ppso/75j6JvBlroK1l8cUXD8dRrMz3JZdcMkzC0t6CUHG8hb7zUCKC3d5xxx3h+bYHz5G0Hjx4cEMbN+nit8qYVn7wgx8EcYJGThOxg1dffTXU59MQStvKAPekBYd/1llnZTfccENtGtjzzjsv22uvvUIaUOfLjHzrrrtudsUVVzTVbsH0PTw37JAhjNVOoi04lszdqFGjgsg7U1Ud/KSNaUXd5BqJPMgxUhTNZ8333tbxRUUROiP6nXPOOdkhhxySfe1rXwticeSRR4ZjuD+ic+bYZyQ7U1wYC4FZ7MiwtYdKZci0KcNnqoGF3pgc1UEjcBRl16NiTo6T0BMdSTTLgK5TDp572W233bLrr78+u/DCC7Nf/epXoaid4yiipx0Cje8YMVD3b4oHpS6UwDBPQXvo+bNm8TOtDhZ6Y3KIdOgXr25e9SCO6jpG4zNGxFNdJ4JZBiTwrFloX0DreNWP8x2B5zvbiRJpUPfDH/6wVprRLMoAsZA+NPK77rrrwnj3jLxHC3u2N8pUmc5BqcvCCy8cMqBKb6i3S575vHnzssmTJ4fnybGmGljojcnB6f3kJz/J/v3vf9cEMYb9iB8OEodKtyZEETorgmUAYb7yyitDlM80vOoX3wykFWmoiJH1/vvvn33nO9/JLr744mzvvffOzj333LCvUVqb5iGtGQqXRo30cAClKZm3ejjmkksuCb9rtN+kiYXemBycI9E84t1IfCTmZASIdOnWxHFsT7EIlAZ53/zmN0OpBYLQWUGO04ZZ8j73uc+FrnR0wUPozzjjjNDfHsExXYd0pm0JYx/QxgKU7vV2ybFk2PQ8nfbVwUJvTA6OD8fYXjE8jpGZ3RighHpRjmcbvy0ScvLdceT8HsFAFBCHzt4j/6202W677bKvf/3r4Zx8Z957BtKhuLnReXXdrCnaj+8j/mzeh+e0wQYbhDEK6CKpNKpPWz1TpanTsjpY6I3JwelJ1KCRAOEkKfokM8BgOXxnKYrD1HXQeI5Sh0b30Cy6N85BlUVninn5DYvOoUGF2MY1Ukd81VVXhXM2Srt4G/36+X137iVlSBfSdqONNgqlTYrqG0G68jyUmeWzqQZ+0sa0wtzsjOoGjQQIh4ojpdsZY4xrW1EcJtfB9dDdShmSosC1TZs2LfTVv+mmm2qi00jAtZ174T6YT0BY8OeHdCRN6AHCOPYa4KgRHEd7C0ZFhEY2btLEQm9MDk5wxx13DPWdjRygBIaGeBQ5M2+7jiuK+Eg8maOcqLkth98fUFzMdKoPPfRQ9o9//CPbfvvtwwiDXG8jlLbjx4+vjb3f1rEmqzXIo4sd9shS//xJP+x28803b91iqoLfHGNyECK6l+EcGwk3wsMyderUIDyLLrpozaFKlPobroOqB+q/ybAwJSnb+qNOlnTUf0qgGVr3/PPPz8aMGRMyTIrsG8F9zJkzJ2QGGOZVFKmUoiiQ1ksssUQYCpeIvpHIgxrikYZt2blJEwu9MTmIkUSkLUeJKNHXPh4Tn6UokWZ8PVtssUV255131oS2Lx076aj/1Hf+W9ey8847B8Eh09RWOtMugIlaqHsmA8bvObYoaV0kSBPSiAwR9qn0rkcir/YWbWWyTHr4rTEmB8dIv3HqOYmA6x0lIkOLZlrc0xCviOC41cBt9913Dw3e+My9SCj7AoQHQeG/+U8JtMbLp0QEIWeM9kbCzbXyHC699NJszz33nO/a65+Lef+5kz6M7cB0tUTueu4xfCdzRRUK6c4xphpY6I3JwelRlIzAIEj1AiQniRPFoRYRrlmZlE022SQ4f6J6vjdy/L0N/8k10V+eAXgk/vfff3/oGTBy5MiGYsN1E5lSP6+GY2zTfZj/gzTRcyUDSnUHPS6g3oZJOzKqf/3rX2sZJ1MNLPTG5OAUWSQm9YLC9pdeeinUGTNYThHhmhU1c70/+9nPspNPPjlEeXyXc1edfW/B/7CQnmSMvvKVr4RxB1gff/zx2bXXXpvdcssttSJkZQBYs5DGRx11VBhUh7YQwLk4RqJm3od0Vrpgl3xXg7xGz5j92AiL07I6WOiNycEpskgo650k3+lah/AwtngRwXFrQRiZL/973/tetscee4Somu04eq17CwRcIs51MBUu86WT6WB2PIYappW40ppjyHyw5trYv88++2Sf/exn57sfFjM/pA22yXq11VYL6c4IhFBvw6QfGS8gQ2Wqg98cY3IQPuqNGQa3kRPEkVIkSj97+tGXAe5phx12CKPSjR49OgyoIkHtTaGX8ABr/pN6eao8mB1PmQ0dR3ojUIjQ3/72tzDfP5kT6M3rTAXSkIXMExnRF154oWG6kc402oP6TIBJGwu9MTmIH7O0DRgwIDjN+ugRx0nXJfoqa1CdooN4ct1f/epXw3S0P/7xj2v3xT32FhJwFj7rP7ketSFQxC/RR4RoPHjWWWeF7nfstxg1h9KZEhLEHqEn/dgewzYG1mGCIY51+lYHC70xOQgNo92pK1e9E2QbDZkYcKQ+E1BUuGacOxP1fPnLX8723XffbObMmb3u4JU+rBEUroHPXI++81n7+Uy9PAP9XHHFFWEcAPbxLDjWtA1pp0VCz2BJ2HP9c+Y7ET82zH49J5M+ftLG5Eh82nKAOEmGcKUPPceVEbqzUSIBRbsHSkmY/Ib0N12H6hENNFT/jFWcr2i+tzN8pjhY6I3JwSnStU6RZ72TnDdvXjZr1qwQ9ZfVQXLdtEPA4RfpHrgepbvFp3swaM7cuXPDUm/DZGZJXxbSW8Jv0sdCb0wOzu/UU08NDhIHWB9ZUhyKc6RotN6BlgVdN/dapHsgXam7J93JiJiuQ7E8GVaqmeozTaQvwyKfe+65NbE31cBP2pgchI+55uUc6+uGZ8+eHcQfoS9r8TKOnQUxLVLdd3xNZc1EFQWqlrBPei6QeYohnSnSp2jfIl8t/LSNaYIpU6YEJ0k3sSKJpDEx2CfdPxk0x2JuhC3BmCZgVLyPfvSj2dChQz9QJGpMUWC+eWYuZHZAC70RtgRjchBvhhDFOVKXKTFnzTJ9+vQwFShib0xRoSvlUkstFey1EXFm1Y3xqoOF3pgcHB/DxeII49bJghb37MORxtuNKRK0cUDIEfp6O6Xuni6W++23X6i/dxVUdbDQG5NDJI9jxFGqEVMc8TD8LcO3qg+yMUUE+yWix17rG40i7Ng2DR+xY0f01cFCb0wrOD8cIeu49TfbaHWvIk/XfZoiQ1965jXAZmOwXcS+rL1GTNexxzImh+jmqaeeqjlCCT3Oke84TkaVY7sjelNkmOufvvSvvfZa65b3wcbfeOONbOrUqcGG48ysSRsLvTE5OL2xY8eGPsZE9Ip6JPRER4zBbkzR0aA59RE9mdjx48dn48aNC6LvDGt1sNAbk4PTW3jhhYMDZEH4KaJnzVzujJjHYDnsM6bIUHTPdMvMzRCD7dKYFJuWfZtq4CdtTA5Oj7p41nEdJg6RIlAaMBHROwoyRQc7pYcIIz3WoxIq7Dq2c5M2FnpjcuQAJfY4Qok6I+IRDeFAKf5knzFFBJtlKlpKpxr1pZeN246rhYXemByc3q677lprcCcniMDTEI+IfsCAAfPtM6ZoYJvYKaPjzZw5s3Xr+7CPIXK32Wab8N1F99XBT9qYHCIhhF6RjurpWTQJCJkAtivSN6aIYKvM78/ENjHYLUPkbr755rbjimGhNyYHQadYEwcokccRslZET70n340pKtguQs+gOTQgjcF2KbbnGH031cBCb0wOok40r4heC9tpdU9xKPWebsBkigw2K6GnJErCjt3KxoUj+upgoTemlSuvvDL0P44dII6TQUaY0AYn6XpNU3SwWfrSz5gxo5ZhZQEGy/nb3/4Wvlvoq4O9ljE5OL4bbrghiLqQI6QIlDpPO0ZTBojiiegpiZo3b17NbrHxyZMnB6HXd1MNLPTGtILjU6RDcaccJHX0gwcPDtE8+1XHaUwRwU4Z3An7jRvk8Z1SKdqb8Fn2bdLHQm9Mjpwg4ChjJ0iUz8x1oMyAMUUF+2XMB+rlGQYXe1UmFRsn4se+2WaqgZ+0MTmKcDTOvRwjqI4e4gyAMUUFe8VWaZCHHasUijWNSm3H1cJCb0wrxx13XKiLbxTR04feztGUBQn9rFmzwlpVUSNGjMj22msvR/MVw0/bmFYYNQzqhZ6W+Iw0pgjfTtIUFeyWZdCgQWFNQ1LW2C7LggsuGCa9idugmPSxxzImJxZx6jD1HWiMRytmkCM1psgsvvjiodEdtgvYNTYt243705v0sdAb0wpRDvXzOEE+K3Kn6F71mnEGwJiiIZvVSI7MvCibJQPL/nfffTfYt6kOFnpjcnCGJ510UpipTmIPiPs777wTiu7ZbqE3ZYBGpQg9jfEk6oj8hAkTsosuuih8ty1XBwu9MTk4w1deeaXW4h4Q+bfffjtEQtR5sp1tHGNMkUHoGbZZo+MJbJdZ7dimYnyTPhZ6Y3JwfCrajB0jGQCKOqnzJMp3FGSKDOLNQtE9NksdPTarJcZCXx0s9MbkyAki9iAnKKFX0T3bFfEbUzSwY2yWha6iCD2fsVvtU2NT23F18JM2JgcHSPe6RRdddL5IR44Rp6nMgNbGFBVslNHxGO9eGVTsmBEemZOe77Gdm7Sx0BvTCo3xFlpooVpdPAv9kPlOUagiImOKCjaqhqSUQtHGRILOevjw4dmoUaPCcc6wVgcLvTE5KsaUmMsR0gqfNS2YcZR2jqYMYKcIPYM9yZ7JABDVq62JM63VwUJvTI5a0iPmiohwhhR9xmODK9o3pohgm8qQ0hhPRfd8l9jzXdtMNbDQG5ODA3zuuedCn3lFQBAPf4vIOwoyZQG71bgQ2C0Lgz9NmTKlliEw1cBCb0wOTm/s2LHBESLqfMdBEhExPriEH7FXMb8xRQT7pIiefvTYsbrYkZl9+umns0svvTR8th1XBz9pYxqAY2QhIqKBnjFlAXHHdmlXgpjTIE8ZVVNNLPTGNABniZNkCFG61hlTBiiel+3SlY41mVW2mepioTemFQ0kooiItbYZUwawVRYEX1VORPSgenqJPp9NNbDQG9PKTjvtFFrYEwXJKU6fPj0bOHBg6xHGFBuJOALP4E/YMplVYB9z0W+88cbBtp2BrQ4WemNycHq77LJLmAxEkQ5OEmiUZ0wZUNdQRJ22JYroWWPPQ4YMyUaOHBmOsdBXBwu9MTmIO45QEZBEnpm+6I9sTBlQVzpQRE8dPSD+ccTvDGx1sNAb04rEnuFuAUfIhDbUdRpTBhBwLZROYc8Sf7apWx3fFf2b9LHQG9PKtddeGwbIUQM8HKG7JpmyIXtdYoklwpqxIADBnzZtWnbHHXeEYxTdm/Sx0BuTg9O77rrrgrAj8CrWfO2110J/ZGPKgkSckihF78DnyZMnZ3fffbeFvmJY6I3JUZE9Ao8TlINUdGRMGUC8WbBfFhrfMQOj9oGqpkx1sNAbk4OgI+xqdY9TZNvLL78chhI1pgxQGoXdahESf20jQyvhN+ljoTcmgkltcIYSehXhG1NGiN7ViwRbjtuduDFedbDQG9PK8ccfH4YNJaLHMYJE35gyQvuSeFKbNddcMzvooINCyZXtujpY6I3JQdxXXHHFIPBygkQ+LEz3aUwZwY4VzfOZkR+HDRtWa4tiqoGF3pgchJ4lrrtUNI/wG1M2EPi4Bwlgz4wNYZGvFhZ6Y3JwiIwgxlr96HGUGifcmDJCVRSt7lUVhdCz8J2MrakGFnpjcnB+Y8aMyWbNmhWcIN8ZPIfGee5Hb8pKnFEl8/rss89m5513XojqJf4mffykjcnBCb7yyiu1Ik3WFHniEN3v2JQVbFfVUmReaZjHIFCUXDmirw4WemNaQdypj5fYyxnquzFlgzEgKJnChlmI8JVxtV1XBwu9MTmKboh65BSJ5onqPamNKSuqi5eo0+pe0T2LqQYWemNycIgf+9jHgkOU6NO1jgiIeb2NKTvYNiVWgwYNCvYu8TfpY6E3Jgend+KJJ4Y+80Q6CDxOEdGX8BtTNiiNUvc61muttVZ28MEH16J6Uw0s9MbkIOwqole/eTtCU3YGDhwY6uiBzKyK8rFtR/TVwUJvTA7iTp08DfBwgqwhHmzEmLKhBqXYNCKPjQOfnZGtDhZ6Y1p56aWXasKOE5w3b16op3c/elNWEHkidxZsm2XmzJkW+YphoTcmB4d46qmnhn7GgGNkGwvRjzFlJC6qJ7p//PHHs/PPPz9st11XBz9pY3IU9cSRjrYZU1Zk07Jj23Q1sdAbY0yiLL744mEOB1NtLPTGGJMo9RG9qSYWemNyaKS0/fbbh5HDcIwsNMTDQbou05QV9R7BnrFx5qLfcMMNQ729qQ72YMbkIOa77LJLGAVP9ZizZ88O+xZddNGwNqbMYNNDhgzJtt122/DdYl8dLPTG5Chyx/kR+bi406RGPNojuKSqOvhJG9MKAo/zkwPUGtE3puww8iNiD2RiHdFXBwu9MTmI+dVXXx3q5fmME2ThsyMfkwLY8/Tp07M777yzZt+mGtiDGZODmP/5z38OXZGIdtSIyZhUQNinTp2a3XvvvcHebePVwUJvTI6iG+owKcJ3/bxJDcSdse41aZNtvDpY6I3JwelRh/nOO+9kCyywQBB7Y1ICoWexwFcPC70xrSDyOEEaLOEQjUkJ2ptQXK8Gea6jrw72ZsbkIPDHHntsNmjQIEc9JkkQ9nXWWSc74IADwnfbeHWw0BuTQ7Sz0korhYiHYntHOyY11PjuYx/7WFhb6KuDhd6YHHWpo7GSYBsUWfS5xjJcp+kfZBOsWSitIiPrNijVwkJvTCvUXRL1qH4ex0jUIyEtKrETNyYG+1UvkvqR8Ypu16bnsGcwJgdHeNxxx2Wvv/56rbGSWt8XWUBx1vQWkPM2JgbblaBjzxMmTMguvPBCZworhp+2MTkI/YwZM0LRPcIJQ4cOzY466qhs8ODB4XsR4bpZvvKVr1jszQdYf/31s9122y2IPQv2PXPmzGAzjuirg4XeGGOMSZgP5bk6Z+uMMcaYRHFEb4wxxiSMhd4YY4xJGAu9McYYkzAWemOMMSZhLPTGGGNMwljojTHGmISx0BtjjDHJkmX/H9simWarGuM3AAAAAElFTkSuQmCC"></strong></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept an indication of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p><br>vertical asymptotes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p>horizontal asymptote <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>uses a valid method to find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the local maximum point <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> For example, uses the axis of symmetry or attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<p><br>local maximum point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)A0</strong></em> for a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> and coordinates not given.</p>
<p><br>three correct branches with correct asymptotic behaviour and the key features in approximately correct relative positions to each other <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> (this can be done at a later stage).</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempts to complete the square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mfenced><mrow><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>1</mn><mo>=</mo><mo>±</mo><msqrt><mn>4</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mo>±</mo><msqrt><mfrac><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac></msqrt></mrow></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msqrt></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>even if <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo></math> (in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo></math>) is missing</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>±</mo><msqrt><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></msqrt></mrow><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>3</mn></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><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> is rejected <em><strong>R1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for concluding that the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> must have the ‘<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo></math>’ sign.<br>The <em><strong>R1</strong> </em>may be awarded earlier for using the condition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>3</mn></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><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>
<p><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> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find <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></math> <em><strong>(M1)</strong></em></p>
<p><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><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mfrac><mrow><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow><mn>2</mn></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><mrow><mn>2</mn><mfenced><mrow><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p>attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mn>2</mn><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mfrac><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mn>2</mn></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mn>2</mn></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mfenced><mn>2</mn></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award all available marks to this stage if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>a</mi><mo>-</mo><mn>7</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to solve their quadratic equation <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>4</mn><mfenced><mn>2</mn></mfenced><mfenced><mn>7</mn></mfenced></msqrt></mrow><mn>4</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mo>±</mo><msqrt><mn>72</mn></msqrt></mrow><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award all available marks to this stage if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msqrt><mn>2</mn></msqrt></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>3</mn></math>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mn>18</mn></msqrt></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>p</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>18</mn></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was generally well done. It was pleasing to see how often candidates presented complete sketches here. Several decided to sketch using the reciprocal function. Occasionally, candidates omitted the upper branches or forgot to calculate the <em>y</em>-coordinate of the maximum.</p>
<p>Part (b): The majority of candidates knew how to start finding the inverse, and those who attempted completing the square or using the quadratic formula to solve for y made good progress (both methods equally seen). Otherwise, they got lost in the algebra. Very few explicitly justified the rejection of the negative root.</p>
<p>Part (c) was well done in general, with some algebraic errors seen in occasions.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>attempts to express the integral in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><msup><mi>u</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>d</mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><msubsup><mfenced open="[" close="]"><msup><mi>u</mi><mi>n</mi></msup></mfenced><mn>1</mn><mn>2</mn></msubsup><mo> </mo><mo> </mo><mo>(</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><msubsup><mfenced open="[" close="]"><mrow><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></msubsup><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone the absence of or incorrect limits up to this point.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><msup><mn>1</mn><mi>n</mi></msup></mrow><mi>n</mi></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correct substitution of <span style="text-decoration:underline;"><strong>their</strong></span> limits for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> into their antiderivative for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> (or given limits for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> into their antiderivative for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><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><mo>=</mo><mo>∫</mo><msup><mtext>sec</mtext><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mi>x</mi><mo> </mo><mtext>sec</mtext><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>applies integration by inspection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><msubsup><mfenced open="[" close="]"><mrow><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></msubsup></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A2</strong></em> if the limits are not stated.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><mfenced><mrow><msup><mtext>sec</mtext><mi>n</mi></msup><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>-</mo><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correct substitution into their antiderivative.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>a vector normal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Pi _p}"><msub><mi>Π</mi><mi>p</mi></msub></math> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A</strong><strong>1)</strong></em></p>
<p><strong>Note:</strong> Allow any scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, including <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 0 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>attempt to find scalar product (or vector product) of direction vector of line with any scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right) = 5"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>∙</mo> <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> <mo>=</mo> <mn>5</mn> </math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ 0 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 5 \\ {{\text{sin}}\,\theta } \\ {{\text{cos}}\,\theta } \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ { - {\text{cos}}\,\theta } \\ {{\text{sin}}\,\theta } \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <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> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>) <em><strong>A1</strong></em></p>
<p>(if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> is the angle between the line and the normal to the plane)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\alpha = \frac{5}{{1 \times \sqrt {25 + {\text{si}}{{\text{n}}^2}\,\theta + {\text{co}}{{\text{s}}^2}\,\theta } }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo>=</mo> <mfrac> <mn>5</mn> <mrow> <mn>1</mn> <mo>×</mo> <msqrt> <mn>25</mn> <mo>+</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </msqrt> </mrow> </mfrac> </math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\alpha = \frac{1}{{1 \times \sqrt {25 + {\text{si}}{{\text{n}}^2}\,\theta + {\text{co}}{{\text{s}}^2}\,\theta } }}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>×</mo> <msqrt> <mn>25</mn> <mo>+</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </msqrt> </mrow> </mfrac> </math>)</span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{cos}}\,\alpha = \frac{5}{{\sqrt {26} }}"> <mo stretchy="false">⇒</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo>=</mo> <mfrac> <mn>5</mn> <mrow> <msqrt> <mn>26</mn> </msqrt> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\alpha = \frac{1}{{\sqrt {26} }}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>α</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>26</mn> </msqrt> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>this is independent of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span>, hence the angle between the line and the plane, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {90 - \alpha } \right)"> <mrow> <mo>(</mo> <mrow> <mn>90</mn> <mo>−</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> </math></span>, is also independent of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> The final <em><strong>R</strong></em> mark is independent, but is conditional on the candidate obtaining a value independent of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta "> <mi>θ</mi> </math></span>.</p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>attempts to calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><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><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> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><msup><mi>m</mi><mn>2</mn></msup></math> <strong>A1</strong></p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>m</mi><mn>2</mn></msup><mo>≥</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>-</mo><msup><mi>m</mi><mn>2</mn></msup><mo><</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> <strong>R1</strong></p>
<p>so <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 <strong>AG</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> is parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> is perpendicular to the normal of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> and so)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong>R1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>4</mn><mo>-</mo><mi>m</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>6</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>since there are no points in common, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> does not lie in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> </p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> into <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><mfenced><mrow><mo>≠</mo><mi>p</mi></mrow></mfenced></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mfenced><mrow><mo>≠</mo><mi>p</mi></mrow></mfenced></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≠</mo><mo>-</mo><mn>5</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>=</mo><mn>3</mn><mo>-</mo><mn>3</mn></math> <em><strong> A1</strong></em></p>
<p>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>. <em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to set equal to a parameter or rearrange cartesian form <em><strong> (M1)</strong></em></p>
<p><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><mo>=</mo><mi>λ</mi><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mi>λ</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mn>3</mn><mo>-</mo><mi>λ</mi></math> OR <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><mfrac><mrow><mi>y</mi><mo>-</mo><mn>0</mn></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><mi>z</mi><mo>-</mo><mn>3</mn></mrow><mrow><mo>-</mo><mn>1</mn></mrow></mfrac></math></p>
<p>correct direction vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> or equivalent seen in vector form <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</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><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> (or equivalent) <em><strong> A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="bold-italic">r</mi></math> is omitted.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use the scalar product formula <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>∙</mo><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><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><mn>6</mn></msqrt><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt><mo> </mo><mi>cos</mi><mo> </mo><mn>45</mn><mo>°</mo></math> <em><strong> (A1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for LHS and <em><strong>A1</strong></em> for RHS</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo>=</mo><mfrac><mrow><mfenced><mo>±</mo></mfenced><msqrt><mn>6</mn></msqrt><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>a</mi><mo>+</mo><mn>2</mn><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><mn>3</mn></msqrt><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for LHS and <em><strong>A1</strong></em> for RHS</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>a</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>a</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p>attempt to solve their quadratic</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>8</mn><mo>±</mo><msqrt><mn>64</mn><mo>+</mo><mn>8</mn></msqrt></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>8</mn><mo>±</mo><msqrt><mn>72</mn></msqrt></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mo>-</mo><mn>4</mn><mo>±</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to equate the parametric forms of <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> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close><mtable><mtr><mtd><mn>2</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>t</mi><mi>a</mi></mtd></mtr><mtr><mtd><mi>λ</mi><mo>=</mo><mn>1</mn><mo>+</mo><mi>t</mi></mtd></mtr><mtr><mtd><mn>3</mn><mo>-</mo><mi>λ</mi><mo>=</mo><mn>2</mn><mo>-</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempt to solve equations by eliminating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>2</mn><mi>t</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>t</mi><mi>a</mi><mo>⇒</mo><mn>1</mn><mo>=</mo><mi>t</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>a</mi><mo>⇒</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>λ</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math></p>
<p>Solutions exist unless <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> This <em><strong>A1</strong></em> is independent of the following marks.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mstyle displaystyle="true"><mi>a</mi></mstyle><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mn>2</mn><mo>-</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mfenced><mrow><mfrac><mi>a</mi><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>2</mn><mi>a</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> for any two correct coordinates seen or final answer in vector form.</p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>no unique point of intersection implies direction vectors of <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> parallel</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> This <em><strong>A1</strong></em> is independent of the following marks.</p>
<p> </p>
<p>attempt to equate the parametric forms of <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> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close><mtable><mtr><mtd><mn>2</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>t</mi><mi>a</mi></mtd></mtr><mtr><mtd><mi>λ</mi><mo>=</mo><mn>1</mn><mo>+</mo><mi>t</mi></mtd></mtr><mtr><mtd><mn>3</mn><mo>-</mo><mi>λ</mi><mo>=</mo><mn>2</mn><mo>-</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempt to solve equations by eliminating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>2</mn><mi>t</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>t</mi><mi>a</mi><mo>⇒</mo><mn>1</mn><mo>=</mo><mi>t</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>a</mi><mo>⇒</mo><mi>a</mi><mo>-</mo><mn>1</mn><mo>=</mo><mi>λ</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mstyle displaystyle="true"><mi>a</mi></mstyle><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mn>2</mn><mo>-</mo><mfrac><mn>1</mn><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mfenced><mrow><mfrac><mi>a</mi><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mi>a</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>2</mn><mi>a</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>a</mi><mo>-</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mfenced></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> for any two correct coordinates seen or final answer in vector form.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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,iVBORw0KGgoAAAANSUhEUgAAAboAAAEiCAYAAACPwherAAAetUlEQVR4Ae2dMWskSZaAi/4XstsYZ7yhfVlydkDWOmfIGBrGWGMtCdpYGEecIWg42jgYEGyfe9QP0IE8OQ0CGWs08o7ppp1BLI2gl2YQsbxSPymUlVUVGZURGfHiSxBVyorMePG9rPwqIiOrZo4FAhCAAAQgYJjAzHDbaBoEIAABCEDAIToOAghAAAIQME0A0ZlOL42DAAQgAAFExzEAAQhAAAKmCSA6o+n98uWLOzs7czc3N0ZbSLMgAAEIhBFAdGGcqiklgnvz5s3iT0S3v7/vLi4uqomfQCEAAQiMTQDRjU10wv2J5F6+fOkODw8fojg+PnYvXrx4+J8nEIAABFojgOgMZVylJsLTRXp3z5490395hAAEINAcAURnJOXX19cLob19+/ZJi6SHh+ieIOEfCECgMQKIzkjCZbhShOZPPpGenazzhzKNNJdmQAACEAgmgOiCUZVbUOQmQlOpidjkT3tz3V5euS0hMghAAALjE0B04zPNvkeZVSmSk1mW/qKi+/Dhg7+a5xCAAASaIoDoDKR7Pp8vROcLTZ5rD89AE2kCBCAAgWgCiC4aXTkb6vU5PyKdbenLz3+d5xCAAARaIYDoDGRapaZN0aFMbhRXIjxCAAItE0B0BrJ/dXW1GKYUscl1Ork2J+tYIAABCEDA8TM9Vg4CkZzcME4vzkpGaQcEIDAWAXp0Y5FkPxCAAAQgUCQBRFdkWggKAhCAAATGIoDoxiLJfiAAAQhAoEgCiK7ItGwXlExG8b8KbLu9sTUEIACBugkgurrztxS9zLqczWaLiSlLL7ICAhCAQIMEEJ2hpMuXOMtvz4no5E9+0YAFAhCAQOsEEJ2hI0C+vFklJ48yhMkCAQhAoHUCiM7IESDX5HzJ6fPuFz0baS7NgAAEIBBMANEFoyq7oHzfpcrNf5ShTP8Xx8tuBdFBAAIQGJ8AohufafY9yrU4X27d5/weXfaUUCEEIFAQAURXUDJiQ9nf318rOhEftxvE0mU7CECgdgKIrvIMym/RdXtwff/L0CYLBCAAgRYJILqKs969naBPcP46bjeoONmEDgEIRBNAdNHoyt1Q5VZuhEQGAQhAIB8BRJePdbaaEF021FQEAQhUQADRVZCkoSEiuqHEKA8BCFgmgOgMZhfRGUwqTYIABKIJILpodOVuiOjKzQ2RQQAC+QkguvzMk9eI6JIjpgIIQKAiAoiuomSFhoroQklRDgIQaIEAojOYZURnMKk0CQIQiCaA6KLRlbshois3N0QGAQjkJ4Do8jNPXiOiS46YCiAAgYoIILqKkhUaKqILJUU5CECgBQKIzmCWEZ3BpNIkCEAgmgCii0ZX7oaIrtzcEBkEIJCfAKLLzzx5jYguOWIqgAAEKiKA6CpKVmioiC6UFOUgAIEWCCA6g1lGdAaTSpMgAIFoAoguGl25GyK6cnNDZBCAQH4CiC4/8+Q1IrrkiKkAAhCoiACiqyhZoaEiulBSlIMABFoggOgMZhnRGUwqTYIABKIJILpodOVuiOjKzQ2RQQAC+QkguvzMk9eI6JIjpgIIQKAiAoiuomSFhoroQkmNUe53d370g1PmD497p+76TvbffX3H7Z2+d4uXxqiefUAAAhsJILqNiOoroCfb+iKvNeI7d3s9d0e7Owvh7Rydu89PmvIvd336ZzfbfeXm109feVKMfyAAgSQEEF0SrNPuFNFNw//u0/+5VwvZ/cmdXP7zWxB37vbytdvd/cWdf/o6TWDUCoHGCSA6gwcAopsqqXfu9v3f3cHOzM12X7vL2zu3kN/eX938I5KbKivUCwFEZ/AYQHRTJvWr+zj/i9uZ7bjd//wfd/rzX93pe4Yrp8wIdUMA0Rk8BhDd1En9p7s8+ZObzZh4MnUmqB8CQgDRGTwOEN3USf3qPp3/4nZnj0OYU0dE/RBomQCiM5h9RDdtUu8+zt3PP/+ve//utduVIcyTd+522pCoHQJNE0B0BtOP6CZM6u07d3LwX4uJKM7pEOaee3X+kXvnJkwLVbdNANEZzD+imyipdx/d+d/+9nTyye0/3OnB92628xdmXk6UFqqFAKIzeAwgutxJ/eo+Xc7dycGeOzr/vVP5nft8/srtyPW6nQP3+t0nenYdQvwLgdQEEF1qwhPsH9FlhH733p3u3X8jyj33HzzZeZIT0enfw9eDZYyTqiDQMAFEZzD5ekI12DSaBAEIQGAwAUQ3GFn5GyC68nNEhBCAQD4CiC4f62w1IbpsqKkIAhCogACiqyBJQ0NEdEOJUR4CELBMANEZzC6iM5hUmgQBCEQTQHTR6MrdENHlzc2HDx/cxcWFu76+dl++fMlbObVBAAIbCSC6jYjqK4Do8uRMpHZ4ePh428Bs5l68eOFEfCwQgEA5BBBdObkYLRJENxrKtTt68+bNE8kpd5EdPbu16HgRAlkJILqsuPNUpifcPLW1W4ty7ns8OztrFwwth0BhBBBdYQkZIxw98Y6xL/axmoBy7nucz+erN+QVCEAgKwFElxV3nsr0xJuntnZrkSFKZd19lMkpLBCAQBkEEF0ZeRg1Cj3pjrpTdrZE4O3bt72i29/f5xrdEi1WQGA6AohuOvbJakZ0ydAu7bg76/K7775j1uUSJVZAYFoCiG5a/klqR3RJsPbuVHpvP/744+Ieul9//XXRw5P76VggAIFyCCC6cnIxWiSIbjSUa3ckQhPWOsNSbimQ63bHx8drt+NFCEAgLwFEl5d3ltoQXRbMi5vFu/fM6b11Nzc3eYKgFghAYCMBRLcRUX0FEF36nInIhLNMSPEXXS/CY4EABMoggOjKyMOoUSC6UXH27mxdz23da707YyUEIJCUAKJLineanSO6tNw39do2vZ42OvYOAQh0CSC6LhED/yO6tEmUbz0Rxuu+vFlvOxDpsUAAAtMSQHTT8k9SO6JLgnWxU51ZKSJbt+iMTK7VraPEaxDIQwDR5eGctRZElw639uZC7pWjV5cuD+wZAkMIILohtCopi+jSJCq0N6e106tTEjxCYFoCiG5a/klqR3RJsLohvTmNgF6dkuARAtMRQHTTsU9WM6IbH+3Q3pxGwAxMJcEjBKYjgOimY5+sZkQ3PtqY3pxGwX11SoJHCExDANFNwz1prYhuXLyxvTmNQnt1m2ZqankeIQCBcQkgunF5FrE3RDduGrbpzWkk+tt1IbM1dRseIQCBcQggunE4FrUXRDdeOrbtzWkkY+1H98cjBCAQTgDRhbOqpiSiGy9VY/TmNBrd18XFha7iEQIQyEAA0WWAnLsKRDcO8RTX1uSHWuVPengsEIBAHgKILg/nrLUgunFwp5gteXV1tfieTOndsUAAAnkIILo8nLPWgui2x53yW01k9qX8YCtf+Lx9ntgDBEIIILoQSpWVQXTbJyyljORXDyRHx8fH2wfKHiAAgY0EEN1GRPUVQHTb5SzH8CK3G2yXI7aGwBACiG4IrUrKIrr4RMkkkRwTRvR2A6mLBQIQSEsA0aXlO8neEV08dr0FQHp1qRe5zUByxcSU1KTZf+sEEJ3BIwDRxSVVJofIJJGcX9WV8lpgHAW2goA9AojOXk4XvQSRHcswAjI5RLjJZJFci05MySnXXG2jHgiUQoCzYSmZGDEOenTDYeoEFJkkknvJOVyau23UB4ESCCC6ErIwcgyIbhjQXBNQVkWl9cuwqTxngQAExiWA6MblWcTeEN2wNJTQo9Ib1Lm3bljuKA2BEAKILoRSZWUQXXjC9BpZCYLRrxzjp3zC80dJCIQQQHQhlCorg+jCE/by5ctivo7Lv7eOIczwHFISApsIILpNhCp8HdGFJU2HLM/OzsI2yFBqykkxGZpHFRCYhACimwR72koR3Wa+OmRZ4rR+vc0hx03rm0lRAgL1E0B09edwqQWIbgnJ0oqShiy7wekQJrMwu2T4HwJxBBBdHLeit0J069NT4pBlN2JmYXaJ8D8E4gkgunh2xW6J6FanpuQhy27U+gsH8p2YLBCAQDwBRBfPrtgtEV1/amRIsOQhy27UtcXbjZ//IVAKAURXSiZGjAPR9cOssYekPVARNAsEIBBHANHFcSt6K0S3nB6dtl/CjeHL0a1fo9cU5ZEFAhAYTgDRDWdW/BaI7mmK9Od35EdOa70RW26DkLzyrSlPc8t/EAghgOhCKFVWBtE9TZhKQoYBa1245aDWzBF3CQQQXQlZGDkGRPcIVIf9Svr2k8fohj3TWw5KvMl9WEsoDYG8BBBdXt5ZakN095hrvi636kBRcXO9bhUh1kNgmQCiW2ZS/RpE55yF63KrDkQdiuV63SpCrIfAUwKI7ikPE/+1Ljq9/0w41HxdbtXB6F+vE6GzQAAC6wkguvV8qny1ddHp77pZ/kYRvV7H/XVVvkUJOjMBRJcZeI7qWhadTDqR9ovsrC8ttdV6LmlfWgKILi3fSfbequha7OW00Hud5E1EpaYIIDpT6bxvTIuik2tx8rM2rf20jfXrkQbfnjRpAgKIbgLoqatsTXR6shfJWZx8sul40Rmm0n4mp2yixestEkB0BrPemuhkQoa02fLkk02HqQheGAgLET8LBCDwSADRPbIw86wl0ek1KgvffLLtASiil9zX+MXV27ad7SGwjgCiW0en0tdaEZ1+S0gLMyxDD0X9KSKYhBKjXAsEEJ3BLLcgOnovqw9c7eW2PJS7mg6vtEgA0RnMunXRqeS4HtV/8OrkHDkOkF0/I9a2RQDRGcy3ZdHpbQRIbv2BK7KT399rdSbqejq82hoBRGcw41ZFp5JjGn3YQevzkucsEGiVAKIzmHmLouOkHXegKreaf109ruVsBYFHAojukYWZZ9ZEpydrhuHiDlH9XT6Ge+P4sVX9BBBd/TlcaoEl0SG5pfRGrWACTxQ2NjJCANEZSaTfDCuiQ3J+Vrd/7stu+72xBwjUQwDR1ZOr4EgtiA7JBad7UEFush+Ei8JGCCA6I4n0m1G76JCcn83xn+sN5Xx7yvhs2WOZBBBdmXnZKqqaRYfktkp98MbILhgVBQ0QQHQGkthtQq2iQ3LdTKb9H9ml5cveyyGA6MrJxWiR1Cg6JDda+gftCNkNwkXhSgkgukoTty7s2kQnswHlHjnuk1uX1XSvIbt0bNlzGQQQXRl5GDWKmkTnT3mXXh3LNASQ3TTcqTUPAUSXh3PWWmoRnU515xs7sh4eKyvzZcevlK/ExAsVEkB0FSZtU8g1iE5PqoeHh46T6qaM5ntd88KHj3zMqSk9AUSXnnH2GkoWnUhN5CYxykmVpTwC/nAyH0LKyw8RDSeA6IYzK36LUkV3c3PjpKcg8cmwJUu5BHzZSd5YIFAzAURXc/ZWxF6i6K6vrxezKiU2OYmylE9AZcds2PJzRYTrCSC69XyqfLU00XHCrPIwWgStH1CQXb05JHLnEJ3Bo6Ak0TG5of4DTG/mpzdefy5bbQGiM5j5EkQnkxj0epzIjkkNdR9oXF+tO3+tR4/oDB4BU4tOh7skjrOzM4OE22wSM2bbzLuFViM6C1nstGFK0elN4HJNR4THYo8Aw9H2cmq9RYjOYIanEJ3/aZ+bwA0eVJ0mSU9djrP9/X3HV7d14PBvcQQQXXEp2T6g3KLzhyq5P277/NWyh6urq4cv4+aWkVqy1maciM5g3nOKjqFKgwfQgCZJb04nHb19+3bAlhSFQD4CiC4f62w15RCdf4I7Pj5mVmW27JZXkQxbyzEgxx3D1uXlh4i4j87kMZBadHJ9RiabyB+zKk0eQlGN8nv3XLeLQshGiQjQo0sEdsrdphKd3EulX8gsw1WczKbMcpl1+9dr+RBUZo5ajArRGcx6CtFpL072zYQTgwfNiE3yby5nWHtEsOwqmgCii0ZX7oZjio5eXLl5Lj0ymZwix6LcgsA9laVny3Z8iM5gfscSHb04gwdH5ib5Q5mMBGSGT3UPBBDdAwo7T7YVnT+jkmtxdo6LqVrif5mAHE8ySsACgZwEEF1O2pnqihWdnJB0uIkZlZmS1VA1MkIgxybHVkNJL6SpiK6QRIwZRozo5Jst5AQk28oEAj51j5kR9qUE/NECmcHLcaZkeExJANGlpDvRvoeITk48essAkwYmSliD1fr33HEbQoMHQOYmI7rMwHNUFyI6GabUb6GXnhwTBXJkhjp8AvTufBo8T0kA0aWkO9G+14lOBCdS02FKkR3DRxMlimoXBPzeHR+4OChSEEB0KahOvM9VopPrcDI8Ka/LcCX3Nk2cKKp/IOAPocvMTI7NBzQ8GYEAohsBYmm76IpOThoqOHnkJ1VKyxjxKAH/3k0ZbZARCBYIbEsA0W1LsMDtVXQiOJ1owpTuAhNFSL0EutePmazSi4mVAwggugGwaimqopNHnWjCJ+NaskecSkA+qMkwphzHDGcqFR5jCCC6GGqFbuP34OTkIBf2EVyhySKsYAL+cCb3eAZjo6BHANF5MGp96gtOZ1OK6FggYIWAfGDTb+2RY1ue8yHOSnbTt4OzYXrGyWqQSSU6tOMPUcqJANElw86OJyQgt8Lor5n7x/yEIVF1BQQQXQVJ8kOUT7H+bQJ9b3ZE5xPjuUUC3VEM7r+zmOXx2oToxmOZdE/ySVbezDo0ue42AUSXNBXsvCACIjxunSkoIYWGgugKTYyGJW9k/aouEZjcLnB1daUv9z4iul4srDRMwB/l0A+BXMMznPCBTUN0A4HlKK7Dk3r9TcQlspNvjwhZEF0IJcpYJOALr29Y32KbadNmAohuM6NsJURkIjR/eDLmFgFEly1lVFQogb6JWnyna6HJyhAWossAeV0Vfb03mVW2aXhy3T4R3To6vNYSAX/SytCRkZY4WW8ropsowyIy/9qbXFeQ3tsYnzoR3URJpdpiCehoib435Fq39PpY2iCA6DLmWd5scqOrzhLTT5jb9N76wtc3c99rrINAywT6Zi+P9QGzZa6ltx3RJc6QvLG6ctNPk6lmhSG6xEll9yYI+Nfx5D2z7SUDE1CMNgLRJUis9NzkU6I/a1Key3f2jTE0uSlkRLeJEK9D4JGADmv6k8Dkw2mO9+pjFDxLSQDRjUS3b1gyp9z8ZiA6nwbPIRBGoG9imL6HU42+hEVGqW0JILpIgnLg64QS/SQogtE3xpSfBhFdZFLZDALfCPRdcpChTRnuRHr1HSaIbkDOpNcmw4/6Y6YqFH0DTCk3vxkal7+O5xCAQByB7tCmvL/0PY/04pjm3grRrSEu4pJPcHIbgD9TUp7LurFnS64JZdBLiG4QLgpDIJhAqaM4wQ1otCCi8xK/Smz6CU56c/LprvQF0ZWeIeKzQEBuRu/OqJYPwbKu1A/BFrjHtKFp0Ym0+npsIgoZnpSZk3Iw17YgutoyRry1E1h1WaPm80jtOfHjb0Z0OnlE5CUHnz+BpHax+QmV54iuS4T/IZCPgJ5rupc8rJ1n8hHdviaTopMhSBk6EKnJRWP/+pocbHqNrZahyKFpRnRDiVEeAukIrLskIh+6ZahTRpakHEsaAlWLTj45ydCiCEsOlr6emkhNZKfDkC3MkkJ0ad4s7BUCYxDQD+J6ztL3qzzKSJMOd4r8apgTMAaT1PuoQnQiM+2hyXBAn9DkIJH18npLUus7QPSN0/ca6yAAgfII6Ad2OX/Jvbj6HtZHWacf2OVcWOPcgSmpTy46+XQjSZM/EZT8ibDkT5PsP+onHl9odPmfHkLK6+la/oMABGoiIOdE6dXpObF7CUbe53o+VAnK6JaeT2tqa+pYRxWddLMVsjxqL6wrsFUS0xO0JNTvneknmBaGHcdIuHIcY1/sAwIQKIuAf25dN8Kl5wHpDWrnQYZL9XwsEvXP1/Lc6rIkOmmsAhr7UQUm0H3gChuRjXOYad7G2Rt7gQAEaiGgI2TaE9TeoIpOzw3bPsp+a1qWRCeg1PhDH/1us8qrJhhWYtWD2Ep7aAcEIDA+gU0jcOvO/7X1/pZENz5O9pibAKLLTZz6IACBkgkgupKzExkboosEx2YQgIBJAojOYFoRncGk0iQIQCCaAKKLRlfuhoiu3NwQGQQgkJ8AosvPPHmNiC45YiqAAAQqIoDoKkpWaKiILpQU5SAAgRYIIDqDWUZ0BpNKkyAAgWgCiC4aXbkbIrpyc0NkEIBAfgJLonv27Jnz/1aF5JfpPu/bplvG/z91eamrb/Fj6D6forzU2Y3D/5+Y7gn4TPqeT8GpLw5d1xePrNPX+x77tukrp+vGKF9iTNq+VY9jtHvVvmV937KufMw2Q+sYo7zsY107QuroK1PqOkTXEbsmvy9h+lrf4xjlZR99+9Z1Y9Sh++p77Nt/iTH1xe6v62uH/3r3+RTlS+RaYkzdXHX/nyJ33Ri6/5cWU188Y+R61X5LXL8kuhKDJKZhBBi6HMaL0hCAgG0CiM5gfhGdwaTSJAhAIJoAootGV+6GiK7c3BAZBCCQnwCiy888eY2ILjliKoAABCoigOgqSlZoqIgulBTlIACBFgggOoNZRnQGk0qTIACBaAKILhpduRsiunJzQ2QQgEB+AoguP/PkNSK65IipAAIQqIgAoqsoWaGhIrpQUpSDAARaIIDoDGYZ0RlMKk2CAASiCSC6aHTlbojoys0NkUEAAvkJILr8zJPXiOiSI6YCCECgIgKIrqJkhYaK6EJJUQ4CEGiBAKIzmGVEZzCpNAkCEIgmgOii0ZW7IaIrNzdEBgEI5CeA6PIzT14jokuOmAogAIGKCCC6ipIVGiqiCyVFOQhAoAUCiM5glhGdwaTSJAhAIJoAootGV+6GiK7c3BAZBCCQnwCiy888eY2ILjliKoAABCoigOgqSlZoqIgulBTlIACBFgggOoNZRnQGk0qTIACBaAKILhpduRsiunJzQ2QQgEB+AoguP/PkNSK65IipAAIQqIgAoqsoWaGhIrpQUpSDAARaIIDoDGYZ0RlMKk2CAASiCSC6aHTlbojoys0NkUEAAvkJILr8zJPXiOiSI6YCCECgIgKIrqJkhYaK6EJJUQ4CEGiBAKIzmGVEZzCpNAkCEIgmgOii0ZW7IaIrNzdEBgEI5CeA6PIzT14jokuOmAogAIGKCCC6ipIVGiqiCyVFOQhAoAUCiM5glhGdwaTSJAhAIJoAootGV+6GiK7c3BAZBCCQnwCiy888eY2ILjliKoAABCoigOgqSlZoqIgulBTlIACBFgggOoNZRnQGk0qTIACBaAKILhpduRsiunJzQ2QQgEB+AoguP/PkNSK65IipAAIQqIgAoqsoWaGhIrpQUpSDAARaIIDoDGYZ0RlMKk2CAASiCSC6aHTlbojoys0NkUEAAvkJILr8zJPXiOiSI6YCCECgIgKIrqJkhYaK6EJJUQ4CEGiBAKIzmGVEZzCpNAkCEIgmgOii0ZW7IaIrNzdEBgEI5CeA6PIzT14jokuOmAogAIGKCCC6ipIVGiqiCyVFOQhAoAUCiM5glhGdwaTSJAhAIJoAootGV+6GiK7c3BAZBCCQnwCiy888eY2ILjliKoAABCoigOgqSlZoqIgulBTlIACBFgggOoNZRnQGk0qTIACBaAKILhpduRsiunJzQ2QQgEB+AoguP/PkNSK65IipAAIQqIgAoqsoWaGhIrpQUpSDAARaIIDoDGYZ0RlMKk2CAASiCSC6aHTlbojoys0NkUEAAvkJILr8zJPXiOiSI6YCCECgIgKIrqJkhYaK6EJJUQ4CEGiBAKIzmGVEZzCpNAkCRgjc3Ny46+vrrK1BdFlx56kM0eXhTC0QgMBwAiI5OUe9efPGifQGL39cupPns8U+Zs9P3Lvf3rm/H+3d/79z4F6/++TuOjtFdB0gFv5FdBaySBsgYJOAik7OUy9evHDz+dx9+fJlYGN/c/OD5262+x/up6P/du8+fXXu9p072d1xs71Td90xHaIbiLeG4oiuhiwRIwTaJOCLTs9V+/v77uLiIhzI53N3tDNzs93X7vJWrfa7Oz/6wUkv7/KPp7tCdE95mPhPDx4TjaEREICAKQJ9otNz1uHhofvw4cPG9v5xeeKez35wR+e/P5a9e+9O9wJ7dFohj9/GgGc8cixwDHAMcAzkPAbk+t3q4cx/uevTP7vZzit3/ll7c3fu9vK12519736a///ma3Q5G0NdvHk4BjgGOAY4BvqOAbl217/cX5/bOTp3n7XA7T/c6cH3buenufuo7tPXnHMMXXoweAoBCEAAAmkJrBu6FOFtHL78dn1OpXb36cK9PvjezXZ/cecyKaVnQXQ9UFgFAQhAAAJpCKwSXeiElPvrc34veM8dnZ6764dJKctxI7plJqyBAAQgAIFEBLqiG3aLQd/1uc2BIrrNjCgBAQhAAAIjEfBFN/ym8ftbCJ5cnwuIC9EFQKIIBCAAAQiMQ0BE9/Lly6ivAbv7OHc/7Tx3B/PfBgWD6AbhojAEIAABCGxDYPVtA+v2eusuT3bvv+br2y1fQ3p1iG4dW16DAAQgAIHqCSC66lNIAyAAAQhAYB0BRLeODq9BAAIQgED1BBBd9SmkARCAAAQgsI7AvwEkhDr4+PqyugAAAABJRU5ErkJggg=="></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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AV}}} = \left( {\begin{array}{*{20}{c}} p \\ p \\ {p - 10} \end{array}} \right)"> <mover> <mrow> <mtext>AV</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mo>−</mo> <mn>10</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} = \left( {\begin{array}{*{20}{c}} 0 \\ {10} \\ { - 10} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} p \\ p \\ {p - 10} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {10\left( {p - 10} \right) + 10p} \\ { - 10p} \\ { - 10p} \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> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mo>−</mo> <mn>10</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>10</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {20p - 100} \\ { - 10p} \\ { - 10p} \end{array}} \right) = - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>20</mn> <mi>p</mi> <mo>−</mo> <mn>100</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <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> <em><strong>AG</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AC}}} \times \overrightarrow {{\text{AV}}} = \left( {\begin{array}{*{20}{c}} {10} \\ 0 \\ { - 10} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} p \\ p \\ {p - 10} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {10p} \\ {100 - 20p} \\ {10p} \end{array}} \right)\left( { = 10\left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right)} \right)"> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> <mo>×</mo> <mover> <mrow> <mtext>AV</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>10</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>10</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mo>−</mo> <mn>10</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>100</mn> <mo>−</mo> <mn>20</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find a scalar product <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right) \bullet 10\left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right) = 100\left( {3{p^2} - 20p} \right)"> <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> <mo>∙</mo> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>100</mn> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>20</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right) = 3{p^2} - 20p"> <mo>−</mo> <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> <mo>∙</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>20</mn> <mi>p</mi> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to find magnitude of either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AV}}} "> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>×</mo> <mover> <mrow> <mtext>AV</mtext> </mrow> <mo>→</mo> </mover> </math></span> or <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> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| { - 10\left( {\begin{array}{*{20}{c}} {10 - 2p} \\ p \\ p \end{array}} \right)} \right| = \left| {10\left( {\begin{array}{*{20}{c}} p \\ {10 - 2p} \\ p \end{array}} \right)} \right| = 10\sqrt {{{\left( {10 - 2p} \right)}^2} + 2{p^2}} "> <mrow> <mo>|</mo> <mrow> <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> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>10</mn> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </msqrt> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="100\left( {3{p^2} - 20p} \right) = 100{\left( {\sqrt {{{\left( {10 - 2p} \right)}^2} + 2{p^2}} } \right)^2}{\text{cos}}\,\theta "> <mn>100</mn> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>20</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>100</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{3{p^2} - 20p}}{{{{\left( {10 - 2p} \right)}^2} + 2{p^2}}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>20</mn> <mi>p</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>10</mn> <mo>−</mo> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for any intermediate step leading to the correct answer.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}"> <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> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Do not allow FT marks from part (a)(i).</p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( {3p - 20} \right) = 0 \Rightarrow p = 0"> <mi>p</mi> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mi>p</mi> <mo>−</mo> <mn>20</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>p</mi> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{20}}{3}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p>coordinates are (0, 0, 0) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{20}}{3}{\text{, }}\frac{{20}}{3}{\text{, }}\frac{{20}}{3}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not allow column vectors for the final <em><strong>A</strong></em> mark.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>two points are mirror images in the plane<br>or opposite sides of the plane<br>or equidistant from the plane<br>or the line connecting the two Vs is perpendicular to the plane <em><strong> R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>geometrical consideration or attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 = \frac{{p\left( {3p - 20} \right)}}{{6{p^2} - 40p + 100}}"> <mo>−</mo> <mn>1</mn> <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> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{10}}{3}{\text{, }}\theta = \pi "> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mn>10</mn> </mrow> <mn>3</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mi>θ</mi> <mo>=</mo> <mi>π</mi> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 180^\circ "> <mi>θ</mi> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p \to \infty \Rightarrow {\text{cos}}\,\theta \to \frac{1}{2}"> <mi>p</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">⇒</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo stretchy="false">→</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p>hence the asymptote has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}"> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>for eliminating one variable from two equations <strong><em>(M1)</em></strong></p>
<p><em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ {2x + 2z = 8} \\ {2x + 3z = 11} \end{array}} \right."> <mrow> <mo>{</mo> <mrow> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>8</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>z</mi> <mo>=</mo> <mn>11</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p>for finding correctly one coordinate</p>
<p><em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ {(2x + 2z = 8)} \\ {z = 3} \end{array}} \right."> <mrow> <mo>{</mo> <mrow> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>z</mi> <mo>=</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> <mo>=</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p>for finding correctly the other two coordinates <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = 1} \\ {y = - 1} \\ {z = 3} \end{array}} \right."> <mo stretchy="false">⇒</mo> <mrow> <mo>{</mo> <mrow> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> <mo>=</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math></span></p>
<p>the intersection point has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }} - 1,{\text{ }}3)"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </math></span></p>
<p><strong>METHOD 2</strong></p>
<p>for eliminating two variables from two equations or using row reduction <strong><em>(M1)</em></strong></p>
<p><em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{l}} {(x + y + z = 3)} \\ { - 2 = 2} \\ {z = 3} \end{array}} \right."> <mrow> <mo>{</mo> <mrow> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>z</mi> <mo>=</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math></span> <strong>or</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&{ - 2}&0 \\ 0&0&1 \end{array}\left| {\begin{array}{*{20}{c}} 3 \\ 2 \\ 3 \end{array}} \right.} \right)"> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mrow> <mo>|</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p>for finding correctly the other coordinates <strong><em>A1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left\{ {\begin{array}{*{20}{l}} {x = 1} \\ {y = - 1} \\ {(z = 3)} \end{array}} \right."> <mo stretchy="false">⇒</mo> <mrow> <mo>{</mo> <mrow> <mtable columnalign="left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </math></span> <strong>or</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\left| {\begin{array}{*{20}{c}} 1 \\ { - 1} \\ 3 \end{array}} \right.} \right)"> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mrow> <mo>|</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>the intersection point has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }} - 1,{\text{ }}3)"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </math></span></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} 1&1&1 \\ 1&{ - 1}&1 \\ 1&1&2 \end{array}} \right| = - 2"> <mrow> <mo>|</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to use Cramer’s rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{\left| {\begin{array}{*{20}{c}} 3&1&1 \\ 5&{ - 1}&1 \\ 6&1&2 \end{array}} \right|}}{{ - 2}} = \frac{{ - 2}}{{ - 2}} = 1"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{\left| {\begin{array}{*{20}{c}} 1&3&1 \\ 1&5&1 \\ 1&6&2 \end{array}} \right|}}{{ - 2}} = \frac{2}{{ - 2}} = - 1"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{\left| {\begin{array}{*{20}{c}} 1&1&3 \\ 1&{ - 1}&5 \\ 1&1&6 \end{array}} \right|}}{{ - 2}} = \frac{{ - 6}}{{ - 2}} = 3"> <mi>z</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>only if candidate attempts to determine at least one of the variables using this method.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img 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"></p>
<p>concave down and symmetrical over correct domain <em><strong>A1</strong></em></p>
<p>indication of maximum and minimum values of the function (correct range) <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> = 0 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> = 0 only if consistent with their graph.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant x \leqslant 5"> <mn>1</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>5</mn> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Allow FT from their graph.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4\,{\text{cos}}\,x + 1"> <mi>y</mi> <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></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4\,{\text{cos}}\,y + 1"> <mi>x</mi> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{x - 1}}{4} = {\text{cos}}\,y"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow y = {\text{arccos}}\left( {\frac{{x - 1}}{4}} \right)"> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {g^{ - 1}}\left( x \right) = {\text{arccos}}\left( {\frac{{x - 1}}{4}} \right)"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>g</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><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> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 12">
<mo>=</mo>
<mo>−</mo>
<mn>12</mn>
</math></span><strong><em>i</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>j</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>k </em></strong><strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 12x - 2y - 3z = d">
<mo>−</mo>
<mn>12</mn>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mi>d</mi>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 12 \times 1 - 2 \times 0 - 3( - 1) = d">
<mo>−</mo>
<mn>12</mn>
<mo>×</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mo>×</mo>
<mn>0</mn>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>d</mi>
</math></span> <strong>(<em>M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow d = - 9">
<mo stretchy="false">⇒</mo>
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>9</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 12x - 2y - 3z = - 9{\text{ }}({\text{or }}12x + 2y + 3z = 9)">
<mo>−</mo>
<mn>12</mn>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>9</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>or </mtext>
</mrow>
<mn>12</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} { - 12} \\ { - 2} \\ { - 3} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \\ { - 1} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} { - 12} \\ { - 2} \\ { - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>∙</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<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>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>∙</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 12x - 2y - 3z = - 9{\text{ }}({\text{or }}12x + 2y + 3z = 9)">
<mo>−</mo>
<mn>12</mn>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>9</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>or </mtext>
</mrow>
<mn>12</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {2A + B} \right) = {\text{cos}}\,2A\,{\text{cos}}\,B - {\text{sin}}\,2A\,{\text{sin}}\,B"> <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> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>A</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>A</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> </math></span> (may be seen later) <em><strong>M1</strong></em></p>
<p>attempt to use any double angle formulae (seen anywhere) <em><strong>M1</strong></em></p>
<p>attempt to find either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,A"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,B"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> </math></span> (seen anywhere) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,A = \frac{2}{3} \Rightarrow {\text{sin}}\,A\left( { = \sqrt {1 - \frac{4}{9}} } \right) = \frac{{\sqrt 5 }}{3}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo stretchy="false">⇒</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>5</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,B = \frac{1}{3} \Rightarrow {\text{cos}}\,B\left( { = \sqrt {1 - \frac{1}{9}} = \frac{{\sqrt 8 }}{3}} \right) = \frac{{2\sqrt 2 }}{3}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo stretchy="false">⇒</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </msqrt> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>8</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2A\left( { = 2\,{\text{co}}{{\text{s}}^2}\,A - 1} \right) = - \frac{1}{9}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,2A\left( { = 2\,{\text{sin}}\,A\,{\text{cos}}\,A} \right) = \frac{{4\sqrt 5 }}{9}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <msqrt> <mn>5</mn> </msqrt> </mrow> <mn>9</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {2A + B} \right) = \left( { - \frac{1}{9}} \right)\left( {\frac{{2\sqrt 2 }}{3}} \right) - \left( {\frac{{4\sqrt 5 }}{9}} \right)\left( {\frac{1}{3}} \right)"> <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> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <msqrt> <mn>5</mn> </msqrt> </mrow> <mn>9</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{{2\sqrt 2 }}{{27}} - \frac{{4\sqrt 5 }}{{27}}"> <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> <em><strong>AG</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>cos <em>θ</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\left( {3i - 4j - 5k} \right) \bullet \left( {5i - 4j + 3k} \right)}}{{\left| {3i - 4j - 5k} \right|\left| {5i - 4j + 3k} \right|}}">
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>i</mi>
<mo>−</mo>
<mn>4</mn>
<mi>j</mi>
<mo>−</mo>
<mn>5</mn>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>∙</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>5</mn>
<mi>i</mi>
<mo>−</mo>
<mn>4</mn>
<mi>j</mi>
<mo>+</mo>
<mn>3</mn>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>i</mi>
<mo>−</mo>
<mn>4</mn>
<mi>j</mi>
<mo>−</mo>
<mn>5</mn>
<mi>k</mi>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mn>5</mn>
<mi>i</mi>
<mo>−</mo>
<mn>4</mn>
<mi>j</mi>
<mo>+</mo>
<mn>3</mn>
<mi>k</mi>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{16}}{{\sqrt {50} \sqrt {50} }}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
</mrow>
<mrow>
<msqrt>
<mn>50</mn>
</msqrt>
<msqrt>
<mn>50</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong></em> for correct numerator and <em><strong>A1</strong> </em>for correct denominator.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{8}{{25}}\left( { = \frac{{16}}{{50}} = 0.32} \right)">
<mo>=</mo>
<mfrac>
<mn>8</mn>
<mrow>
<mn>25</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
</mrow>
<mrow>
<mn>50</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.32</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AD}}} = - ">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mo>−</mo>
</math></span><strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + 10">
<mo>+</mo>
<mn>10</mn>
</math></span><strong><em>j</em></strong> – 7<strong><em>k</em></strong> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area}} = \left| {\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AD}}} } \right|{\text{ = }}\sqrt {{1^2} + {{10}^2} + {7^2}} ">
<mrow>
<mtext>area</mtext>
</mrow>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mtext> = </mtext>
</mrow>
<msqrt>
<mrow>
<msup>
<mn>1</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>7</mn>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 5\sqrt 6 \left( {\sqrt {150} } \right)">
<mo>=</mo>
<mn>5</mn>
<msqrt>
<mn>6</mn>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>150</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \bullet \overrightarrow {{\text{AD}}} = - 4 - 2 - 6">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>∙</mo>
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
<mo>−</mo>
<mn>2</mn>
<mo>−</mo>
<mn>6</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 12">
<mo>=</mo>
<mo>−</mo>
<mn>12</mn>
</math></span></p>
<p>considering the sign of the answer</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \bullet \overrightarrow {{\text{AD}}} < 0">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>∙</mo>
<mover>
<mrow>
<mtext>AD</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo><</mo>
<mn>0</mn>
</math></span>, therefore angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{D\hat AB}}">
<mrow>
<mrow>
<mi mathvariant="normal">D</mi>
<mrow>
<mover>
<mi mathvariant="normal">A</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
</math></span> is obtuse <strong><em>M1</em></strong></p>
<p>(as it is a parallelogram), <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 <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BA}}} \bullet \overrightarrow {{\text{BC}}} = + 4 + 2 + 6">
<mover>
<mrow>
<mtext>BA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>∙</mo>
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mo>+</mo>
<mn>4</mn>
<mo>+</mo>
<mn>2</mn>
<mo>+</mo>
<mn>6</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 12">
<mo>=</mo>
<mn>12</mn>
</math></span> considering the sign of the answer <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{BA}}} \bullet \overrightarrow {{\text{BC}}} > 0 \Rightarrow {\rm{A\hat BC}}">
<mover>
<mrow>
<mtext>BA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>∙</mo>
<mover>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>></mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<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 <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>attempt to use a right angled triangle <em><strong> M1</strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p>correct placement of all three values and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> seen in the triangle <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo><</mo><mn>0</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cosec</mtext><mo> </mo><mi>θ</mi><mo>></mo><mn>0</mn></math> puts <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> in the second quadrant) <em><strong> R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1R0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math> seen as the final answer<br> The <em><strong>R1</strong></em> should be awarded independently for a negative value only given as a final answer.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>Attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><msup><mtext>cot</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><msup><mtext>cosec</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><msup><mtext>cot</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>9</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><msup><mo> </mo><mn>2</mn></msup><mi>θ</mi><mo>=</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mo>±</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo><</mo><mn>0</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cosec</mtext><mo> </mo><mi>θ</mi><mo>></mo><mn>0</mn></math> puts <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> in the second quadrant) <em><strong> R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1R0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math> seen as the final answer<br> The <em><strong>R1</strong></em> should be awarded independently for a negative value only given as a final answer.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sin</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>sin</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mtext>cos</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>9</mn></mfrac><mo>+</mo><msup><mtext>cos</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cos</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>5</mn><mn>9</mn></mfrac></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cos</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mo>±</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cos</mtext><mo> </mo><mi>θ</mi><mo><</mo><mn>0</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cosec</mtext><mo> </mo><mi>θ</mi><mo>></mo><mn>0</mn></math> puts <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> in the second quadrant) <em><strong> R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cos</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A1R0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>5</mn></msqrt><mn>2</mn></mfrac></math> seen as the final answer<br> The <em><strong>R1</strong></em> should be awarded independently for a negative value only given as a final answer.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><em><strong>a </strong></em>•<em><strong> </strong></em><em><strong>b</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) \bullet \left( {\begin{array}{*{20}{c}} { - 3} \\ {k + 2} \\ k \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>
<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>
<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></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 6 + k\left( {k + 2} \right) - k">
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
<mo>+</mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>k</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>a </strong></em>•<em><strong> </strong></em><em><strong>b</strong></em> = 0 <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} + k - 6 = 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>k</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>attempt at solving their quadratic equation <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {k + 3} \right)\left( {k - 2} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = - 3{\text{,}}\,\,2">
<mi>k</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Attempt at solving using |<em><strong>a</strong></em>||<em><strong>b</strong></em>| = |<em><strong>a</strong></em> × <em><strong>b</strong></em>| will be <em><strong>M1A0A0A0</strong></em> if neither answer found <em><strong>M1(A1)A1A0</strong></em><br>for one correct answer and <strong><em>M1(A1)A1A1</em></strong> for two correct answers.</p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 0 \\ 2 \\ { - 2} \end{array}} \right)">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note: </strong>Accept row vectors or equivalent.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ { - 2} \end{array}} \right)">
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note: </strong>Accept row vectors or equivalent.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt at vector product using <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> and <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>. <em><strong>(M1)</strong></em></p>
<p>±(2<em><strong>i</strong></em> + 6<em><strong>j</strong></em> +6<em><strong>k</strong></em>) <em><strong>A1</strong></em></p>
<p>attempt to use area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left| {\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} } \right|">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt {76} }}{2}\,\,\,\left( { = \sqrt {19} } \right)">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>76</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msqrt>
<mn>19</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \bullet \overrightarrow {{\text{AC}}} = \left| {\overrightarrow {{\text{AB}}} } \right|\left| {\overrightarrow {{\text{AC}}} } \right|{\text{cos}}\,\theta ">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>∙</mo>
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ 2 \\ { - 2} \end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ { - 2} \end{array}} \right) = \sqrt {{0^2} + {2^2} + {{\left( { - 2} \right)}^2}} \sqrt {{3^2} + {1^2} + {{\left( { - 2} \right)}^2}} {\text{cos}}\,\theta ">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>⋅</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mn>0</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<msqrt>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>1</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 = \sqrt 8 \sqrt {14} \,{\text{cos}}\,\theta ">
<mn>6</mn>
<mo>=</mo>
<msqrt>
<mn>8</mn>
</msqrt>
<msqrt>
<mn>14</mn>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{6}{{\sqrt 8 \sqrt {14} }} = \frac{6}{{\sqrt {112} }}">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mn>6</mn>
<mrow>
<msqrt>
<mn>8</mn>
</msqrt>
<msqrt>
<mn>14</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>6</mn>
<mrow>
<msqrt>
<mn>112</mn>
</msqrt>
</mrow>
</mfrac>
</math></span></p>
<p>attempt to use area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left| {\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} } \right|{\text{sin}}\,\theta ">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\sqrt 8 \sqrt {14} \sqrt {1 - \frac{{36}}{{112}}} \,\left( { = \frac{1}{2}\sqrt 8 \sqrt {14} \sqrt {\frac{{76}}{{112}}} } \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msqrt>
<mn>8</mn>
</msqrt>
<msqrt>
<mn>14</mn>
</msqrt>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>36</mn>
</mrow>
<mrow>
<mn>112</mn>
</mrow>
</mfrac>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msqrt>
<mn>8</mn>
</msqrt>
<msqrt>
<mn>14</mn>
</msqrt>
<msqrt>
<mfrac>
<mrow>
<mn>76</mn>
</mrow>
<mrow>
<mn>112</mn>
</mrow>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt {76} }}{2}\,\,\,\left( { = \sqrt {19} } \right)">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>76</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msqrt>
<mn>19</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sec ^2}x = {\tan ^2}x + 1"> <mrow> <msup> <mi>sec</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>=</mo> <mrow> <msup> <mi>tan</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\tan ^2}x + 2\tan x + 1 = 0"> <mrow> <msup> <mi>tan</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>1</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\tan x + 1)^2} = 0"> <mrow> <mo stretchy="false">(</mo> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = - 1"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4},{\text{ }}\frac{{7\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{{{\cos }^2}x}} + \frac{{2\sin x}}{{\cos x}} = 0"> <mfrac> <mn>1</mn> <mrow> <mrow> <msup> <mrow> <mi>cos</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2\sin x\cos x = 0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = - 1"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \frac{{3\pi }}{2},{\text{ }}\frac{{7\pi }}{2}"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4},{\text{ }}\frac{{7\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1A0 </strong></em>if extra solutions given or if solutions given in degrees (or both).</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>setting at least two components of <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> equal <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>+</mo><mn>2</mn><mi>λ</mi><mo>=</mo><mn>2</mn><mo>+</mo><mi>μ</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>1</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>=</mo><mo>-</mo><mi>μ</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>2</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>λ</mi><mo>=</mo><mn>4</mn><mo>+</mo><mi>μ</mi><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>3</mn></mfenced></math></p>
<p>attempt to solve two of the equations eg. adding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>2</mn></mfenced></math> <em><strong>M1</strong></em></p>
<p>gives a contradiction (no solution), eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>2</mn></math> <em><strong>R1</strong></em></p>
<p>so <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 <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> For an error within the equations award <em><strong>M0M1R0</strong></em>.<br><strong>Note:</strong> The contradiction must be correct to award the <em><strong>R1</strong></em>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><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 parallel, so <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 either identical or distinct. <em><strong>R1</strong></em></p>
<p>Attempt to subtract two position vectors from each line,</p>
<p>e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><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><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><mfenced><mrow><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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>k</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> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><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 parallel (as <math xmlns="http://www.w3.org/1998/Math/MathML"><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> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><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>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>4</mn></mrow></mfenced></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math></p>
<p>Attempt to find vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mfenced><mrow><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>Distance required is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">v</mi><mo>×</mo><mover><mtext>AB</mtext><mo>→</mo></mover></mrow></mfenced><mfenced open="|" close="|"><mi mathvariant="bold-italic">v</mi></mfenced></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mfenced open="|" close="|"><mrow><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><mo>×</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mfenced open="|" close="|"><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p>minimum distance is <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>18</mn></msqrt><mfenced><mrow><mo>=</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><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 parallel (as <math xmlns="http://www.w3.org/1998/Math/MathML"><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> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><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>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> be a fixed point on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> be a general point on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>+</mo><mi>μ</mi><mo>,</mo><mo> </mo><mo>-</mo><mi>μ</mi><mo>,</mo><mo> </mo><mn>4</mn><mo>+</mo><mi>μ</mi></mrow></mfenced></math></p>
<p>attempt to find vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</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><mo> </mo><mfenced><mrow><mi>μ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mover><mtext>AB</mtext><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>1</mn><mo>+</mo><mi>μ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mo>-</mo><mi>μ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>5</mn><mo>+</mo><mi>μ</mi></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>3</mn><msup><mi>μ</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>μ</mi><mo>+</mo><mn>30</mn></msqrt></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>null <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mover><mtext>AB</mtext><mo>→</mo></mover></mfenced><mo>=</mo><msqrt><mn>3</mn><msup><mfenced><mrow><mi>μ</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>18</mn></msqrt></math> to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>minimum distance is <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>18</mn></msqrt><mfenced><mrow><mo>=</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>+</mo><mi>μ</mi><mo>,</mo><mo> </mo><mo>-</mo><mi>μ</mi><mo>,</mo><mo> </mo><mn>4</mn><mo>+</mo><mi>μ</mi></mrow></mfenced></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p>(or let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>4</mn></mrow></mfenced></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mi>λ</mi><mo>,</mo><mo> </mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>λ</mi></mrow></mfenced></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</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><mo> </mo><mfenced><mrow><mi>μ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></mrow></mfenced></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi>λ</mi><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>μ</mi><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi>μ</mi><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mi>μ</mi><mo>+</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>·</mo><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><mo>=</mo><mn>0</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi>λ</mi><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mo>·</mo><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><mo>=</mo><mn>0</mn></math>) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>minimum distance is <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>18</mn></msqrt><mfenced><mrow><mo>=</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,2x = - {\text{sin}}\,b">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</math></span></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,2x = {\text{sin}}\left( { - b} \right)">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,2x = {\text{sin}}\left( {\pi + b} \right)">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>π</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,2x = {\text{sin}}\left( {2\pi - b} \right)">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>π</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> … <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for any one of the above, <em><strong>A1</strong> </em>for having final two.</p>
<p><strong>OR</strong></p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for one of the angles shown with b clearly labelled, <em><strong>A1</strong></em> for both angles shown. Do not award <em><strong>A1</strong></em> if an angle is shown in the second quadrant and subsequent <em><strong>A1</strong></em> marks not awarded.</p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = \pi + b">
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mi>π</mi>
<mo>+</mo>
<mi>b</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = 2\pi - b">
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>π</mi>
<mo>−</mo>
<mi>b</mi>
</math></span> <em><strong>(A1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2} + \frac{b}{2},\,\,x = \pi - \frac{b}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>b</mi>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mi>π</mi>
<mo>−</mo>
<mfrac>
<mi>b</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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 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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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{cos}}\,2x = 4 - 11\,{\text{cos}}\,x"> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <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>attempt to form a quadratic in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {2\,{\text{co}}{{\text{s}}^2}\,x - 1} \right) = 4 - 11\,{\text{cos}}\,x"> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <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> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {6\,{\text{co}}{{\text{s}}^2}\,x + 11\,{\text{cos}}\,x - 7 = 0} \right)"> <mrow> <mo>(</mo> <mrow> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>7</mn> <mo>=</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>valid attempt to solve their quadratic <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3\,{\text{cos}}\,x + 7} \right)\left( {2\,{\text{cos}}\,x - 1} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>7</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = \frac{1}{2}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{3}{\text{,}}\,\,\frac{{5\pi }}{3}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Ignore any “extra” solutions.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider (±) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_{\frac{\pi }{3}}^{\frac{{5\pi }}{3}} {\left( {4 - 11\,{\text{cos}}\,x - 3\,{\text{cos}}\,2x} \right)} \,{\text{d}}x"> <munderover> <mo>∫</mo> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \pm \right)\left[ {4x - 11\,{\text{sin}}\,x - \frac{3}{2}{\text{sin}}\,2x} \right]_{\frac{\pi }{3}}^{\frac{{5\pi }}{3}}"> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mo>)</mo> </mrow> <msubsup> <mrow> <mo>[</mo> <mrow> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </msubsup> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Ignore lack of or incorrect limits at this stage.</p>
<p>attempt to substitute their limits into their integral <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{20\pi }}{3} - 11\,{\text{sin}}\frac{{5\pi }}{3} - \frac{3}{2}{\text{sin}}\frac{{10\pi }}{3} - \left( {\frac{{4\pi }}{3} - 11\,{\text{sin}}\frac{\pi }{3} - \frac{3}{2}{\text{sin}}\frac{{2\pi }}{3}} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>10</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>−</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>−</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{16\pi }}{3} + \frac{{11\sqrt 3 }}{2} + \frac{{3\sqrt 3 }}{4} + \frac{{11\sqrt 3 }}{2} + \frac{{3\sqrt 3 }}{4}"> <mo>=</mo> <mfrac> <mrow> <mn>16</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>11</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>11</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{16\pi }}{3} + \frac{{25\sqrt 3 }}{2}"> <mo>=</mo> <mfrac> <mrow> <mn>16</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>25</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate both functions and equate <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6\,{\text{sin}}\,2x = 11\,{\text{sin}}\,x"> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p>attempt to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11\,{\text{sin}}\,x + 12\,{\text{sin}}\,x\,{\text{cos}}\,x = 0"> <mn>11</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>12</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x\left( {11 + 12\,{\text{cos}}\,x} \right) = 0"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mo>+</mo> <mn>12</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x = - \frac{{11}}{{12}}\,\,\left( {{\text{or}}\,\,{\text{sin}}\,x = 0\,} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>or</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace"></mspace> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow y = 4 - 11\left( { - \frac{{11}}{{12}}} \right)"> <mo stretchy="false">⇒</mo> <mi>y</mi> <mo>=</mo> <mn>4</mn> <mo>−</mo> <mn>11</mn> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{169}}{{12}}\,\left( { = 14\frac{1}{{12}}} \right)"> <mi>y</mi> <mo>=</mo> <mfrac> <mrow> <mn>169</mn> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>14</mn> <mfrac> <mn>1</mn> <mrow> <mn>12</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><strong>METHOD 1</strong></p>
<p>from vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">P</mi></math>, draws a line parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mi>QR</mi></mfenced></math> that meets <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mi>SR</mi></mfenced></math> at a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">X</mi></math> <strong>(M1)</strong></p>
<p>uses the sine rule in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔPSX</mi></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>PS</mi><mrow><mi>sin</mi><mo> </mo><mi>β</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>sin</mi><mo> </mo><mfenced><mrow><mn>180</mn><mo>°</mo><mo>-</mo><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mfenced><mrow><mn>180</mn><mo>°</mo><mo>-</mo><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mo>=</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></math> <strong>(A1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PS</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi></mrow><mrow><mi>sin</mi><mo> </mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></mrow></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>let the height of quadrilateral <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PQRS</mi></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>PS</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>α</mi></math> <strong>A1</strong></p>
<p>attempts to find a second expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>PS</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>α</mi></mrow></mfenced><mo> </mo><mi>tan</mi><mo> </mo><mi>β</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PS</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>α</mi><mo>=</mo><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>PS</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>α</mi></mrow></mfenced><mo> </mo><mi>tan</mi><mo> </mo><mi>β</mi></math></p>
<p>writes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>β</mi></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>sin</mi><mo> </mo><mi>β</mi></mrow><mrow><mi>cos</mi><mo> </mo><mi>β</mi></mrow></mfrac></math>, multiplies through by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>β</mi></math> and expands the RHS <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PS</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>α</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>β</mi><mo>=</mo><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi><mo>-</mo><mi>PS</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>α</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PS</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi></mrow><mrow><mi>sin</mi><mo> </mo><mi>α</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>β</mi><mo>+</mo><mi>cos</mi><mo> </mo><mi>α</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PS</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo> </mo><mi>sin</mi><mo> </mo><mi>β</mi></mrow><mrow><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></mrow></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[5 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>b</mi><mtext>i</mtext></mrow></mfenced><mfenced><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>2</mn><mi>b</mi></mrow></mfenced><mtext>i</mtext></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mtext>i</mtext><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mtext>i</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mi>b</mi><mo>+</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mtext>i</mtext><mfenced><mrow><mo>-</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> and A1 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>b</mi><mtext>i</mtext><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup><mtext>i</mtext></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><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><mtext>arctan</mtext><mfenced><mfrac><mrow><mo>-</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfrac></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>EITHER</strong><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mfenced><mrow><mo>-</mo><mi>b</mi></mrow></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>≠</mo><mn>0</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>) <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) was generally well done with many completely correct answers seen. Part (b) proved to be challenging with many candidates incorrectly equating the ratio of their imaginary and real parts to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi>π</mi><mn>4</mn></mfrac></math>. Stronger candidates realized that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi>π</mi><mn>4</mn></mfrac></math>, it forms an isosceles right-angled triangle and equated the real and imaginary parts to obtain the value of <em>b</em> .</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><em><strong>a</strong></em> • <em><strong>b</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 \times 0} \right) + \left( {1 \times - t} \right) + \left( {t \times 4t} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>×</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>×</mo>
<mo>−</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>×</mo>
<mn>4</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - t + 4{t^2}">
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that<em><strong> a</strong></em> • <em><strong>b</strong></em> = |<em><strong>a</strong></em>||<em><strong>b</strong></em>|cos <em>θ </em> <em><strong>(M1)</strong></em></p>
<p><em><strong>a</strong></em> • <em><strong>b</strong></em> < 0 or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - t + 4{t^2}">
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> < 0 or cos <em>θ </em>< 0 <em><strong> R1</strong></em></p>
<p><strong>Note:</strong> Allow ≤ for <em><strong>R1</strong></em>.</p>
<p> </p>
<p>attempt to solve using sketch or sign diagram <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < t < \frac{1}{4}">
<mn>0</mn>
<mo><</mo>
<mi>t</mi>
<mo><</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use the chain rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></mfrac></math> or equivalent seen</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>''</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><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><mo>=</mo></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mn>1</mn></msup><mfrac><mrow><mn>1</mn><mo>!</mo></mrow><mrow><mn>0</mn><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><mn>2</mn></mrow></msup></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R0</strong></em> for not starting at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>. Award subsequent marks as appropriate.</p>
<p> </p>
<p>assume true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, (so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</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>k</mi></mrow></msup></math>) <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>M1</strong></em> for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>” or “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> is true”. Subsequent marks can still be awarded.</p>
<p> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><msup><mi>f</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>d</mi><mfenced><mrow><msup><mi>f</mi><mfenced><mi>k</mi></mfenced></msup><mfenced><mi>x</mi></mfenced></mrow></mfenced></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></mfenced><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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext><mo>=</mo><msup><mi>f</mi><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>k</mi></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><strong>Note:</strong> The following <em><strong>A</strong></em> marks can be awarded in any order.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><mfenced><mfrac><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow><mn>2</mn></mfrac></mfenced><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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for isolating <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> correctly.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for multiplying top and bottom by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>k</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for leading coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mi>k</mi></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>k</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1</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>k</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><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><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></msup><mo>=</mo><mtext>RHS</mtext></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>since true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>, and true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> if true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, the statement is true for all, <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> by mathematical induction <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>To obtain the final <em><strong>R1</strong></em>, at least four of the previous marks must have been awarded.</p>
<p> </p>
<p><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math></p>
<p>using product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><mfenced><mrow><msqrt><mn>1</mn><mo>+</mo><mi>x</mi><mo> </mo></msqrt><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">+</mi><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msqrt><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></msqrt></mrow></mfrac><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><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><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>h</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math></p>
<p>equating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>f</mi><mo>'</mo><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to apply binomial theorem for rational exponents <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mfenced><mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math><em><strong> A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempt to set equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> and solve <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>m</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>m</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </strong></em>or <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>m</mi><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>x</mi><mi>h</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mo>…</mo></math></p>
<p>equating <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> coefficient to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>=</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p>using product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mfenced><mi>x</mi></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mn>2</mn><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>+</mo><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mi>g</mi><mfenced><mi>x</mi></mfenced></math><em><strong> A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mi>g</mi><mo>''</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mn>2</mn><mi>g</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mi>f</mi><mo>'</mo><mfenced><mn>0</mn></mfenced><mo>+</mo><mi>g</mi><mfenced><mn>0</mn></mfenced><mi>f</mi><mo>''</mo><mfenced><mn>0</mn></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>×</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>m</mi><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>m</mi><mo>-</mo><mn>15</mn><mo>=</mo><mn>0</mn></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> </strong></em>or <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> A1</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>sin</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>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempting to factorise <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempting to use the quadratic formula <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><msqrt><msup><mn>5</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>2</mn></msqrt></mrow><mn>4</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><mn>3</mn></mrow><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><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></p>
<p>use of the correct formula for area and arc length <em><strong>(M1)</strong></em></p>
<p>perimeter is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r\theta + 2r">
<mi>r</mi>
<mi>θ</mi>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong> </em>independent of previous <em><strong>M1</strong></em>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{r^2}\left( 1 \right) = r\left( 1 \right) + 2r">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>r</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} - 6r = 0">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 6">
<mi>r</mi>
<mo>=</mo>
<mn>6</mn>
</math></span> (as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> > 0) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award final <em><strong>A1</strong></em> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 0">
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> is included.</p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>recognition that the angle between the normal and the line is 60° (seen anywhere) <em><strong>R1</strong></em></p>
<p>attempt to use the formula for the scalar product <em><strong>M1</strong></em></p>
<p>cos 60° = <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\left| {\left( {\begin{array}{*{20}{c}} 2 \\ 1 \\ 2 \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \\ p \end{array}} \right)} \right|}}{{\sqrt 9 \times \sqrt {1 + 4 + {p^2}} }}"> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>∙</mo> <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> <mo>|</mo> </mrow> </mrow> <mrow> <msqrt> <mn>9</mn> </msqrt> <mo>×</mo> <msqrt> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} = \frac{{\left| {2p} \right|}}{{3\sqrt {5 + {p^2}} }}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mn>3</mn> <msqrt> <mn>5</mn> <mo>+</mo> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\sqrt {5 + {p^2}} = 4\left| p \right|"> <mn>3</mn> <msqrt> <mn>5</mn> <mo>+</mo> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </msqrt> <mo>=</mo> <mn>4</mn> <mrow> <mo>|</mo> <mi>p</mi> <mo>|</mo> </mrow> </math></span></p>
<p>attempt to square both sides <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9\left( {5 + {p^2}} \right) = 16{p^2} \Rightarrow 7{p^2} = 45"> <mn>9</mn> <mrow> <mo>(</mo> <mrow> <mn>5</mn> <mo>+</mo> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>16</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">⇒</mo> <mn>7</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>45</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \pm 3\sqrt {\frac{5}{7}} "> <mi>p</mi> <mo>=</mo> <mo>±</mo> <mn>3</mn> <msqrt> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </msqrt> </math></span> (or equivalent) <em><strong>A1A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
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