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<h2>HL Paper 3</h2><div class="specification">
<p>This question will diagonalize a matrix and apply this to the transformation of a curve.</p>
<p>Let the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M = \left( {\begin{array}{*{20}{c}} {\tfrac{5}{2}}&{\tfrac{1}{2}} \\ {\tfrac{1}{2}}&{\tfrac{5}{2}} \end{array}} \right)">
<mi>M</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}}&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\tfrac{{ - 1}}{{\sqrt 2 }}}&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right) = {{\mathbf{R}}^{ - 1}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{R}}\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right)">
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 3 }}}&0 \\ 0&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} u \\ v \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>u</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>v</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Hence state the geometrical shape represented by</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M">
<mi>M</mi>
</math></span>. For each eigenvalue find the set of associated eigenvectors.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the matrix equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x\,\,\,y} \right){\mathbf{M}}\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( 6 \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</math></span> is equivalent to the Cartesian equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{2}{x^2} + xy + \frac{5}{2}{y^2} = 6">
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>6</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{{ - 1}}{{\sqrt 2 }}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{1}{{\sqrt 2 }}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> are unit eigenvectors and that they correspond to different eigenvalues.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</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="M\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\frac{{ - 1}}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\frac{{ - 1}}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right)">
<mi>M</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find matrix <strong>R</strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{M}} = {{\mathbf{R}}^{ - 1}}\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right){\mathbf{R}}">
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.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="\left( {\begin{array}{*{20}{c}} X&Y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} x&y \end{array}} \right){{\mathbf{R}}^{ - 1}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the Cartesian equation satisfied by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation satisfied by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> and state the geometric shape that this curve represents.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State geometrically what transformation the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{R}}">
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
</math></span> represents.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the curve in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y">
<mi>Y</mi>
</math></span> in part (e) (ii), giving a reason.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the curve in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> in part (b).</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equations of two lines of symmetry for the curve in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> in part (b).</p>
<div class="marks">[2]</div>
<div class="question_part_label">i.</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="\left| {\begin{array}{*{20}{c}} {\frac{5}{2} - \lambda }&{\frac{1}{2}} \\ {\frac{1}{2}}&{\frac{5}{2} - \lambda } \end{array}} \right| = 0 \Rightarrow {\left( {\frac{5}{2} - \lambda } \right)^2} - {\left( {\frac{1}{2}} \right)^2} = 0 \Rightarrow \frac{5}{2} - \lambda = \pm \frac{1}{2} \Rightarrow \lambda = 2\,\,{\text{or}}\,\,{\text{3}}">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>λ</mi>
<mo>=</mo>
<mo>±</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">⇒</mo>
<mi>λ</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>or</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>3</mtext>
</mrow>
</math></span> <em><strong> M1M1A1A1</strong></em></p>
<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> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{1}{2}}&{\frac{1}{2}} \\ {\frac{1}{2}}&{\frac{1}{2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right) \Rightarrow q = - p">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>q</mi>
<mo>=</mo>
<mo>−</mo>
<mi>p</mi>
</math></span> eigenvalues are of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right)">
<mi>t</mi>
<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>
</mtable>
</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="\lambda = 3">
<mi>λ</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - \frac{1}{2}}&{\frac{1}{2}} \\ {\frac{1}{2}}&{ - \frac{1}{2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right) \Rightarrow q = p">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>q</mi>
<mo>=</mo>
<mi>p</mi>
</math></span> eigenvalues are of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t\left( {\begin{array}{*{20}{c}} 1 \\ {1} \end{array}} \right)">
<mi>t</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><em><strong>[8 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="\left( {x\,\,\,y} \right)\left( {\begin{array}{*{20}{c}} {\frac{5}{2}}&{\frac{1}{2}} \\ {\frac{1}{2}}&{\frac{5}{2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( 6 \right) \Rightarrow \left( {\begin{array}{*{20}{c}} {\frac{5}{2}x + \frac{1}{2}y}&{\frac{1}{2}x + \frac{5}{2}y} \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( 6 \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>y</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mi>y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<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=" \Rightarrow \left( {\frac{5}{2}{x^2} + \frac{1}{2}xy + \frac{1}{2}xy + \frac{5}{2}{y^2}} \right) = \left( 6 \right) \Rightarrow \frac{5}{2}{x^2} + xy + \frac{5}{2}{y^2} = 6.">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>6.</mn>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 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="\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{{ - 1}}{{\sqrt 2 }}} \end{array}} \right) = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<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>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> corresponding to <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{1}{{\sqrt 2 }}} \end{array}} \right) = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> corresponding to <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> <em><strong>R1R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{{ - 1}}{{\sqrt 2 }}} \end{array}} \right) = 2\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{{ - 1}}{{\sqrt 2 }}} \end{array}} \right)\,\,{\text{and}}\,\,M\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{1}{{\sqrt 2 }}} \end{array}} \right) = 3\left( {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}} \\ {\frac{1}{{\sqrt 2 }}} \end{array}} \right) \Rightarrow M\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\frac{{ - 1}}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\frac{{ - 1}}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right)">
<mi>M</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
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</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
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<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
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<mtr>
<mtd>
<mrow>
<mfrac>
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<msqrt>
<mn>2</mn>
</msqrt>
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</mfrac>
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</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
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<mrow>
<msqrt>
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</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>and</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>M</mi>
<mrow>
<mo>(</mo>
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<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
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</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
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</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mi>M</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
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</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em><span style="font-family: 'Verdana',sans-serif;">A1AG</span></em></strong></span></p>
<p><strong><em><span style="font-size: 10.5pt;font-family: 'Verdana',sans-serif;color: black;">[1 mark]</span></em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Determinant is 1. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{R}} = \left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}}&{\tfrac{{ - 1}}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}}&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right)">
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{M}}{{\mathbf{R}}^{ - 1}} = {{\mathbf{R}}^{ - 1}}\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right)">
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> so post multiplying by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{R}}">
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
</math></span> gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\mathbf{M}} = {{\mathbf{R}}^{ - 1}}\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right){\mathbf{R}}">
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
</math></span> <em><strong>M1AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}}&{\tfrac{{ - 1}}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}}&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right) \Rightarrow \left( {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}x - \tfrac{1}{{\sqrt 2 }}y} \\ {\tfrac{1}{{\sqrt 2 }}x + \tfrac{1}{{\sqrt 2 }}y} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right) \Rightarrow \left( {\begin{array}{*{20}{c}} X&Y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}x - \tfrac{1}{{\sqrt 2 }}y}&{\tfrac{1}{{\sqrt 2 }}x + \tfrac{1}{{\sqrt 2 }}y} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>x</mi>
<mo>−</mo>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>y</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>x</mi>
<mo>+</mo>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>x</mi>
<mo>−</mo>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>y</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>x</mi>
<mo>+</mo>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x&y \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}}&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \\ {\begin{array}{*{20}{c}} {\tfrac{{ - 1}}{{\sqrt 2 }}}&{\tfrac{1}{{\sqrt 2 }}} \end{array}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\tfrac{1}{{\sqrt 2 }}x - \tfrac{1}{{\sqrt 2 }}y}&{\tfrac{1}{{\sqrt 2 }}x + \tfrac{1}{{\sqrt 2 }}y} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>x</mi>
<mo>−</mo>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>y</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>x</mi>
<mo>+</mo>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mstyle>
<mi>y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> completing the proof <em><strong>A1AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x&y \end{array}} \right){\mathbf{M}}\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( 6 \right) \Rightarrow \left( {\begin{array}{*{20}{c}} x&y \end{array}} \right){{\mathbf{R}}^{ - 1}}\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right){\mathbf{R}}\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( 6 \right) \Rightarrow \left( {\begin{array}{*{20}{c}} X&Y \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&3 \end{array}} \right)\left( {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right) = \left( 6 \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mrow>
<mi mathvariant="bold">M</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mrow>
<mi mathvariant="bold">R</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>X</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>Y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {2{X^2} + 3{Y^2}} \right) = \left( 6 \right) \Rightarrow \frac{{{X^2}}}{3} + \frac{{{Y^2}}}{2} = 1">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>Y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>X</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>Y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{X}{{\sqrt 3 }} = u{\text{,}}\,\,\frac{Y}{{\sqrt 2 }} = v \Rightarrow {u^2} + {v^2} = 1">
<mfrac>
<mi>X</mi>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mi>u</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>Y</mi>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mi>v</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>, a circle (centre at the origin radius of 1) <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A rotation about the origin through an angle of 45° anticlockwise. <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>an ellipse, since the matrix represents a vertical and a horizontal stretch <em><strong>R1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>an ellipse <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span> <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">i.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p>A graphic designer, Ben, wants to create an animation in which a sequence of squares is created by a composition of successive enlargements and translations and then rotated about the origin and reduced in size.</p>
<p>Ben outlines his plan with the following storyboards.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The first four frames of the animation are shown below in greater detail.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAxwAAACsCAYAAADxJargAAAgAElEQVR4Ae2de3cU17nm8xXmg8yav2fOmslaM5NzVjI5KyeOg3Nyzoyds06WY5IYGwcHm9icGDtgDDEYHINtnPiCLYTBmJttBBJIGBBIIGwhLpIA3ZC6WwJdW91dl2fWW63qi7rV6u5d3XV7Cvaq3dVVe7/72W+p96/fvau/A25UgApQASpABagAFaACVIAKUIEaKfCdGpXLYqkAFaACVIAKUAEqQAWoABWgAiBw0AmoABWgAlSAClABKkAFqAAVqJkCBI6aScuCqQAVoAJUgApQASpABagAFSBw0AeoABWgAlSAClABKkAFqAAVqJkCBI6aScuCqQAVoAJUgApQASpABagAFSBw0AeoABWgAlSAClABKkAFqAAVqJkCBI6aScuCqQAVoAJUgApQASpABagAFagaOOLxONWjAnkKJJNJzMzM5B0Lw4uOjo4wNJNtpAJlK2CaJm7cuIHBwcGyr+GJVCDsCnR3d2Pv3r1hl4HtD6gCBI6AdqwbzSJwuKE666QC3lOAwOG9PqFF3legqakJGzZsQEtLi/eNpYVUoEIFqgaO6enpCqvi6WFQIJVKQXzDThIJy81PTEwETgZGOALXpWyQogIEDkUBeXloFdi0aVNo286GB1sBAkew+5etq4MCBI46iMwqfKUAgcNX3UVjPaQAgcNDnUFTHFWAwOGonCwsDAqMjIzg4MGDsNcxETjC0OtsYyUKEDgqUYvnUoGsAgSOrBbMBUuBsoHj5s2buHTpUqb1nFKVkYKZECpw/vx5q9WyyO/w4cMhVIBNpgJLK0DgWFobvkMFSilA4CilDt/zswJlA4c08sCBA5m2EjgyUjATQgUaGxszrW5ra8vkmaECVAAgcNALqEB1ChA4qtONV3lfgbKBQxYDyxMUZJPHHco3u9yoQFgVsKdTSfs5pSqsXsB2L6UAgWMpZXicCpRWgMBRWh++618FygYOmbcu0GFvjHDYSnAfdgUIHGH3ALZ/sQIEjsWK8DUVKE8BAkd5OvEs/ylQNnAsblruN7yL3+NrKhAmBQgcYepttnWxArquW1OoNE2z3jIMw9r39PRgYGBg8el87TEF5GtE66tEUwPMeSQt+5LQTcCUrpRklme0nJZOOpLQMS+XGQnoehxIzgOaARhlFlZelYE7i8ARuC5lgxYUqBo4GOGgD1GBtAL2AnLqQQXCpoANF/IFVHt7O7766iuMjY1BjsvUWwKH9z1Cgw5dqMJIAUbcgg9dIMEENF2HoWswDB2GaS6bTN1EOunQDB0pgRU9BQMaoGkwNB26SeDI9YpIJJL7EgSOPDn4IkAKEDgC1JlsijsKMMLhju6s1X0FBCzsJNOo5F4YHh62Ih6nTp0icLjfRctaYJoSoTIge5gpJCQoYaSgp5Iw41PoPvkZhi6dxNDF5dPIhWYMX2jBcLvsT2P4QhsGL5xC74VTuHe+GbHudqvsZY0KyQnHjx+3wFz29kbgsJXgPmgKEDiC1qNsT90VYISj7pKzQo8oIJBhp/n5eev3aeS1TLOSx0Xfvn3bI5bSjCUVMAyYhpmePmXCmlIlEQ0zlQBig9j30m/Qf+Jj9J/8dNl0++Q+9J3aj95Tn+LOic9w98Rh9J7aj1vNn+Hbv25F63tbASM9aWtJe0L0RktLC9atWwfZyyZRQQJHiBwgZE0lcISsw9lc5xXgE9uc15Ql+kMBe0qVvX6joaEhY7gMnu7cuZN5zYxHFTA0QDcAPb1eQ3bplRgGzIkBHN/9ErRYHxKRoWVTMtqHeKwf8dhtaKN3YdwbQmL8DhLRu5hqO44zDe8sLArxqBZ1Nkue/CmAYQOHVE/gqHMnsLq6KUDgqJvUrCioCjDCEdSeZbuqVUCiHAIcg4OD1RbB6+qkgL1oPL3Uf2F9uKyzEPKIDePUrk1IRgYxGxtbNs1H5ZwIpmMxxKNRJCJRTE6MIB4dQ/L0UbTvfXdhJXqdGufDaggcPuw0mlyWAgSOsmTiSVRgaQW4hmNpbfhOOBUgcPio3+XhYnayqENIQ9ZzABgfxKndG5GMDGMuFlk2JSP3kIiMYjYWxUwsgtnxEUw8GMTMxDASrY043/AGIE/D4rakAgSOJaXhGz5XgMDh8w6k+e4rwF8ad78PaIG3FCBweKs/SlmTgvyTZ1UtbGYSJkyLQfT7w/h8z+uYiQ1jZnxk2TQfHYE+OorU6DjikSjmo6OYid2zoh3J1v04/8lfGOGwdV5iT+BYQhge9r0CBA7fdyEb4LYCjHC43QOs32sKEDi81iNL26Ob8qhaw/p5DOuBtaY8xta0HotrTAzgy3c2IR4dwmzs3rJpZnwQicgwEmNRzEXHEI+OYDYqU6qimG9txLkGAsfSPZF+h8CxnEJ8368KEDj82nO02zMKEDg80xU0xCMKEDg80hHlmCGUIT/yJ7+zYWWM9G/9WVOq7qJ518tIRAYwF723bJqeGMJcbMg6TyIbc7ERKz8fjWDm7D58bQFHJpZSjnWhO4fAEbouD02DCRyh6Wo2tFYKEDhqpSzL9asCBA7/9ZxMoxLksJjDNn98ECd3b0IiOoA5CyAEIkqlUev92diINZVKzp2NjkEWk0+dkwjHm4D83ge3JRUgcCwpDd/wuQIEDp93IM13XwE+pcr9PqAF3lKAwOGt/ijHGgJHOSrV/hwCR+01dqOG+7c68KDvyrJp8s63bphXlzoJHHWRmZUEWQFGOILcu2xbNQoQOKpRzd1rCBzu6m/XTuCwlQjWvv/YbiQmY8umwZZPgtXwnNYQOHLEYJYKVKMAIxzVqMZrgqwAgcN/vUvg8EafETi80Q9OW0HgAAgcTnsVywudAoxwhK7L2eBlFCBwLCOQB98mcHijUwgc3ugHp60gcBA4nPYplhcSBbq7uzMtJXBkpGCGClgKEDj85wgEDm/0GYHDG/3gtBUEDgKH0z7F8kKgwK5du3Dw4EGkUin09/db+RA0u6ZNnL53F9827MDNYx94Nn3b8AamhvpqqkNQCidw+K8nCRze6DMChzf6wWkrCBwVAMelS5fwxRdfZPpgeno6k2eGCoRJAflAGBkZgR3lYIRDvfdHu87i1MuPY/zmFc+ms2+tx/DFZvXGhqAEAof/OpnA4Y0+I3B4ox+ctoLAUQFwnD17Ftu2bbP6IB6PIxKJON0fLI8K+EIBiWoIfMt9IBuBQ73botcu4evXnsRMbMx6jv/UeNTa269z97l5ec6//dq+xt7bx3P3uflyr7Wv6fpgM8aunlNvbAhKIHD4r5MJHPXvs4GBAbS1tUH29kbgsJUI1p7AUQFwHD58GOvWrct4ACMcGSmYCbkCfEqVugMIcJzZuHKZHxUr9YNjtX+PwFHYz4ZhQNd1KwlkyGtJku/p6cHg4GDhRSE9Ij93lwKgSfsNHUno1i97J00TCUMDdMM6wZC365jEHPlRcbHLsk3qNpMwjHmYsT6cePtPiEfvYi4qvxpeOs2PxTAfuYfp8WE8uD+AmYk+3B8fhRYbRPzCPrR+tB0wpXXcbAW6urrQ0tJivwSBIyNFoDIEjgqAY3HPEzgWK8LXYVXAnloV1vY70e5oTzrCUfpXjGsPFaXqJ3AU9rSmpYeosp5pfHwc3//+9619MplEb28v7t69W3hRSI9YEQTTzI7uRQdDBveS5gF9CkjGAYGPeibpQ0n6Qr3SpXoS0BNApA+n3nwZ2tgdpCKDyyZtbAz66BCSkUHr18n1yG1o4xPQRoYx234Ebfv/mm5/SH2gWLP3vPtu5rB8lmzYsCHzmhnvKNC++VFcfXtN1an1uX9A9wfrl0384b8ifU7gKCIKD4VSAUY41LvdnlJVasDv9nsEjsJ+tiMaiUQCjz/+ODo7OzEmg05dR0NDg/VQhcKrwnnElHiGaViAIV/yS7RDt2ADMObG8c3xBgy2ncDtM0fqmgZOH8GdM0fR15pOt09/gf4zn6Pv9CEMHXkHB555BNePvY+e4x8tn459guvH5LwP0XPsQ9w6+jf0HD+AG0f3oWP382h590+AkQynAyzR6sWfH4xwLCGUy4cjV5qX/dG+Uj/sd+vA6y63wP3q+Tsc7vcBLfC5AlzDod6BkWsXrUXjbkNFqfoJHIX9LFOnBC4EOJ5++mkrqtHa2mpNqzp37hwjHLmSmTnAYQIy7E6ZJqygx+Qw9r34BIa/asCdM5/XNd09fRh3Th9BX6skyR9H35nD6G87isFjf8Px9Y9jrPMkxrrOLJtGu1ox1tWMsa4WRK6cRqyzGWOd53Cvow2Ri0dx/9YFaKagFrelFCBwLKWMu8cJHOr6EzjUNWQJIVdg8TdUIZejquYzwlGVbK5fZEc4ZD83N2et85O9bNevX89bDOu6sW4bIGENKy1MpbLtkTlVU8M4suV3wPAVzIyP1jXNR8YQj4xZD1+YGR/DXDS2kI9ipvcbtOzZnp7iJQs9lktyggCFKdOz9PQ0LVm8Iik5g4SpIWG3m/uiChA4isri+kECh3oXEDjUNWQJIVeAEQ51B+CicXUNvVQCn1K1RG/YA3Z7Vbj9enoMh/78PDB2EzPjw3VNyegQEtEhxGNDiI/L2ot7mIsOYnZ8GPG+y2jZsxUwBBNso5fb6zCRTrIM3W4izHnopmaxxxLq8DDAReMe9QICh3rHEDjUNWQJIVeAwKHuAFw0rq6hl0ogcBT2hgy85Yt++2lQ6deGNTQ3JqM4tnk9cO8u5qNjdU0CGPMCGbERzMWGkYyMIh4dwuz4PcT7utCyZ9vCugs7VFFiLwEOM9tOu73Wc6ms6I60mlspBRjhKKWOe+8RONS1J3Coa8gSQq4AgUPdATilSl1DL5VA4CjWGwujcRlzywhcyMNMA4c+HcH+bS8gcb8f0+NjdU1z0SgkzcZGMRu7h4RMsYoOWY+ojvdeXQCO9NPIirUq75i0beFBXLlYYsdEBEaskEfeRXyRqwCBI1cN7+QJHOp9QeBQ15AlhFwBruFQdwBOqVLX0EslEDiK9UZ6DYcs2ZCnU0m4wzANa9KRPjWGz7c+Dy12s+6/RTNrwUYUM7FRzMRGkLAiHINWxCPeKxGO1wFD1mWUsYbD1C2Ikv43JFlsJROs0nxltVvK4bakAgSOJaVx9Q0Ch7r8BA51DVlCyBVghEPdAeQpVfJL46WeEuX2e3xKVfn9TOAo1Cr3sbgyAJcBvAF7StUYjm55HojcRDw6UteUjmxIdCMd4Zi3ftxPIhyjiPdeRst7soZDKw84LLQQsFqI5lgypMFD2mxNrSqUhkdyFCBw5IjhoSyBQ70zCBzqGrKEkCvACIe6AzDCoa6hl0ogcBT2hoH0t/75o245asKciuDzrX8Aon11h+7ZcVm7IWkUc7ExxGPD1tSquVjUAo7m97YAErngVhcFCBx1kbniSggcFUtWcAGBo0ASHqAClSnAXxqvTK9iZ3PReDFV/HuMwFHYdwSOQk14pFABAkehJl44QuBQ7wUCh7qGLCHkCjDCoe4AXDSurqGXSiBwFPYGgaNQEx4pVIDAUaiJF44QONR7gcChriFLCLkCXMOh7gCcUqWuoZdKIHAU9gaBo1ATHilUgMBRqIkXjhA41HuBwKGuIUsIuQJtbW0hV0C9+Vw0rq6hl0ogcBT2BoGjUBMeKVSAwFGoiReOEDjUe4HAoa4hSwi5AoxwqDsAIxzqGnqpBAJHYW8QOAo14ZFCBQgchZp44QiBQ70XCBzqGrKEkCtA4FB3AC4aV9fQSyUQOAp7g8BRqAmPFCpA4CjUxAtHCBzqvUDgUNeQJYRMAZlCtXPnzkyrCRwZKarOcNF41dJ58kICR2G3EDgKNQn7kaamJmzcuBF9fX0ZKQgcGSk8lSFwqHcHgUNdQ5YQMgUOHz6M3bt34/79+5BH4jY2NoZMAeebyylVzmvqZokEjkL1CRyFmoT9yPr1663PkpaWFkuKxn37sGHDhrDLkmn//MQoevZuWDJdb9i05HulrqvmvfbNj+KbPc9Vndr+8MNMu8KaKRs4du3alfet7vT0dFg1Y7tDrkB/f38eZDDCoe4QXDSurqFbJRiGAQGMVCoFyWuaBl3XcePGDQwODrpllufqJXB4rktcN0gAQ76wkkiHvTHCYSsBPOi7guTMJBLT94unyVjx40udr3BcbOGmpkDZwCHTSFavXm3VJjSee4OomcCrqYC/FeDvcKj3HyMc6hq6UYLAhWwCGvIZcfr0aczMzFgAcvPmTYyMjLhhlifrJHB4sls8ZxSBI9slMshPCFR4IBE4sv1Sba5s4JCpIxLlsDdGOGwluA+7AoxwqHsA13Coa+hGCRLZENiQ9Itf/AJbt26FfDbI63feeQd37txxwyxP1kng8GS3eM4oAke2SwgcWS2CkCsbOBY3lsCxWBG+DqsCBA71nudTqtQ1dKMEmTplT6X65JNPIFGNs2fPWsdkStXAwIAbZnmyTgKHJ7vFc0YROLJdQuDIahGEHIEjCL3INriqAIFDXX5OqVLX0I0SJMIh0CERDZk+dfv2bczPz1um9PT0YGhoyA2zPFkngcOT3eI5owgc2S4hcGS1CEKOwBGEXmQbXFWAvzSuLj8Xjatr6JUSBEJk46Lx/B4hcOTrwVfFFSBwZHUhcGS1CEKOwBGEXmQbXFWAEQ51+RnhUNfQKyUQOIr3BIGjuC48mq8AgSOrB4Ejq0UQcgSOIPQi2+CqAnxKlbr8XDSurqFXSiBwFO8JAkdxXXg0XwECR1YPAkdWiyDkCBxB6EW2wVUF5Alu3NQU4KJxNf28dDWBo3hvEDiK68Kj+QoQOLJ6EDiyWgQhR+AIQi+yDa4qwAiHuvycUqWuoVdKEOCQxDUc+T1C4MjXg6+KK0DgyOpC4MhqEYQcgSMIvcg2uKoA13Coy89F4+oaeqkEAkdhbxA4CjXhkUIFCBxZTQgcWS2CkCNwBKEX2QZXFWCEQ11+RjjUNfRSCQSOwt4gcBRqwiOFChA4spoQOLJaBCFH4AhCL7INrirACIe6/Fw0rq6hl0ogcBT2BoGjUBMeKVSAwJHVhMCR1SIIOQJHEHqRbXBVAQKHuvxcNK6uoZdKIHAU9gaBo1ATHilUgMCR1YTAkdUiCDkCRxB6kW1wVQECh7r8nFKlrqGXSiBwFPYGgaNQk7AfSaVSkJS7ETiyahA4sloEIUfgCEIvsg2uKsBfGleXn4vG1TX0UgkEjsLeIHAUahL2I3v27MHx48fzZCBwZOUgcGS1CEKOwBGEXmQbXFWAEQ51+RnhUNfQSyUQOAp7g8BRqEnYj3R1dWH9+vWZKIc8SnrDhg1hlyXTfgJHRopAZAgcgehGNsJNBfiUKnX1uWhcXUMvlUDgKOwNAkehJmE/0tLSkgccogcjHFmvIHBktQhCjsARhF5kG+quwO7duzN1MsKRkaLqDBeNVy2dJy8kcBR2C4GjUBMeKVSAwJHVhMCR1SIIOQJHEHqRbairAv39/Vi3bp1VZ3d3Nw4+swLtO5+30vk31uLiWy9A9nLsws7n8/Jfb382c9zOy17Ok/OXykt59jm5deTmVetbXEc96/vq+X9B44r/jAvSTo+l9h3PWTZ9teYhjF09V1df83plZsqAZphIwURKT+DB/Ri0ZAowgOs9NzB8dwAwdSZLA0NksZIOzepazUzBgAbjwSCOvP4H6JF+zMTG6prmoqNIRoYRj97DdCyGRGQEU9FJzMcmoN88j5b3NqX7z+vOGBD7CBzZjiRwZLUIQq4i4Ni3b1+mzdPT05k8M1QgTAqcOHECL774IkZGRqxmn3r5cUyNR61BwlzsHpgq12Dg7HFcfDMNcWHyJd+31ZhDKjVnwQagY9265xGdeAAdQPf1axgevgFTN5l0E4YhyYBm6kiZSSsv4KGZGszJERzcsg7JsT7MRwbqmuaiw5iP3sVs7A4mxwcxH+vF9PgYpkYGoN1oR+ueHYAhqMStHgoQOLIqEziyWgQhVzZwyJN49u7da7VZ5h2efWs9rh/+G24efd/aM1+dDpc/3IrUHOHNzzfTyT+txP379wkaCrBF4PDpHWACugnEDR2vbHkNB48eRSw6Dkzcx6P/7b/grZ//T3z2zMNMlgYrcPCZR3DgmRU48LuHcfCZf8aHv/kR9q/5GY6v/Hvs/Lv/hKbVP8L+VT+pa/pw9c/x8dM/xb6nfox9Tz2Ez3/7f9Cw6hF8+OQK7F/9Exx66d8BY96nDuo/swkc2T4jcGS1CEKubOAQ2Hj22WczT1P47JffxcilZiZFDVo3rkRyZjIIvhTaNpx85VdWhIORjcojG7ZmAhwylYqbzxQwdBiGjngijvMXL+CjhkYMj4zBTCRxu/0UWv78DEbOHGTK0WCo9QCG2vZj+MznGGw7hIG2Q4h++Vd8sfZnmO44gomu1rqmyDfnMH61DRNdZzBxtRVTV5ow0fU17l08jVhnMyZvXYRmpKeA+cw7fWkugSPbbQSOrBZByJUNHNLYeDyeafOxp/6R3+gqfKNrD7Q63v4jElMTGV2Z8Z8CMqXK7k/uq4MORjj85/disWYa0I0UYMqPl+lIpFLQDMA0gVsXW9G5bzvmoiNM0RHEo7JOYhBzsQHMjd9FXI7dH8VUbASJO9/g2LYXAG0S1uJymPXbm9JfJgzpS9OEaU33MtOLTTQNuqFbU+T86aH+s5rAke0zAkdWiyDkKgKO3AYTOKobWC0ekBI4cr3Kn3lGONTvBUY4/On7MtEmZQ1HBTi0hYFremV0/8VzuPzxDsSjo0yWBiOYF+iIDSEuayWiY5iNjVrrJeZvd+PYtv8ANHem15owocO0lrJLzlqxYQpD6jANeYdbvRQgcGSVJnBktQhCrmrgOLLqh/xWlxGOINwDym0gcKgDx52vv+SicWVPdKEAWR0uSUaopnw3L9+USx64fbENlz/eibnxYabxYczGRjC78JkxOy7RjihmYxFMjUcwf/sajm7/I6DN1L8TDR2wUrrfrP60+1SmzEGepMVF4/XqGAJHVmkCR1aLIOSqBg5GONQHWRLtYITD/7cRF42r3wucUuXP+0BDEtaEG4EMCzrS06mkNTcvn8fX+9/FfHSUSSIcAhsWfA0hHhtAMnIPcQtCRpC8fRXHt/0B0B644AiaBRQSrZpbSMkF9rBCG7rBp1TVsVcIHFmxCRxZLYKQI3A4EKVYPE2qktcEDv/fRoxwOAMcXDTuw3tB13MiHOlRqinrOYxZ3Ll4Elc+3gb5Np9pGNPj9zA1MYbp8VHMWBGOyMKUqlHM3/4WR7e/AOguPEBEHnkrYamFyJQFjguRKzksy8U5pap+9yaBI6s1gSOrRRByVQMHp1SpD7IY4QjCLQRw0bj6vSARjvPb1gTDIULbCpl4I+sAJNSho6/jDDoadnDq7cKXWvLDepLk774AmER+ZD3HXGwIydtdOLZtPaC7s4YjtC7rwYYTOLKdQuDIahGEXNXAwSlV6oMsAkcQbiGAEQ71e4GLxoNwLxA4SkW3CRxB8PHat4HAkdWYwJHVIgg5AgenVAXBj11tA4GDwOGqA3qmcgIHgcMzzuhbQwgc2a4jcGS1CEKuauDglCr1QRYjHEG4hQAuGle/F7hoPAj3AoGDwBEEP65vGyKRSF6FBI6sHASOrBZByFUNHJxSpT7IInAE4RbilKpSg6xy3+OUqiDcCwSOUv7OKVVB8HHn2/BP//RPeYUSOLJyEDiyWgQhR+DglKog+LGrbeCUKnX4JnC46sIOVU7gIHA45EohKWb9+vV4//33M61tampC4/e+w+RBDQR+uKkpUDVwcEqV+iBLPpz4WFw1B/bC1XxKlfq9IMDBp1R5wZtVbCBwEDhU/Cd817a0tOC73/0uZC9bKpXClXefQ2J2OvQpOTOJ8WvnwucUAW5x1cDBKVXqgywCRzDuLEY41O8FRjiCcC8QOAgcQfDj+rYhHo/nVdi5+1kkJmNMkzFErjTnacMX/laAwMEpVf72YA9Yz0XjzgDHxTfXeaA3aUL1ChA4CBzVew+vTCtA4MjCFoEjWHdF1cDBKVXqgyxGOIJxMzHCoX4vMMIRhHuBwEHgCIIfu9uGK7vXMLqxEOEhcLjri07XXjVwcEqV+iCLwOG0O9envLa2Nhw+fDhTGYFD/V4gcGTcyVcZwzCg6wZM04SmJSHIkTI0mKaGvsutuPjJG/ylcf7SuK982m1jCRyMcLjtg7Wqn8DBKVW18q1Al/vYY49B13V89NFH2PXY/+agSvE+4qJxf94uAhyGYULmob/313fR0LgPN/t6EZ+bwo2Lzeho2MF7g8DhT+d2yWpOqSJwuOR6Na+2auDglCr1b3UZ4ai5f9ekAoluNDY2WsAhFTDCoX4vMMJRE1eteaES2ZAIh6ZpmJ2dxpm2M+i+0QNdS+DrIw24xAhHBrj4Oxw1d8dAVMAIB4EjEI5cpBFlA0dzczNefvnlTBGcUqU+yCJwZNzJd5n+/v6MzVw0rn4vCHBw0XjGpXyTkQiHPMpzcnISz69bi8tXr+BCx0UAOm62n0LXgV2ZAXep9Q1heI/A4Ru3dtVQAgeBw1UHrGHlZQOH2PDGG29YpsgHzNEnv88PEsVpJASOGnp2HYtmhMMZ4Ljwxto69hqrckIBmVZoLKzhiM/PovPKZWimYUU4uIYj/74gcDjhccEvg1OqCBxB9fKygUPmqj/99NMZHTilKv/DpNpv6PjDfxmX8m2GwKF+L3BKlW/dP8dwPqWq1OcAgSPHVZhdUgFGOAgcSzqHz98oGzgWt5NTqtQHWYxwLPYqf77mL42r3wtcNO5P38+3msBB4Mj3CL6qXAECB4Gjcq/xxxUEDgemRZX6kFnuPUY4/HGjlLKSEQ5ngINTqkp5mR/eI3CU+nvPCIcffNh9GzmlisDhvhfWxoKqgYNTqtQHWfLhROCojWPXs1QuGle/F7hovJ4eW6u6CBwEjlr5VnjKZYSDwBFUb68aODilSn2QReAIxm3FCIf6vcA1HEG4FwgcBI4g+LG7bSBwENRO3X4AABlbSURBVDjc9cDa1U7g4JSq2nlXSEomcBA4QuLqyzSTwEHgWMZF+PayCnBKFYFjWSfx6QlVAwenVKkPshjh8Olds8hsLhpXvxe4aHyRU/nyJYGDwOFLx3XN6OvXr6OpqSmvfkY4CBx5DhGgF1UDB6dUqQ+yCBzBuJMY4VC/FzilKgj3AoGDwBEEP65fG+Q3zXbv3m39eKbU2rhvH66+vQaJyeygO8z5yJXm+nUGa6q5AgQOTqmquZMFvQICB4Ej6D5eXvsIHASO8jyFZ2UV6O3tRX9fX+YAp1RlYYvAkXGLQGSqBg5OqVIfZDHCEYh7CHxKlfq9wKdUBeFeIHAQOILgx/VrQ0tLC7Zt25ZXIadUETjyHCJAL6oGDk6pUh9kETiCcScxwqF+L3BKVRDuBQIHgSMIfuxuGwgcBA53PbB2tRM4OKWqdt4VkpK5aNwZ4Di/bU1IPCaozSRwEDiC6tv1axenVBE46udt9a2pauDglCr1QRYjHPV19lrVxgiH+r3ACEetvLOe5RI4CBz19Ldg1sUIB4EjmJ4NVA0cnFKlPsgicATjtiJwqN8LBA5/3guGYcA0zAXjDeimAc00AOjou9yGSw07UGoQHqb34tF7kCRtnh0fxnx0FPHYEOZiQ0je7sKxbesBfdqfjkCrHVOAwEHgcMyZPFYQgYNTqjzmkv4zh4vGnQGOi2+u81/nh9xiTdMgScDj/v1xfH70MHSkgeNW+ym0f/IGgWPhM4bAEfKbpczmc0oVgaNMV/HdaVUDB6dUqQ+yGOHw3f1iGazrOgYGBjLGM8Khfi8wwpFxJ19lTNOEJPk9gf2f7sPV7m/Q1f0NdC2BtsOfoLPxTQIHgcNXPu22sYxwEDjc9sFa1V81cHBKlfogi8BRK7eubbmbNm3C+++/j3g8blXERePq94IABxeN19Zva1G6wIam6RAIn52dRuOB/Zidn4Npamg++AGnVOVE0BnhqIUHBq9MAgeBI3henW4RgSPnA8GN+cQdb/8RiamJoPpXINsloHHs2DFrGklHRwcan3kEU+NRfpOrcC8xwuHPW0VAQ9ZwSIRj5cpf4c233kT/3dtIzM+i/9JpXOYajszfBQKHP3283lZzShWBo94+V6/6ygaO7u5uHDlyJGMXp1Spf6vLCEfGnXyVaWpqwrlz5zA/P2/ZzSlV6vcCgcNXt0CBsempVbJ6w4C2sIajr/MMOhp2Zgbcbnyh46U6CRwFbsMDRRRghIPAUcQtAnGobODo7e2FTCWR7e7duzj4xN/zg0ThG137g5ARDv/fR1w07gxwcNG43+8FPhbX/rtebE/g8Lt/18d+AgeBoz6eVv9aygYOmTpy/vz5jIWMcKgPsuRDicCRcSnfZhjhUL8XGOHwrfvnGE7gKAYa9jECR46rMLukApxSReBY0jl8/kbZwLFv375MhEPazEXj6oMsAofP754F87loXP1e4KLxINwLBA4bLortCRxB8PHat4ERDgJH7b3MnRrKBo7F5hE41AdZBI7FXuXP14xwqN8LjHD40/fzrSZwFAMN+xiBI99b+Kq4AgQOAkdxz/D/UQKHA+sw7A+UavacUuX/m4jAQeDwvxc70QICR6nPAAKHEz4WrDKmp6exf//+zCPWpXWcUkXgCJaXZ1tD4CBwZL2BuaoU4KJxZ4CDi8arcj8PXUTgIHB4yB19Ykrjvn1IJpOWtZ2dnWCEg8DhE9et2EwCB4GjYqfhBfkKMMLhDHBceGNtvrB85TMFCBwEDp+5rMvmtrS0oK2tLc8KAgeBI88hAvSCwEHgCJA7u9MUAgeBwx3P81qtBA4Ch9d80tv2jI+P43pPTybCIdYSOAgc3vba6q0jcBA4qvceXmkpwKdUOQMc57etoUf5WgECB4HD1w7sCeO5hoPA4QlHrIERBA4CRw3cKlxFMsLhDHBwSpXf7xsCB4HD7z7svv2McBA43PfC2lhA4CBw1MazQlQqF407AxxcNO73m4bAQeDwuw+7bz+Bg8DhvhfWxgICB4GjNp4VolIZ4XAGOBjh8PtNQ+AgcPjdh923n1OqCBzue2FtLCBwEDhq41khKpXAQeAIkbuXaCqBg8BRwj34VlkKMMJB4CjLUXx4EoGDwOFDt/WWyVw07gxwcNG4t/y6HGtM04RhmNappqnDgImUqQPQ0d9xGh0NO1BqEB6m9/jDf+V4FM8hcBA4gnoXEDgIHEH17bq1ixEOZ4CDU6rq5rKOVSTAoesGZH/5cgeaTzcjaWiAqaG/s5XAkfP5QuBwzO0CXRCnVBE4gurgBI6cDwQ3vm3rePuPSExNBNW/QtEuLhp3Bji4aNx/t4v8QrKuS0QDmJ6exMHPP4NmGjCNJHovt+LSvp2McCx8xhA4/OffbljMCAeBww2/q0edBA4CRz38LNB1MMLhDHAwwuG/28Qw0tGNNHQYOHDoIJJ6CrqWwGd73sClT94gcBA4/OfYLlpM4CBwuOh+Na2awEHgqKmDBbHw+/fvo7W11WqaDLS+eulxTI1HObBSuJcGzh4HgcN/d4sFHAtrOFJaAu0dF6FZazg09F9uQ/vH23lfEDj859guWswpVQQOF92vplUTOBQGSU5MweKUqpr6d80Kb29vt8ru7+/HZ2v/lYMqxftIgIOLxmvmrnUqmE+pKvWZwClVdXJDn1fDCAeBw+cuvKT5BA7FgVKpD5hy3iNwLOmbnn6js7MzYx+nVHFKVcYZQp0hcJT6m0/gCPXNUXbjCRwEjrKdxWcnlg0c8k1uR0dHpnnHnvpHfqvrAKwQODIu5dsMgYPA4VvnddRwAgeBw1GHCmVhnFJF4Aiq45cNHCLA7t27LR1aWlqw97H/TuAgcAT1vqioXXxKlTPAwadUVeR2HjyZwEHg8KBbetikRCKBQ4cO5VnICAeBI88hAvSiKuCQ9h9Z9UMCB4EjQLdC9U1hhMMZ4OCi8ep90BtXEjgIHN7wRL9YYWgpfPvNNxlzBwYGQOAgcGQcImCZioAjt+2cUqU+yJIPJ06pyvUqf+b5S+Pq9wIXjfvT9/OtJnAQOPI9gq+WV6CrqytzUnx2BpxSReDIOETAMgQOB6IUpT5klnuPwOH/O4oRDmeAgxEOv98LBI5Sf++5aNzv/u28/dPT0zh+/DhGRkYyhTPCQeDIOEPAMgQOAkfAXLr+zSFwEDjq73VerJHAQeDwol/6yyZGOAgc/vLY8q0lcBA4yvcWnllUAS4adwY4uGi8qHv56CCBg8DhI3f1qKmMcBA4POqaymYROAgcyk4U9gIY4XAGODilyu93EoGDwOF3H3bffgIHgcN9L6yNBQQOAkdtPCtEpXLRuDPAwV8a9/tNQ+AgcPjdh923n1OqCBzue2FtLCBwEDhq41khKpURDmeAgxEOv980BA4Ch9992H37GeEgcLjvhbWxgMBB4KiNZ4WoVAIHgSNE7l6iqQQOAkcJ9+BbZSlA4CBwlOUoPjyJwEHg8KHbestkLhp3Bji4aNxbfl25NQQOAkflXsMr8hXglCoCR75HBOcVgYPAERxvdqkljHA4AxycUuWSA6tUawAwrf+AacI0TRhWSqD/cgs6P9mGuehIVWk2Mox47B5kPzM2ZO1lQB8fv4fZ2Ahmo8OYk3x02Hot+blY8bqscnLskDKrtUtsmZ8Yw/z4KKbHhiwbyylLbJ6LjiEeiWA2NoxUZBDx8WHExweRut2Fo9v/AzAeVN0bor2u61YfVF3IwoVSlr0ZhnRy9ZuUtThVX1rwr2SEg8ARVC8ncBA4gurbdWsXF407AxxcNF43l3WuIh0wTECXEgU8DEBPGTCNOG5dOomrDTugjfWVmXqhjeWn5OgtGON3YMRuQxu9heRYL5KRPiQj/UiO9VkpYR3vQ2q0F6nRW0gtKkPK1Md6MT96yyrDfDBonVetXcZYLxIjN5Aau4XEvZvQI+W1T+xOjd2Ffm8IqUgvzNEepGJi93VovRfx1fYXAXOs6r6xwUDTNEjKhYZKC80tS/L260rLkfMFgsQe2VTLqqZ+v11D4CBw+M1ny7WXwEHgKNdXeN4SCjDC4QxwMMKxhIN5+PBzzz6HZ577vZXWPvsc1q5ZhxeeX481z/wWLz7xf7Hh4f+FV1c+UkZagVdX/rQgbXriYWx8/CHIfvOvV2Dz4yuw+VcrsGXlz6z9n/79IWz85U+w9df/jNeeeMQ69ppVX35ZUsbmlT/Fnx5/yEqv/aYcm+ScQru2/PZnGTtff+rneHXlw2W07xFs/PUKbHrin7Hl8X/FppUr8OdfPYQ//uon2PzUz7Ht336IDSv+Bzas+Rf8/ve/ryqtXbvWuu53v/sdJNmvqylvzZo1kPTss89aZT333HNV2SR1r1u3zipj9erVVpnyupRNHnb3upjGKVUEjro4mguVEDgIHC64XbCqJHAQOILl0RW0xgTku2s7wgEDMHQDpqnjZuc5tH70Nq6daCgjNeLaif15qafpU3Sf2I/rpz7DF7texZVD76PnZCN6mhqt/bWvPsE3xz7C5c//iu6mfbh2cj+uNe1DT1N+OVKuVVbTAWv/4R9X49tjH5dhk9hdxK4vpa4DuHbiU7zwLz9Ez1dyzvJttGw88Sl6vjyI7qZPreu+bf4cV08fxrUvPkXTri1AKlaB+PmnSvRAogmS/va3v+W/WeErKSOZTFoRiTNnzihFS2ybpqam0NTUpFRWhc3w5emMcBA4fOm4ZRhN4CBwlOEmPKWUAlw07gxwcNF4KS/z6Hu6LN2QOfpCGqaVNMNAyjRx69Z1DPffSC/wkCUBJZO8KWsFcpJpwNQ1mIYGQ09ZyV5aIPtiSdft6T855UiZpm6Vo2vJhfK0Zeyx7S1iFwwYYpepwy6vdNvsskSsFGAKoqVTEgaSMC1ISxkmZsTOImseyjkmA3uZupQ7bamc65Y6x/Y4e3rWUueVc9yekpVKpTLrTJa6zq43DHsBsJdeeglXr17NNJfAQeDIOEPAMgQOAkfAXLr+zWGEwxng4JSq+vuuao0ydjYNE7qm497AEN7a8SZm5+IWcHzT04ORkeF8iMgFity8RQ/2wHxhb3GCQAwwcGcQHRc7qi7L0EzLzmQiiQvn2vGXHX+BtgAfAiNWJcbCfjm7JKqT0jE9OY2/7PwLjh4+Vp1d0j4AKV0HNBO3++9g+84dGWiQvpGBugzMy9kEOGQbHR3FE088gebm5rzL7AF+3sESLwQOpH6JlkxMTJQ4s/RbUm8ikcCvf/1rPProo+jr6yt9QYjeXb9+vdVaey8vOKWKwBHUW4DAQeAIqm8rtau7uxvlpsZVD+H0n3+HM689hTOvrVrIr0Lbq79By9ZnrGN2XvZyjhzP5LeszlzbsmU1Dr34y6LltLz6JJpf/H/WuXKe1GeXk7vPtSFt1yoc/sMvkLmmzGut8u1z7f2C3V++8ptMO0udt5xdYp/ocPSFx3DkhV+UrXk5fXPo0CFPlydt2L59e1EblZy3jhfrAhuGgbnZOWzc8Aru9t/G+fMXIMevXb+OwaFBGPIN/nLJNGAUSTKIluNSjjUdB+by/4qUIxEY3dAxOzuLaz3XcKLpBCYmxq2ydDNdh5RcYGeRsuQbf03XrIH9Vye+wuvbXi+8rmhZ8gSvTCDIitDIg4Q1Q4Oe0rDrL29h3QvrMr1XKSDYEY729nYcOHAAAwMDebBSSXnSRimvp6cH7733HiKRSKYsKaeSzY5qyDUCQTJVi1tagb179+Lw4cOWb9uaMMJB4LB9wQ/7tra2ss0kcBA4ynYWnlhcgdxwePEzKjv6wQcfVHZBGWfv2rWrjLPKP+XWrVvln1zGmTK/u6NDvsF2bnNaR6c1lJZ+73vfc67BLpSUknUDhmF9g711y1Zc+7YbX7edtb66v95jA4cMq5f5V2RgbwGIoSOlJTE9M4WW083W1C2Bh+XSYnjRZEqWqSOZSuBU8ykcOPgpEol5pI+nYUTeL7CyiF0CKJKGhgZx7Xo3Ptr7YeF1xY7IY2sXnugl4CHjdtuuG9dv4sD+A/j3f3sM8Xg8M60qd3rUct1rA8fNmzdx4cIFfPjhh5lylrt28ftS79zcHLq6unDp0iV88cUXVrRDzqsEXOR8KUtsk/JOnjyZAZfFdfrl9c6dO2Gnbdu2WXnZV5ukrB07dlhJvoA4uervINAhkY6wp/bNj1pfylSrLa+r3i/L1e7hhx/G+++/X9btS+AgcJTlKDxpaQWcHigvXVP178iHvpOb0+U5aZtdlh9slMGlnzdrMJnSkJxPYnBwCJcvX4ZMW5LRdM+NHtwdHiz2XX/hMWvdQv66DClbBrf2QDoWixWNgiyGi/SAOL8s+U5eyhG9xUaJLs3OzljgYtUjMQoBmcWxiiJ22fY8mHyAzsudVhSg4LrF5Vjlp9e6GEg/sjYNHRp0I2VFhOLz87jT35sZkEs7bA3K8RGxS7aZmRkLOGQalJRhb2ldsq/t48X2EuGQTfYSMZmcnMyUlVtmsWsXH5M2SJqfn8+LlCw+Lwivxa9Ut1dffVW1CHR2diqXcfHiReUyVB9e8ODBAytSp2rI8ePHVYtAS0uLUhn9/f3W3x+lQgDs2bNHtQjr759KIefPn89cvmnTpkx+uUzVwHF81Q8wNdxflzQ5cLMu9dSrPbn1XNz1H0hMVT8/drkO5vvOKyBTO3784x9nCj537lwm70Rm48aNyn8Qcu2QgdoPfvAD3L9/P/ewUv6VV16xHpmpVMiii3fv3o3GxsZFR6t7KQNLiR7IH3mnNvn2UR4T6tQm/SFtzp2/7VTZdStH1wDNgKkZ0GWdxMJ4NpGI4/qtG+gdulPeeupFyzekmNykWVOrTOtpWDKsLpUErXOvLZaXqIwctwb9so5C2lDGdbnnJFMp65pESpZ9L1+nVR806EhY8Q9pg4EUdKSQNE3MG+nnfeWCQSWDewEOOV8G97LZAGL7Qm659rGl9na9UoZd3lLnlnPctqUSG8opN4jnOAEcQdSFbfK/At+5du0aqkmH16zA6S2rHUvvPfGjJcv6Yu3PlnyvlA2lyix1Xan3vtzwq6psWarMHY/8V3RfvlRVHyzVb2+++aaj5R08eNDR8vx+2wjRy1QDmdcsmxPfBOVqInOcnfimzC5TBt8yLcLJwXdvby/eeecduwrlvbRX5jI7BRzT09PWN0FORp8ENMVGmbLi1CZ9I74UtE0Gljdu3MDg4GDQmsb2UIGSCvTduoljx+RBAtVtmzdvti6Uv9kfffRRxX8f5HNJ9W+zfBkiZah8qy9/0+X3VlS3J598UqkIsUPWyoyMjFRdjvztl8iCrEeqdpOogPwGjfzNr3aTMsQOlX6RNReVRCUW2yprw+woy4YNGyqypeoIh4poixsgr50cYNnlOznYsMuUcLWTm4rjLGWH02XKwIFbVgGZYvDpp59m/vg4Eb7Olg5rUaWTcCDTIeRDzMm1JvKBKoNvpzaZ3y0fcqdOnXKkSPn7JPZJXzm1yR97+eBxEjgkwuH031Kn2qtSDoFDRT1e60cF5G/N0NCQZbp8SVftZg8G7cinDOoq3bZu3VrpJQXnywJ/lS/TZHD+29/+tqDcSg6IjuWuD1iqXAEN+SyQL6Gq3WSdTdOJE0rQInU70Zaf/vSnmS87q2mPfKknbVEZH2/ZsgUdly5ZY6C21tayzSgbOKSz7t69mylYpfMyheRkxDnlcX5ObmKvk9NIxMZagJH9TblTbZdBkdP9c+XKFafMC1w5Kjdu4MRgg6iATC9ihIN+EFIF5PPc/ga4GgnsCIcMTuVLooaGhoqKkc9++WV4lalwAk+yaFglKiA6yEBfZXwj169cuVJpPCN6yJdFlTxNabHg0hcSiVaJRksZquMygU+JMMi062o3sUEiPtVGa+R6gWH5okzKkSfilbuVDRzyzaM4oL2pCmeXY+8FDIQindzEwU6cOOFYkTKwlJu/2o4qZoiETWUKlFObRCOkTCc3IWKVG81JW7xYlsq3QF5sD21yTwEn/7a41woCh5vas25/K2BHOKQV9toXP7ZI/pY5PU6sVgcn7PBKJFq1LXK9U58zldpSNnDI4F3C//ZWaUX2daX2TgOHUK0kpzZpszwe0EnHW7t2LVatWuWYA8gCZrHPqbnwop1Mn5FpWrXoc6f6xs1ynJy242Y7WLd7Cjj5N8W9VmRrZoQjqwVzVKASBXKBo5LreC4V8LoCZQOHDNxzHy1Wi8Gnk3AgwsuCKyefICTQpToHr5hDqIQtF5cnkSKx0ckBjER2JHTGrbgCjHAU14VHy1PAyXu1vBprfxaBo/Yas4ZgKkDgCGa/slVA2cCxWKxaAMfiOviaCvhBAQKHH3rJmzYWW2MWBAAhcHjT32iV9xUgcHi/j2hhdQoQOKrTjVdRgYwCXDSekYKZChQoBhbiS0EYcBA4KnAEnkoFchQIwv2f0xxmqUBGgaqBQx61yY0KFFNAFrrZSX5TIugbIxxB7+HatG8xcMhrmbYpye8bgcPvPUj73VKAwOGW8qy31gpUDRyLPyxrbSjL94cCS/nFUsf90arSVnLReGl9+G5xBXLvCcnLQznksdtO/pp58Zprf5TAUXuNWUMwFSBwBLNf2SqFNRwUjwpQASpABagAFaACVIAKUAEqsJwCVUc4liuY71MBKkAFqAAVoAJUgApQASpABQgc9AEqQAWoABWgAlSAClABKkAFaqbA/wcrcSwzfPWddgAAAABJRU5ErkJggg=="></p>
<p>The sides of each successive square are one half the size of the adjacent larger square. Let the sequence of squares be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mi>U</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>U</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>…</mo></math></p>
<p>The first square, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math>, has sides of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mtext>cm</mtext></math>.</p>
</div>
<div class="specification">
<p>Ben decides the animation will continue as long as the width of the square is greater than the width of one pixel.</p>
</div>
<div class="specification">
<p>Ben decides to generate the squares using the transformation</p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><msub><mi>x</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>y</mi><mi>n</mi></msub></mtd></mtr></mtable></mfenced><mo>=</mo><msub><mi mathvariant="bold-italic">A</mi><mi>n</mi></msub><mfenced><mtable><mtr><mtd><msub><mi>x</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>y</mi><mn>0</mn></msub></mtd></mtr></mtable></mfenced><mo>+</mo><msub><mi mathvariant="bold-italic">b</mi><mi>n</mi></msub></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mi>n</mi></msub></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>2</mn></math> matrix that represents an enlargement, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">b</mi><mi>n</mi></msub></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>1</mn></math> column vector that represents a translation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mi>y</mi><mn>0</mn></msub></mrow></mfenced></math> is a point in <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">U</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>,</mo><mo> </mo><msub><mi>y</mi><mi>n</mi></msub></mrow></mfenced></math> is its image in <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">U</mi><mi>n</mi></msub></math>.</p>
</div>
<div class="specification">
<p>By considering the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><msub><mi>y</mi><mn>0</mn></msub></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>,</p>
</div>
<div class="specification">
<p>Once the image of squares has been produced, Ben wants to continue the animation by rotating the image counter clockwise about the origin and having it reduce in size during the rotation.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>θ</mi></msub></math> be the enlargement matrix used when the original sequence of squares has been rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> degrees.</p>
<p>Ben decides the enlargement scale factor, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>, should be a linear function of the angle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>, and after a rotation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> the sequence of squares should be half of its original length.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the width of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>n</mi></msub></math> in centimetres.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given the width of a pixel is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>025</mn><mo> </mo><mtext>cm</mtext></math>, find the number of squares in the final image.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mn>1</mn></msub></math>.</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>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mi>n</mi></msub></math>, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>state the coordinates, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced></math>, of its image in <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>1</mn></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">b</mi><mn>1</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">b</mi><mi>n</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mo>-</mo><mi>n</mi></mrow></msup></mrow></mfenced></mtd></mtr><mtr><mtd><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mn>2</mn><mrow><mo>-</mo><mi>n</mi></mrow></msup></mrow></mfenced></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the coordinates of the top left-hand corner in <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>7</mn></msub></math>.</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, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>, in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>θ</mi></mfenced><mo>=</mo><mi>m</mi><mi>θ</mi><mo>+</mo><mi>c</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>θ</mi></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> after it is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>135</mn><mo>°</mo></math> and enlarged.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> at which the enlargement scale factor equals zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After the enlargement scale factor equals zero, Ben continues to rotate the image for another two revolutions.</p>
<p>Describe the animation for these two revolutions, stating the final position of the sequence of squares.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mfenced></math> <strong><em>M</em></strong><em><strong>1A</strong></em><em><strong>1</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac><mo>></mo><mn>0</mn><mo>.</mo><mn>025</mn></math> <em><strong>(A</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>n</mi></msup><mo><</mo><mn>160</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>≤</mo><mn>7</mn></math> <em><strong>(A</strong></em><em><strong>1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept equations in place of inequalities. </p>
<p> </p>
<p>Hence there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> squares <em><strong>A</strong></em><em><strong>1</strong></em></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A</strong></em><em><strong>1</strong></em></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>n</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>4</mn></mrow></mfenced></math> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mn>1</mn></msub><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><msub><mi mathvariant="bold-italic">b</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">b</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Recognise the geometric series <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">b</mi><mi>n</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>4</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mo>…</mo></mtd></mtr><mtr><mtd><mn>4</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mo>…</mo></mtd></mtr></mtable></mfenced></math> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p>Each component is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mstyle></mrow></mfenced></mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><msup><mn>2</mn><mi>n</mi></msup></mstyle></mfrac></mrow></mfenced></mrow></mfenced></math> <em><strong>M</strong></em><em><strong>1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>n</mi></msup></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mn>8</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mrow></mfenced></mtd></mtr></mtable></mfenced></math> <em><strong>M</strong></em><em><strong>1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>7</mn><mo>.</mo><mn>9375</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>.</mo><mn>96875</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>θ</mi></mfenced><mo>=</mo><mi>m</mi><mi>θ</mi><mo>+</mo><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>c</mi><mo>=</mo><mn>1</mn></math> <em><strong>M</strong></em><em><strong>1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mn>360</mn></mfenced><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"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>360</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>⇒</mo><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>720</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>θ</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mi>θ</mi><mn>720</mn></mfrac><mo>+</mo><mn>1</mn></math></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>θ</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mfrac><mi>θ</mi><mn>720</mn></mfrac><mo>+</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mfrac><mi>θ</mi><mn>720</mn></mfrac><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mfrac><mn>135</mn><mn>720</mn></mfrac><mo>+</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mfrac><mn>135</mn><mn>720</mn></mfrac><mo>+</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>cos</mi><mo> </mo><mn>135</mn><mo>°</mo></mtd><mtd><mo>-</mo><mi>sin</mi><mo> </mo><mn>135</mn><mo>°</mo></mtd></mtr><mtr><mtd><mi>sin</mi><mo> </mo><mn>135</mn><mo>°</mo></mtd><mtd><mi>cos</mi><mo> </mo><mn>135</mn><mo>°</mo></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>M1A1A1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>15</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>720</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The image will expand from zero (accept equivalent answers)</p>
<p>It will rotate counter clockwise</p>
<p>The design will (re)appear in the opposite (third) quadrant <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any two of the above</p>
<p> </p>
<p>Its final position will be in the opposite (third) quadrant or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>˚</mo></math> from its original position or equivalent statement. <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">h.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.iii.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">A</mtext></math>. The shortest time from sunrise to sunset at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">A</mtext></math> must be found.</strong></p>
<p>Radians should be used throughout this question. All values given in the question should be treated as exact.</p>
<p>Mars completes a full orbit of the Sun in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>669</mn></math> Martian days, which is one Martian year.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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6/Kndt35N7de3L//n159OiRPH36tHVhShie05Mj4AgkBgHaAHOhwVcq2UTKCVSQj2RGbm6u9O/fT8aNG6sOqQTYKyoslNKSEtm3d6/sZ4hzebkcOnhQFi5YIKNGjZQPPviLRjq2zOM+FusC4oR65ckRcAQSi8C8edny9TejZcOGDbJz504pLS3RUAMQIBzCWxWoxkZ1IseMB3FiRB5mOxYUJM5TAlVXJ0eOtIzE0zAFhCs4KAcPHpCKlrhPtbWMvKuVEyeOy6nTp+RcwzlBzbpw8YJcu3atlUA9ePBAnj55Ik+fxAjUs2fP5Oat2zJ48EftqtWJRcjv5ghkBgIou2beSyaRcgLVUr6uXbuqitIPP4zXaR12bN8uu/Lz3yFPZfv3K3mKNaaMzqnQ41u3bJF5OTka9ZhRN/SKnThlRsX1t+wdBDCB9+v3gSxetChGoHbskD2lpRqfCSdyCBT+ScR7Onf2nJImfKHU56kBs12MQNWfqdfzUKwgT4dra6WyqrKVOB0oL5PycmJAlbeoUJVSe7hWjh2PmQghUJcuX5Yb16/L7Vu35P69e/IY5enJE3ne3CzPnz9vXb94/kJKiou7He6kdxD3f3UEUh8BI1KYyyFSiBk9mZxAiehkof36fygL5s8XyND2bduUPBUWFGiDt3fPnlblCfKEtE8cmNhw5iNyrK5OThw/LpUVFTJixDBVo9zXoSeLrd870xFYtixP1V8IFJP85ufn69x0FlUcPyUlUGfOqMqEozeKE+rTWZzGW4gTyhOj6Rh9p7Gdamq0HkOaiAO1f/8+2btvr5SV7W814xH7CQKF+e7iRRzHL8ud27fl7t27arp79vSpPG9+Ls0tBOrFixfy8uVLefHipTx69Fg7WD3dsGd6+fD3z2wEjEghZqBM9VTKeAKFcvTJx4Nl3NixsnnTJtm2davs3LFDCnbvluKiIu3VmtkuJE/0VuvqjipxOnnyhJw+TW+3Xo4fO6bTSEyePLGn8szv6whkNAI0jjSMc2bP1ol8N6xfrxP9FhTs1ulWiBLOSLkTx4kDFQtwySg7zHUQKZZTp06q2Q6ShVoFKcJJnDpOwEwIFNHHiZSMc/r+/TEChToFOTtx8oSqWYy8u3r1ity+dVMe3L8vDx8+kGfNz+Tli5fy+vVrJU6vXr0SW56/eCGrV6/ysAYZXYL95ZOFAEo1ShRhEHpi4FZGEygcSP/617+oc/iwYcNl+bI8iZruIE/lZWXq82TKE+TJVCcaYpxQz5zBYfWsnD/fIFs2b5bBgwdpo56sguL/4whkCgIQFiT6+bm5SqDo+DDh7549e5Ts4PStKtGxYxqnCZJUf/q0hiqAUBG76fSpU+poHiNPRzS2U001o+4qta7j54jJHgJVuqdU9uyNTedSUXFI6o4fUwLFoJDLVy7LrVs35e7dO/L48SNpbn4mz583K4H66c1P8ub1GyVSr16/VhIFgTp3rkEYmOLJEXAEkoOAOZon2h85KQTq008/1h4jRCNe+uab0e0ej3dNd/choeP8PWXKZMmdN08gUCtXLG9Vn/BVwHRHI3rwwAH1d6KBxWxnyhPkCdXJiFNTE7FgmtSHit7xhx/+NemjArqLi1/vCHQUAcgESpAtCxbMl2fPnnb08i6fN3fuHJ0mBQK1auVK2YpqvHOnkqdy6mpVlU7ua/PcoQpjqouZ606oyQ6fJ44TNJPgmKo+VVSoDxUEDMWJBQLFpMLl5ftVnaqurlLydKr+tFy8dFGu37ihy/0H95VAvXzxXH76+Sf5+aef5Oeff5Y3b36Sn376WV6/eSMvX70SCNSTp89k5MgRPdIj7jKofqEj0McRYJAXHa9EkqikEShIFA1sNP3rX/8j7yNY0WsS8Xv27NlqaoM8TZ82TUaNGiVbN29+x3E8NN0RlRgH03jkiZ4o5Al/CKZv4J4sjOQjRo0nR6CvIUBdZvnjjz9aX80I1T/+8Vvrvp7YgLBBopYvWyYbN26Ubdu2SWFhgTDRrxKoysqWgJeHVVkyH6djdcfUXAdpOnbsuBIolCpUJ52qhZAFmO/KyrTjBIEq279PSRQEinOo/yhY5xvPy/Xr1zXe08NHD+Xp0yeqPP3yyy/yUwt5+uXnn+Vvf/ubEqrXb17Li5cvlTw9fPxYo6ajZHtyBByB5CGAGS+RJCppBAr/BIjS77///g5aNLrr1q1NqgKF+kQjPHPGdJkxfbpMmThRQxbg+0TYAnyfouoTDqY0ePRml+flybFjdao8QZ5QnS5duihXr1wRRvOtXbOmlUQNGNA/oYz3HfD8hyPQCwhQh6k/8YgSajJ1vacSDSDmcXUeX7VK1V78FguLCqWktET9lmJTr1QpicJficCYR48cVfJEnCdIE/tRnmLkKRbvqbz8gKpYOIzTecKBHBKFMzl+VYRHgHyhpDddaJIbN67LvXt3VXlitB0+T3//+9/lV5Zff1Xy9DOESpWoN/L0WbM8evxYHjx8pPdfsmRJT8Hk93UEHIE2EDASZWGH2jitQ7uTRqAgSvRYo2Y8Glxkfxrk8Jj1ZtnPMnXqFEGtItHrZR8NtR2nUed6U7PseDwUCgp2aZgBJh5dv25dq/N4/s6dYiPviPlETxSn0lB9ouFGXZoxdaqaBFCeUJ0gT9evX5ObN26oH1WoQk2fPj3eY/g+RyAtETACFdbXeC9i9TpUqaL7qNd0oGgbrC63R8CQ34cPHyYrVqxQ893mzZt0BF5JSUkL+SHkQCzoJYQHkgRZgkAdOXJUag8f1n0cQ1E6dPCQEqQD5Qdk3759UlpaKqWle9Sfat9e1Kf9GkiTgJrVNbXqlM68d5cuX5KbN28ogUJ9evX6lZrsfv01Rp5++/U3JVOmSL14+UqePH0qjx4/kRs3bsqJkyd1xG483HyfI+AI9CwCzKeHEtXd0fJJJVAQDBpKS/RgIVBGiKxBhjzZfs6lwea3XWvn0/iSIFbciwbYfDC4BjJl97T/ZE2E8VkzZ8qa1auFETw4fVvoAoJmEk/GzHeEJsD3ydQnfKOys+ZKbk6Omu1MeeLdmECUBp5ReUagsrOy9Lm6m1Hh8/u2I9DbCFjnhTpGfY1Xz6JkiWeO7qMOU2+pP+Fx+x19T4JaEqtt7dq1sm3bVp0iaffuXQKBgvxAnlCLIEf4NUGUwoW6rMpTi78TqhMqEyPtcELnPizFJSWqau2xjlRlpSpXOJ83nDunc+kx392jRw/lefMz+enNG/n73/4mv/36qy5s4wOFOY/wBc+am1WBunf/gVy7fl0uX76i7x19P//tCDgCyUGA6dW6G+IgqQQK4hOa8Wh4WYwQxWuEDUp6qZAoUrzzaXBpiOOZFewerG0I9MoVK9TUtnHDBo39xOg7M98ZgTLnccx3AE2sJ8IV0CAz2g7THROTYrYz8kQgvQcP7us9jUR98slgdyYPM8G3+wQC1Dkzv5t6xG9LUbLE/ug+CJR1jOw66jntQrzEvFeTJk6UDRvWy9atW3TKpULCjezZowoSIQhiBAq/Jpa389hVVwfO4gcP6mg7zHOEKygqKpJdu/KVkDGNE41rQWGBhkXgnrU1selbzjL3XSNz312U27dvxwjU82Z58+a1Eqh//Pab/Pbbr4LyBIF6+fKVPH32TB48fCh37t6TW7duydVr13TiYTDzjlW8XPZ9jkDPI4Apj4DX3UlJJVA8KA2sESUaSghPPEKEqmQEy3qpkC9SvPPZb2YA6xXTWEcTds9hw76U9giU+T8xXQuhC1qdx08c12dn1B2mO1Of8IW4ffuWzr5OHJgn+Dk8uK8xpSBRw4d/2W2mG30P/+0IpBIC1FdTpYxERckSzxvdR92OkqV4++xdZ8yYLosWLZRNmza+Q6BQn1CS6PQwig6TW0VFpRxCaarAx4no4hVy8NAhNdtprKeyMlWtIF+QJpTodWvX6n1xUEfhYoQfIQy4Dv9HwpUwCfHVK1eVQOED9eJ5s5KlX36JOY2/ef1K4z81P3+u6xiBeiS379yVxqYmOX++UU6eOiUTJ07wjpVlrK8dgV5AADNed4LaJp1A0WuF6Jj5DsyihAiCRe+MBpnG1Xq67yNQ3AvTHdcYmYr6U0Cghg79TP0naCw3MYonCJ6Jie5P/k9M3VBXp6Y5whYwDUSr79PVK+r3hOmOQHo6jQMjcpqJBfNCJxBmuDX+FZ4cgb6OAPXN6mmULPHu0X3xyFK8fYYbBAo/RMgNihEqEea2WLDLMp1qCQJF5+ctYWI+u9jCPshTGYEyW0x3XM+9tmzZrKr0srw8WbBgQWuIBAgWRIzgnMSNIo4Tsaiu37gu9+/fE0IY4AeFCtX8vFlePH8uz541y+MnT+Thw0eqPt26fUcamy4ogTpz9qwSqPHjxzmBsoz1tSPQCwgsXLhAOz9d/eukEygelAYWgmM9zyiB4ripVPZiNKrWMEfPt3OiayNinG+pMwRq1apVMn9+rk7xkrdkiY74wbwXEigz3929c0dVJ9Sn5mdEIn4hb16/ljdv3mhsKfwxPDkCfQEB6q2Z06PvQ52zY0aW6NRYitbJeGQp3j67fubM6bJixXLZzlyVTPRdVKxO36hPNmIOgoQvFKoRC2Y6XQ6wJsp4zGynQTJLS2X37t1qssMsuHzZcg0xAIlaunSpbNmyRckVBMwc0gmLQFTzixcvqkkOZ/IHDx/I3Xt3lUBR/58+faYE6tat23Lnzl1punBBHdBP19dru4dSNm7cd06gLGN97Qj0AgJEKO/OaLxeIVAm9UdH1RlpogGGYFmiwUaRao9AcS3H7Z5cyz1ojMPUUQKFic98mGw94YfxGv24PQJlk4i+evlSHUsJqMfoPgLweXIE+gICNkCD+hWSIwgTdRfFmMQx6q11lKibHGefdWrikaV4+ww3CNTChQtl1+5dUlhYqOQpZr7bp75M+DOpea4cEmVkab/OZYfiRIgCCBRkCwJVXFysRIw5MNesWaPz6tFxWrlypY7027ptq+zM36mO5ZgFCYFAEE6UaObVu3Tpkty4cUODaqJKManw/fv35d69e+rrdPXqNbl06bISKOJH1dRUq8JN4M8hQz71YJqWsb52BJKMgPlDd+dve4VAWUNrDx5VlGhoaURpaFnMb4ptGuXo+XYf692G14UNPOdBoPCBIoJxeyY8I03heuyYb2XIkM86rEC9fvVKWBjp111vf3tHXzsCqYAA9co6QlbfqLNGnuwZ+U3HhnM4bnW0qwQK4pOVlaXkCfIDecJ8t29/jEBBjlSBalGdiOWkcZ00IOa+WHwnneMuFrIAEpa/a5dAlNatW/cOiVq/PuaovmPHdiVQ3LemlgElR5VEMSEx9ZpJhRsaGlSRun7tmpIoFCcUqouXLsm5hgY5ceKkEjyel1ApECgw8eQIOAK9gwBuNwxK6U5KCoHqzgMm+loLomlO5G35QIXESbdz58no0aNl7NgxGgG51Yk84gP16GEsKjEyPr4QmPKmTZvaLZkw0Rj4/RyBdEUAwkIIEg2cWVKio+TwLwwJFCqUmuzKy1RpgkShNrHEtt+GLSguKlYCRTRzopoTBJfJflGjMOlt2bpZzXuENiAuHA7l3BsfK/WrOnRQTp0+FYt4fuK4OphrqIOGBh2th7+Tmu0qK/T/8bGEQK1YvtwJVLoWQn/utEcA9QkH8u6Y7wAh4wgUL40TOaSIxjJeGANG4S1ZvPgdE968nGwZM2aM9mCJLk6cGaZuIYAmo/CYUPTe3bvqB6WO5E+eqGMp++hp+nDltK9z/gIpgAAN3rjvxqrpDVITcx5/a74ra1GeIFGmPO1vIU979zLZcCzcAY7hqEFKaAoLBZUJJ3JI0/r161rahpgCtX3HdlWMUKtQvbi2vLxMCRHz5FVVVWrMKZQp/KOYc49Ji4l6zm/2V1RW6HWbN/MfG3QqGuJZeXIEHIHkI4DyRKiS7qaMJFA0lmPGfNtuIM1d+fnq72RK1LjvvpPd+buECYTxl4BEEZSPKOQ4khOB3EIZMBqPcAaoUQTk9Iayu8XUr7jpnzwAABLmSURBVHcEYgiY3wLkiYjhe/fGpluJESaUJ/yeIE+oT2+VJ4gTZIeFDhLEq6SkWE2BxH/buXOHENWc8AiQqI0bN8QUqC1bZNv2GIHC2bywsEjn3WPuvZLiIindQ+Ty2BQytbWxufcgTIQ/qaqqFggdxGvHjh3aWVu+fLm2K6NGjUxIA+7lwhFwBDqHAMQJ5/FEiBoZSaAIoNW/fz+ZO2e2zJ09W5Yvy9NI5DaVS2lJiRIfnE6LmGOrpESdwC2UAU6x06ZOlX79PhSCbFowTSNRjMhDeSKoJv5WqFOeHAFHIDEIMJULYQzUdKdz1uEcHiNNMdPdgZbfMadxOjGY74w4saZOa7TxwkJ1IofgEPMJAoUpD9P+5k2bBMUI8x4LBGp3QUErgSICejEkqrREZy/geXgOZi/A4ZyAnvv271eiBIni/jayl+C6TCfhyRFwBJKHQCLJE0+dkQSKFx83bqx8883XkjsvR74ePVrWrF6l4QYKdu9WWd9iQf0pmObx49pbRZkaO2aMDBo0QFCrbCJhNefdvKlqFERsxIhhySsd/k+OQAYgQKcmJyenxe/pz+QJFSocgacj7vbvU9UJ5QkTXAnmu5ISjUAeU6B2KklSArVhgypQbG/atEmJDwSK0AmoUawhXPn5+brwPBApopkXFRfrUlBQoP+3bTvXbdPRfUwdtXDBApk+bZqb9TOgnPorpg4CqE05OVkJU57szTKWQJkz+fTp0wTzHCpUe9O52Hx4pkItyM1V4kTvFhKVk52t07togL3rTNXQJIMGDZTm5mbD2teOgCOQAARQkKlzKD6Qo1B9MuJUTgiDFmdyzGixsAU4j++LEagW8sRIQpQlU6DMR8lUKPyVbHsTalQLgYIUQaQYTQeRYm0L8anYx5rpZjAJ5uXlqV8l7cTo0V/JnDmzEoCE38IRcATeh4BNHEydTYTZLvy/jCVQgHDkcK0MHNBffhj/vfon2ITCjJIxMx6NcMWhQ61TujAKiCkdmDCYIdlnz55RvyiGaNOoE2gTk96PP/6gPhUh2L7tCDgCiUEAMx7hTSBP+DxhujMHcraNSNm+GIGKqVCqQOH/VFSk5AniAxmCPKE44f8U84HaIIQyYMGkB4HavGWLnoc5jqVVmdq2rZVA2f04DvnCYXX+/Pkyd84cVZ/69/tQJx1PDBJ+F0fAEYiHACIJdY/Rdj1lLs9oAgXoa9eukc8++0xjQoVTuhQXFanPBAqTTipcUSHVVVVy+HCt1NUd1YmFcSg3EkVwzaLCQvn886HyyScf6ySFiWa78QqJ73MEMhEBon8zOCMWHPNt2IIDjMJrCaCppKrFNwpfJBQrG33HaDo6SqhPO/F/2rZNOzwQHggTMaHWrl2rC0SNbdQo9q9bv07PaVWnNm9WpWnL1q0auZy4bzikEzF99ZrVGtF8AQRq7hwdQWgBgTMx3/ydHYGeRoDvLkIHxKknVKfw+TOeQAEGJIoAmUwgGlWh8IWikcYXitAFmPJCEoUSdfr0KY1MfOL4cfWtIkzC4MEfqc21u3EmwszybUfAEYghQCP50aCBqhy9VaGIQo76FJu6JbY/Zr4zBYrQBRrMsrBQyROmth07dqppDvUpFsZgvcaBIswJbcPq1av/vOixta0K1QZG7W1sUazWrVPytGxZnixevCg2mjd3ngYSpVGvqqrwbHQEHIEEIxASJ/ydkvHtdQLVkolMtcLIvClTJmuPFMdSFCVMcqhQmPKiJMrMeSdOHNcROwMH9lfpn4zsjcxMcHn02zkCKY0AU6h8/vkQ2b//7Qg8tlWF0nhQhDKIBdM053GN/VRSKgUFNvoO5/GtapYjJhxKE0E016xZrWFOIE9M68LMBYygMzLFOezXYy2BN/XYqlV63pIli2XJkiVC/LhZs2bJ7NmzdNCKq08pXaT84dIQgd781jqBCgoMzqmTJ09U6Y/o4QxjDk15NMwhicI3iulgUK+GD/8yrp01zFxiTxA+nn2eHAFHoPsIjBs3TufGU0fyFiLFNsQJ0mQLo+4w32G603AEOoEwfk+b1OzGiDsCaGJyY4EwoSAtW7ZMmLWANdHDV66IkSaIE/uWL1+hDuIrVqwQYjwtXrxYFi1aJBConJxsJVAzZkyXSRMnyAcf/CVuG9F9FPwOjkDmIcD3GhMdqm6yFKcoyk6gooiIyOPHjzRQJorUZ599okE358yeLVlz50p2VpZMnjxJRo4coUORc3NzOuwQCnmCRJltlgLgyRFwBLqOAHWISP84buPjtJd4T8yNp7GfYk7jGlqgqEjJEwEwIVCY7ogwDoHCx4npW9auWyurVq2U5SuWqwluydIlSoQgQ0uXLJG8pUslL2+ZEiZG1dkCaVq8hHOXKHlauHCB5MzLkay5czTWHOTpk48Hy/z587r+on6lI+AIqPgQ/Y7iLN5byQlUHOQJuvfVV6P0CHZUMoxZ1PGjYNqGxsbzal/tqpIUMmcIFffrzUIQBwLf5QikDQLnzp0VzOcExtyzd29sipbSEo3NxJx5qjoV7FaSRfBLRtsxbQsj6zC7QZpQmVauXC55eUtVeWINEVq0aGEriWJ6J1WYFi6URYsXyaLFi1X9Wrhoocybl6OhTLKzs3R75ozpGmwXl4ARI4br9FFdbS/SJiP8QR2BHkKAUXSmNqWSJccJVJwMhzxBono60aBCzhhqSS+aNb+9oe1p5P3+fQ0Bwg6gCusUKyXFUsxSXCTEeSLGU4w4xcx1Gq5g40Y102F2wxSnClPeUlm69K2KlJs7TyBHEKl5ufOUGEGUlCzlZMv8+bnCOZjqsrLm6gKBwt9pxvTpOon4mG+/lf79PxRXm/taifP36WkEqDMmZtg3OdXqkROoSCmA6UJmkq0IRckUNl1XpiKZ4z8dgXYQmD17pqo9+fk7paDFVEevdcuWLRrfCX/F1atWyXoNUbBGnb1RnhYtXCiEGYAQQY4WLFjQSo6YqQCSBEHKzskWCBJLSJjYJkTBnDmzlTRNmzpFCNDLDAe0JckYDdQOLH7IEUgbBExpgjClKmkKwXQCFaIhojIhEmFvpiiZ4nlg4qnGvnsTI/9vRyCKAPUGEjXsyy9kS8scdkz8jeLECDsIFH5MaopbtEiJE+oS06tAkrKzYwQJRSlGkrJk9qxZSpZYQ5KiCw7iqjhhsps2VZdJkybKN1+PVuUJRdmTI+AIxEcg/NbhG5wOpCl8EydQARpkJj3GVGv0QlZuDug8Y7JVsgAq33QEUhYBgl1iNoMoEW6AEXUQJ9QmlCYIE6pTzASXqxHCiRLOABFTkiBFDByZOXOGTJ8+XYnUrFkzBcI0a+YMXU+fNlWmTpmioU8YWDJ1ymSBPH0+dIgMGNDPOzwpW0L8wXoTARRZQgAhDPC9TWdrixOooCRBSshQiFSqJlQoTHvmN2WMHZKVys+dqnj6c/VNBM6cqde6/P3349S/CWdw5qGbn5vbSoZmzpihI2tnzZypJrdpU2OEaMaMGEHCj4mJf6dMnqzHcQg3lQkzHdM3sUyeNEkmTJigUcYJoPvtt19756ZvFit/qy4g0Je/WU6gggIBI8ZnIp2SsXlYPOTPzH0QKleo0ikn/VkTjQAN9zffjNZJvSdM+FGJE0oTJEiJz+RJqhgZKVJCNGWKkqGJEyfIlMmTZMKPP8jECRM0jhNEadLEifLjjz/qOcx3ab+HDftS6x+jdD05ApmMAN8k6+SbWY7val+0mjiBainpkA0ICMQjnVNIqMLCS4F2H6p0zll/9q4icPXqFTXpEQV81KiRSp5+/OEHGT/+e10gS5AkSBbrH8aPl+/HjROCdKJgsZ8Jx5l7D9IEgRr//fd67IsvPpd+/T7Q0AfeYelqDvl16YoAVg++mZjkwk58XyVM0XxyAtWCiA2XjAKU7r9NPqVAY+4zmzMF3lWqdM9df/6OIkBDD5EaMuRTjQj+6aefyMiRw2Xcd9/J2DFjlDB9991Y3f5u7FjdD4limzXL+O/5PUZGDB+uc11Sl5YtW+pKb0czwc9LewRMXWrre5JpbiROoFqKNGpNMmI/9XYNivYYeG8Weg9Oqno7d/z/k4EAnYpdu/Llq69Gaodi0KAB8vHgj5RcffnFFxpPatTIETJ8+DAZOmSIfD50qAwcOEA4D+f0OXNmeby2ZGSU/0evIhCSJXP4NjcXzHFu0RBxAiWiSgy9yUyV4HnvUIaNkiqvLL3ajvmf9yACdCjM7A2pysqao1M1MaqOZcOG9VJeXqaEyT8YPZgRfuteQyCsAyhLUbKE+wd1xNOfEXACJaIj2ig0nt4iAKmyDwsj/qKVCrWK45lKOt8i5VuOgCPgCKQHArTZdIjNZynsLGOB4ZiTpY7nZcYTKNg36hMFx9P7EbAKSGXD7EcFNL8qei9GrLy3/n4s/QxHwBFwBBKNgClKKEdGlMz/lY4wHWJrp70D3D30M55AUcggABQ6T11DIKywRqzCCmv+Vda7cay7hrNf5Qg4Ao6AIUAnlQ4tZIjOa7RDa0QJ9wzv0BpqiV1nPIGCkVP4PPUMAqZYUcmp0FRyCKupVlbJnVz1DP5+V0fAEUhfBEKSFO2c0kmlPWW/K0q9k8cZTaAonHzIYeiekouAqVYQp3jkCmJrypU1Dm6bT24e+b85Ao5AzyJgvqbWDkaVpHgkydWkns2Tztw9owkUzJ0C6in1EIAs2chA63mZI7upV/EIlpsHUy8v/YkcgUxFwBQk80cygmQuDhZCxpR4G/HmvknpUWIymkBReFE3PKUXAqZeRQlWaB60nps1TKGK5SQrvfLbn9YRSEUEjByZemQdvfbaISNIriKlYo52/pnSkkA9e/ZUTW//+tf/dP6NW67g44uS4Uy/yxCm9IUdadys90eDZ34E1sChgDnRSuks9odzBHoEAep+qICHLgahCu6dtB6BP61umpYEKhEIm0NzIu7l90hPBMz/gMaSRpLFJPawFxmaDENFKyRbTsTTswxEn5p8dOIcRSX9f8cjRaFiZCY16roRI9oAaxdskAv38eQIGAIZSaBoIKkoVApPjsD7EKC8WANscn2UbFk8LMpVqGyFhItr7D6snXS9D/nkHyePyD/IsafURCCsQ2F9DAlRqBRRJ21QSkiKwvroJrXUzOtUf6peI1CY4RYsmK9EhgLOUlFxSPH6xz9+e+c3OznGbOqY7aImPH5PnTql9V7c9/fff28TexpHGknvabYJkR/oBgJmPqShb6uBpyEPSRfln322oITRwNvCfcIPh5OvbmRQO5eCt7VHfHTB3FPiEQjLMttWzlmHRIj6EKpD5E1IhtrroHj7nvh88zu+i0CvEKg//vhDyVB9/enWp4EgUTkgTySO2W8jVHZ+lEBBrIx8cS1kChLVVqIC8oHy5AikCgKhysUHBZIfflTM5GwEK0q++G3HbG1+XeF98P2LfrycjL0tBWBlBMrWYO8YyZ/KTbxyCn7Rskp7a1jaOiRBlNdoWQ3LqatDb8unb6UWAr1CoOJBgLJE5bp+/VrrYYgQyzffjH6HEIUECqWJ64xctV7cxgaVkfOp/J4cgb6CAB/4KDEKiZNtG7kK11Ey1tZHzq6JqmN2b9ahX1j0eex3KisDvIO9f3TNsVR69lDpNGzDtY1SDfMn3Lb8jK47Ux7ilYWoWsozeXIE+iICvUqgID0oRwUFu1sbrZAIQY5Ql1hCk1xIoMgUU69o8LhXeI9optHTQRL25Ag4Au0jEH6Mw+2oOhZ+lPmgRj/I0d9RYtLe7+i13fkdPmdb29y/veeBXGzYsP4ddbAj79zR526LvMR7ptDZua37t/We7A9VnjB/XW1rv174UUfAEOgVAoUJD1UJdQnywxI10/GARpRoPNi2ZPujYQy4z7p1a7UB5N7xEg0UjYcnR8ARSG0EombN8CPf2e33qTFGNN5HoCAthYUF7xCojqhuHX1eN1eldpn0p3MEQgR6hUChEKEqhclIkalHRrLwZYIUQbjYR7JzowTK7mdkLCRdHKNn5f4MhpKvHQFHIIoARKottQfTlCdHwBFwBAyBXiFQ+DnRSJnDOEQIxYh9RqBC0mRkCvMcKSRQ5jtl13EcJQqCZoTLXtbXjoAj4Ai0h0CUQJlinUq+T+09vx9zBByB5CHQKwSK14PkWE8PdQlSxRqSZAQrdCgP94UEinvZtXY/VKuo+pQ8SP2fHAFHIF0RCAmUq9Xpmov+3I5AchDoNQKVnNfzf3EEHAFHoOMIQKAYYo/PkidHwBFwBNpDwAlUe+j4MUfAEcgoBHwEWkZlt7+sI9AtBJxAdQs+v9gRcAQcAUfAEXAEMhEBJ1CZmOv+zo6AI+AIOAKOgCPQLQScQHULPr/YEXAEHAFHwBFwBDIRASdQmZjr/s6OgCPgCDgCjoAj0C0EnEB1Cz6/2BFwBBwBR8ARcAQyEQEnUJmY6/7OjoAj4Ag4Ao6AI9AtBJxAdQs+v9gRcAQcAUfAEXAEMhGB/w+tjnZX8+bFTAAAAABJRU5ErkJggg=="></p>
<p>On day <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo> </mo></math>, the length of time, in hours, from the start of the Martian day until sunrise at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> can be modelled by a function, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, where</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi>b</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mi>c</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is shown for one Martian year.</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="specification">
<p>Mars completes a full rotation on its axis in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn></math> hours and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math> minutes.</p>
</div>
<div class="specification">
<p>The time of sunrise on Mars depends on the angle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math>, at which it tilts towards the Sun. During a Martian year, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math> varies from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>0</mn><mo>.</mo><mn>440</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>440</mn></math> radians.</p>
<p>The angle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>, through which Mars rotates on its axis from the start of a Martian day to the moment of sunrise, at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ω</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>839</mn><mo> </mo><mi>tan</mi><mo> </mo><mi>δ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>ω</mi><mo>≤</mo><mi>π</mi></math>.</p>
</div>
<div class="specification">
<p>Use your answers to parts (b) and (c) to find</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> be the length of time, in hours, from the start of the Martian day until <strong>sunset</strong> at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> on day <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> can be modelled by the function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn><mo>)</mo><mo>+</mo><mn>18</mn><mo>.</mo><mn>65</mn></math>.</p>
<p>The length of time between sunrise and sunset at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, can be modelled by the function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn><mo>)</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>)</mo><mo>+</mo><mi>d</mi></math>.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn><mo>)</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>)</mo></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>d</mi></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Im</mtext><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>−</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> are complex functions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≈</mo><mn>0</mn><mo>.</mo><mn>00939</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle through which Mars rotates on its axis each hour.</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 the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>98</mn></math>, correct to three significant figures.</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 minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn></math>, correct to two significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> in exponential form, with a constant modulus.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find an equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo> </mo><mi>sin</mi><mo>(</mo><mi>q</mi><mi>t</mi><mo>+</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in hours, the shortest time from sunrise to sunset at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> that is predicted by this model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognition that period <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>669</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>669</mn></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>00939190</mn><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for a correct expression leading to the given value or for a correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> to 4 sf or greater accuracy.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≈</mo><mn>0</mn><mo>.</mo><mn>00939</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>length of day<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>24</mn><mfrac><mn>2</mn><mn>3</mn></mfrac></math> hours <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>666</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mover><mn>6</mn><mo>¯</mo></mover></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>667</mn></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mrow><mn>24</mn><mstyle displaystyle="true"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mn>360</mn><mrow><mn>24</mn><mstyle displaystyle="true"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow></mfrac></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>255</mn></math> radians <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>254723</mn><mo>…</mo><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>37</mn></mfrac><mo>,</mo><mo> </mo><mn>14</mn><mo>.</mo><mn>5945</mn><mo>…</mo><mo>°</mo></mrow></mfenced></math> <em><strong>A1</strong></em><br><br></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>substitution of either value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math> into equation <em><strong>(M1)</strong></em></p>
<p>correct use of arccos to find a value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Both <em><strong>(M1)</strong></em> lines may be seen in either part (c)(i) or part (c)(ii).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ω</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>839</mn><mo> </mo><mi>tan</mi><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>440</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>97684</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>1</mn><mo>.</mo><mn>98</mn></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> For substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>98</mn></math> award <em><strong>M0A0</strong></em>.</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>440</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>16</mn><mo> </mo><mo> </mo><mo>(</mo><mn>1</mn><mo>.</mo><mn>16474</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mtext>max</mtext></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>97684</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>25472</mn><mo>…</mo></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>7</mn><mo>.</mo><mn>76</mn><mo> </mo><mtext>hours</mtext><mo> </mo><mo> </mo><mo>(</mo><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>70</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>98</mn></math>.<br><br></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mtext>min</mtext></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>16474</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>25472</mn><mo>…</mo></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>57</mn><mo> </mo><mtext>hours</mtext><mo> </mo><mo> </mo><mo>(</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>55</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>56</mn></math> from use of rounded values.<br><br></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>-</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>1</mn><mo>.</mo><mn>59408</mn><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for substituting their values into a correct expression. Award <em><strong>A1</strong></em> for a correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> from their expression which has at least 3 significant figures and rounds correctly to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>6</mn></math>.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>1</mn><mo>.</mo><mn>6</mn></math> (correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> sf) <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>+</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo></mrow><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>12</mn><mo>.</mo><mn>333</mn><mo>…</mo></mrow><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo><mo>+</mo><mn>1</mn><mo>.</mo><mn>59408</mn><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>59408</mn><mo>…</mo></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>.</mo><mn>17</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>16666</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>16</mn></math> from use of rounded values. Follow through on their answers to part (d) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>6</mn></math>.<br><br><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>18</mn><mo>.</mo><mn>65</mn><mo>-</mo><mn>6</mn><mo>.</mo><mn>16666</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn><mo>.</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>12</mn><mo>.</mo><mn>4833</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Follow through for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn><mo>.</mo><mn>65</mn></math> minus their answer to part (f).<br><br><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at least one expression in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><msup><mtext>e</mtext><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced><mtext>i</mtext></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn></mrow></mfenced><mtext>i</mtext></mrow></msup><mi mathvariant="normal">,</mi><mo> </mo><mo> </mo><msub><mi mathvariant="normal">z</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi></mrow></mfenced><mtext>i</mtext></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn></mrow></mfenced><mtext>i</mtext></mrow></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi></mrow></mfenced><mtext>i</mtext></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mo>.</mo><mn>83</mn><mtext>i</mtext></mrow></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>6</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mfenced><mrow><mn>3</mn><mo>.</mo><mn>06249</mn><mo>…</mo><msup><mtext>e</mtext><mrow><mn>2</mn><mo>.</mo><mn>99086</mn><mo>…</mo><mtext>i</mtext></mrow></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>06249</mn><mo>.</mo><mo>.</mo><mo>.</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>150729</mn><mo>.</mo><mo>.</mo><mo>.</mo></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>99086</mn><mo>.</mo><mo>.</mo><mo>.</mo></math> <em><strong>(M1)</strong></em><em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> variables (or equivalent) must be seen.<br><br><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>06</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>99</mn><mo>)</mo><mo>+</mo><mn>12</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>06248</mn><mo>…</mo><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>99086</mn><mo>…</mo><mo>)</mo><mo>+</mo><mn>12</mn><mo>.</mo><mn>4833</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>Note:</strong> Accept equivalent forms, e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>06</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>151</mn><mo>)</mo><mo>+</mo><mn>12</mn><mo>.</mo><mn>5</mn></math>.<br>Follow through on their answer to part (g) replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>.</mo><mn>5</mn></math>.</p>
<p> <em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>shortest time between sunrise and sunset</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>.</mo><mn>4833</mn><mo>…</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>06249</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>42</mn></math> hours <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>9</mn><mo>.</mo><mn>420843</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>44</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> sf values.<br><em><strong><br></strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.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.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.iii.</div>
</div>
<br><hr><br>