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<h2>SL Paper 1</h2><div class="specification">
<p>The volume of a hemisphere, <em>V</em>, is given by the formula</p>
<p style="text-align: center;"><em>V</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{4\,{S^3}}}{{243\,\pi }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>S</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>243</mn>
<mspace width="thinmathspace"></mspace>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</msqrt>
</math></span>,</p>
<p>where <em>S</em> is the total surface area.</p>
<p>The total surface area of a given hemisphere is 350 cm<sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the volume of this hemisphere in cm<sup>3</sup>.</p>
<p>Give your answer correct to <strong>one decimal place</strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down your answer to part (a) correct to the nearest integer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down your answer to <strong>part (b)</strong> in the form <em>a</em> × 10<sup><em>k</em></sup> , where 1 ≤ <em>a</em> < 10 and <em>k </em>∈<em> </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{Z}">
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{4\,{{\left( {350} \right)}^3}}}{{243\,\pi }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>350</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>243</mn>
<mspace width="thinmathspace"></mspace>
<mi>π</mi>
</mrow>
</mfrac>
</msqrt>
</math></span> <em><strong>OR </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{171500\,000}}{{763.407\, \ldots }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>171500</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</mrow>
<mrow>
<mn>763.407</mn>
<mspace width="thinmathspace"></mspace>
<mo>…</mo>
</mrow>
</mfrac>
</msqrt>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <strong>(M1)</strong> for substitution of 350 into volume formula.</p>
<p> </p>
<p>= 473.973… <em><strong>(A1)</strong></em> </p>
<p>= 474 (cm<sup>3</sup>) <em><strong>(A1)</strong></em><strong>(ft) <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>The final<strong> (A1)(ft) </strong>is awarded for rounding <strong>their</strong> answer to 1 decimal place provided the unrounded answer is seen.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>474 (cm<sup>3</sup>) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C1)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>4.74 × 10<sup>2</sup> (cm<sup>3</sup>) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Follow through from <strong>part (b) only</strong>.</p>
<p>Award<em><strong> (A0)(A0)</strong></em> for answers of the type 0.474 × 10<sup>3</sup>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, in terms of an angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta = \frac{3}{4}"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \theta > 0"> <mi>tan</mi> <mo></mo> <mi>θ</mi> <mo>></mo> <mn>0</mn> </math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \theta "> <mi>tan</mi> <mo></mo> <mi>θ</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{\cos x}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>. The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \theta "> <mi>x</mi> <mo>=</mo> <mi>θ</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> is rotated 360° about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>evidence of summing to 1 <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum {p = 1} "> <mo>∑</mo> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></span></p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta + 2\cos 2\theta = 1"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct equation in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta "> <mi>cos</mi> <mo></mo> <mi>θ</mi> </math></span> <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta + 2(2{\cos ^2}\theta - 1) = 1,{\text{ }}4{\cos ^2}\theta + \cos \theta - 3 = 0"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mrow> <msup> <mi>cos</mi> <mn>2</mn> </msup> </mrow> <mi>θ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>4</mn> <mrow> <msup> <mi>cos</mi> <mn>2</mn> </msup> </mrow> <mi>θ</mi> <mo>+</mo> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>evidence of valid approach to solve quadratic <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>factorizing equation set equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0,{\text{ }}\frac{{ - 1 \pm \sqrt {1 - 4 \times 4 \times ( - 3)} }}{8}"> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>±</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mn>8</mn> </mfrac> </math></span></p>
<p>correct working, clearly leading to required answer <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(4\cos \theta - 3)(\cos \theta + 1),{\text{ }}\frac{{ - 1 \pm 7}}{8}"> <mo stretchy="false">(</mo> <mn>4</mn> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>±</mo> <mn>7</mn> </mrow> <mn>8</mn> </mfrac> </math></span></p>
<p>correct reason for rejecting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta \ne - 1"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> </math></span> <strong><em>R1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta "> <mi>cos</mi> <mo></mo> <mi>θ</mi> </math></span> is a probability (value must lie between 0 and 1), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta > 0"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>></mo> <mn>0</mn> </math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>R0 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta \ne - 1"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>≠</mo> <mo>−</mo> <mn>1</mn> </math></span> without a reason.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta = \frac{3}{4}"> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span> <em><strong>AG N0</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>sketch of right triangle with sides 3 and 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\sin ^2}x + {\cos ^2}x = 1"> <mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mi>cos</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct working </p>
<p><strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>missing side <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 7 ,{\text{ }}\frac{{\frac{{\sqrt 7 }}{4}}}{{\frac{3}{4}}}"> <mo>=</mo> <msqrt> <mn>7</mn> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>4</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \theta = \frac{{\sqrt 7 }}{3}"> <mi>tan</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^2}"> <mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \int_\theta ^{\frac{\pi }{4}} {{f^2},{\text{ }}\int {{{\left( {\frac{1}{{\cos x}}} \right)}^2}} } "> <mi>π</mi> <msubsup> <mo>∫</mo> <mi>θ</mi> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mrow> </math></span></p>
<p>correct substitution of both limits and function <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \int_\theta ^{\frac{\pi }{4}} {{{\left( {\frac{1}{{\cos x}}} \right)}^2}{\text{d}}x} "> <mi>π</mi> <msubsup> <mo>∫</mo> <mi>θ</mi> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p>correct integration <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x"> <mi>tan</mi> <mo></mo> <mi>x</mi> </math></span></p>
<p>substituting <strong>their </strong>limits into <strong>their </strong>integrated function and subtracting <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \frac{\pi }{4} - \tan \theta "> <mi>tan</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mi>tan</mi> <mo></mo> <mi>θ</mi> </math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if they substitute into original or differentiated function.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \frac{\pi }{4} = 1"> <mi>tan</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - \tan \theta "> <mn>1</mn> <mo>−</mo> <mi>tan</mi> <mo></mo> <mi>θ</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi - \frac{{\pi \sqrt 7 }}{3}"> <mi>V</mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mfrac> <mrow> <mi>π</mi> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>3</mn> </mfrac> </math></span> <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The diameter of a spherical planet is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo> </mo><mtext>km</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the radius of the planet.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The volume of the planet can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>k</mi></msup></mrow></mfenced><mo> </mo><msup><mtext>km</mtext><mn>3</mn></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30000</mn><mo> </mo><mfenced><mtext>km</mtext></mfenced></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>∙</mo><msup><mn>10</mn><mn>4</mn></msup></math>) <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><msup><mfenced><mrow><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></mrow></mfenced><mn>3</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><msup><mfenced><mn>30000</mn></mfenced><mn>3</mn></msup></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><mo>×</mo><mn>27</mn><mo>×</mo><msup><mn>10</mn><mn>12</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>36</mn><mo>×</mo><msup><mn>10</mn><mn>12</mn></msup></mrow></mfenced></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><mo>×</mo><mn>27000000000000</mn></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>36</mn><mo>×</mo><msup><mn>10</mn><mn>13</mn></msup></mrow></mfenced><mo> </mo><mfenced><msup><mtext>km</mtext><mn>3</mn></msup></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>6</mn><mo>,</mo><mo> </mo><mi>k</mi><mo>=</mo><mn>13</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> passes through the points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}(0,{\text{ }}1,{\text{ }}8)">
<mrow>
<mtext>A</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>8</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}(3,{\text{ }}5,{\text{ }}2)">
<mrow>
<mtext>B</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> are perpendicular, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2">
<mi>p</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {AB} ">
<mover>
<mrow>
<mi>A</mi>
<mi>B</mi>
</mrow>
<mo>→</mo>
</mover>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down a vector equation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A second line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, has equation <strong><em>r</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ {13} \\ { - 14} \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> are perpendicular, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2">
<mi>p</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> intersect at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C(9,{\text{ }}13,{\text{ }}z)">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>z</mi>
<mo stretchy="false">)</mo>
</math></span>. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a unit vector in the direction of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find one point on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> which is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 5 ">
<msqrt>
<mn>5</mn>
</msqrt>
</math></span> units from C.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A - B,\,\, - \left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 8 \hfill \\ \end{gathered} \right) + \left( \begin{gathered} 3 \hfill \\ 5 \hfill \\ 2 \hfill \\ \end{gathered} \right)">
<mi>A</mi>
<mo>−</mo>
<mi>B</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {AB} = \left( \begin{gathered} 3 \hfill \\ 4 \hfill \\ - 6 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mi>A</mi>
<mi>B</mi>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>any </strong>correct equation in the form <strong><em>r</em></strong> = <strong><em>a</em></strong> + <em>t</em><strong><em>b</em></strong> (any parameter for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>) <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p>where <strong><em>a</em></strong> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 0 \hfill \\ 1 \hfill \\ 8 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 3 \hfill \\ 5 \hfill \\ 2 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>, and <strong><em>b </em></strong>is a scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 3 \hfill \\ 4 \hfill \\ - 6 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p> </p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong><em>r</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ 1 \\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 3 \\ 4 \\ { - 6} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>r</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {3 + 3t} \\ {5 + 4t} \\ {2 - 6t} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<mi>t</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mo>+</mo>
<mn>4</mn>
<mi>t</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mn>6</mn>
<mi>t</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>r</em></strong> = <strong><em>j</em></strong> + 8<strong><em>k</em></strong> + <em>t</em>(3<strong><em>i</em></strong> + 4<strong><em>j</em></strong> – 6<strong><em>k</em></strong>)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em> </strong>for the form <strong><em>a</em></strong> + <em>t</em><strong><em>b</em></strong>, <strong><em>A1</em></strong> for the form <strong><em>L</em></strong> = <strong><em>a</em></strong> + <em>t</em><strong><em>b</em></strong>, <strong><em>A0</em></strong> for the form <strong><em>r</em></strong> = <strong><em>b</em></strong> + <em>t</em><strong><em>a</em></strong>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \bullet b = 0">
<mi>a</mi>
<mo>∙</mo>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>choosing correct direction vectors (may be seen in scalar product) <strong><em>A1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ 4 \\ { - 6} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} 3 \\ 4 \\ { - 6} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>∙</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>correct working/equation <strong><em>A1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3p - 6 = 0">
<mn>3</mn>
<mi>p</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2">
<mi>p</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>AG</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1} = \left( {\begin{array}{*{20}{c}} 9 \\ {13} \\ z \end{array}} \right),{\text{ }}{L_1} = {L_2}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>L</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span></p>
<p>one correct equation (must be different parameters if both lines used) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3t = 9,{\text{ }}1 + 2s = 9,{\text{ }}5 + 4t = 13,{\text{ }}3t = 1 + 2s">
<mn>3</mn>
<mi>t</mi>
<mo>=</mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>s</mi>
<mo>=</mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>5</mn>
<mo>+</mo>
<mn>4</mn>
<mi>t</mi>
<mo>=</mo>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>s</mi>
</math></span></p>
<p>one correct value <strong><em>A1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 3,{\text{ }}s = 4,{\text{ }}t = 2">
<mi>t</mi>
<mo>=</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>s</mi>
<mo>=</mo>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>t</mi>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p>valid approach to substitute their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> value <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 + 3( - 6),{\text{ }} - 14 + 4(1)">
<mn>8</mn>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>6</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>14</mn>
<mo>+</mo>
<mn>4</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = - 10">
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>10</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\vec d} \right| = \sqrt {{2^2} + 1} \,\,\left( { = \sqrt 5 } \right)">
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<mover>
<mi>d</mi>
<mo stretchy="false">→</mo>
</mover>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{\sqrt 5 }}\left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ 1 \end{array}} \right)\,\,\,\,\,\left( {{\text{accept}}\left( {\begin{array}{*{20}{c}} {\frac{2}{{\sqrt 5 }}} \\ {\frac{0}{{\sqrt 5 }}} \\ {\frac{1}{{\sqrt 5 }}} \end{array}} \right)} \right)">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>0</mn>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using unit vector) </strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 9 \\ {13} \\ { - 10} \end{array}} \right) \pm \sqrt 5 \hat d">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>±</mo>
<msqrt>
<mn>5</mn>
</msqrt>
<mrow>
<mover>
<mi>d</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 9 \\ {13} \\ { - 10} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ 1 \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} 9 \\ {13} \\ { - 10} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ 1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>one correct point <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(11,{\text{ }}13,{\text{ }} - 9),{\text{ }}(7,{\text{ }}13,{\text{ }} - 11)">
<mo stretchy="false">(</mo>
<mn>11</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>7</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>11</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong>METHOD 2 (distance between points) </strong></p>
<p>attempt to use distance between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1 + 2s,{\text{ }}13,{\text{ }} - 14 + s)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>s</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>14</mn>
<mo>+</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(9,{\text{ }}13,{\text{ }} - 10)">
<mo stretchy="false">(</mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(2s - 8)^2} + {0^2} + {(s - 4)^2} = 5">
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>s</mi>
<mo>−</mo>
<mn>8</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>0</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo>−</mo>
<mn>4</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p>solving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5{s^2} - 40s + 75 = 0">
<mn>5</mn>
<mrow>
<msup>
<mi>s</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>40</mn>
<mi>s</mi>
<mo>+</mo>
<mn>75</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> leading to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = 5">
<mi>s</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = 3">
<mi>s</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>one correct point <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(11,{\text{ }}13,{\text{ }} - 9),{\text{ }}(7,{\text{ }}13,{\text{ }} - 11)">
<mo stretchy="false">(</mo>
<mn>11</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>7</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>11</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> have position vectors <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2} \\ 4 \\ { - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ 8 \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> respectively.</p>
<p>Point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}">
<mrow>
<mtext>C</mtext>
</mrow>
</math></span> has position vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1} \\ k \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{O}}">
<mrow>
<mtext>O</mtext>
</mrow>
</math></span> be the origin.</p>
</div>
<div class="specification">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OA</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OB</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\widehat {\text{O}}{\text{C}} = {\text{B}}\widehat {\text{O}}{\text{C}}"> <mrow> <mtext>A</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <mrow> <mtext>B</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> </math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AOC}}"> <mrow> <mtext>AOC</mtext> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OA</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span> or into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OB}}} \bullet \overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OB</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span> (in (ii)) <em><strong>(A1)</strong></em> </p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \times \left( { - 1} \right) + 4 \times k"> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>4</mn> <mo>×</mo> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 \times \left( { - 1} \right) + 8 \times k"> <mn>6</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>8</mn> <mo>×</mo> <mi>k</mi> </math></span></p>
<p>correct expression <em><strong>A1</strong></em><em><strong> N1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + 4k"> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 2"> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </math></span></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct expression <em><strong>A1</strong></em><em><strong> N1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8k - 6"> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 + 8k"> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>8</mn> <mi>k</mi> </math></span></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding magnitudes (seen anywhere) <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( { - 2} \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 4} \right)}^2}} \,\,\left( { = 6} \right)"> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( 6 \right)}^2} + {{\left( 8 \right)}^2} + {0^2}} \,\,\left( { = 10} \right)"> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct substitution of their values into formula for angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AOC}}"> <mrow> <mtext>AOC</mtext> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{2 + 4k}}{{\sqrt {{{\left( { - 2} \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 4} \right)}^2}} \left| {\overrightarrow {{\text{OC}}} } \right|}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p>correct substitution of their values into formula for angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BOC}}"> <mrow> <mtext>BOC</mtext> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{8k - 6}}{{\sqrt {{{\left( 6 \right)}^2} + {{\left( 8 \right)}^2} + {0^2}} \left| {\overrightarrow {{\text{OC}}} } \right|}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p>recognizing that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,{\text{A}}\widehat {\text{O}}{\text{C}} = {\text{cos}}\,{\text{B}}\widehat {\text{O}}{\text{C}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>B</mtext> </mrow> <mrow> <mover> <mtext>O</mtext> <mo>^</mo> </mover> </mrow> <mrow> <mtext>C</mtext> </mrow> </math></span> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2 + 4k}}{{\left| {\overrightarrow {{\text{OC}}} } \right|\sqrt {{{\left( { - 2} \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 4} \right)}^2}} }} = \frac{{8k - 6}}{{\left| {\overrightarrow {{\text{OC}}} } \right|\sqrt {{6^2} + {{\left( 8 \right)}^2} + {0^2}} }}"> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <msqrt> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2 + 4k}}{{6\sqrt {1 + {k^2}} }} = \frac{{8k - 6}}{{10\sqrt {1 + {k^2}} }}"> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mrow> <mn>6</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mn>10</mn> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span></p>
<p>correct working (without radicals) <em><strong>(A2)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10\left( {2 + 4k} \right) = 6\left( {8k - 6} \right)"> <mn>10</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mn>4</mn> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mi>k</mi> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11{k^2} - 79k + 14 = 0"> <mn>11</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>79</mn> <mi>k</mi> <mo>+</mo> <mn>14</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working clearly leading to the required answer <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20 + 36 = 48k"><mn>20</mn><mo>+</mo><mn>36</mn><mo>=</mo><mn>48</mn><mi>k</mi><mo>-</mo><mn>40</mn><mi>k</mi></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="56 = 8k"> <mn>56</mn> <mo>=</mo> <mn>8</mn> <mi>k</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{2}{{11}}"> <mi>k</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mrow> <mn>11</mn> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {k - 7} \right)\left( {11k - 2} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>7</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 7"> <mi>k</mi> <mo>=</mo> <mn>7</mn> </math></span> <em><strong>AG</strong></em><em><strong> N0</strong></em></p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding magnitude of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OC}}} "> <mover> <mrow> <mtext>OC</mtext> </mrow> <mo>→</mo> </mover> </math></span> (seen anywhere) <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( { - 1} \right)}^2} + {7^2} + {0^2}} "> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {50} "> <msqrt> <mn>50</mn> </msqrt> </math></span></p>
<p>valid attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{2 + 28}}{{6\sqrt {{{\left( { - 1} \right)}^2} + {7^2} + {0^2}} }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>28</mn> </mrow> <mrow> <mn>6</mn> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{56 - 6}}{{10\sqrt {{{\left( { - 1} \right)}^2} + {7^2} + {0^2}} }}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>56</mn> <mo>−</mo> <mn>6</mn> </mrow> <mrow> <mn>10</mn> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>7</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\sqrt {26} } \right)^2} = {6^2} + {\left( {\sqrt {50} } \right)^2} - 2\left( 6 \right)\sqrt {50} \,{\text{cos}}\,\theta "> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>26</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>50</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> <msqrt> <mn>50</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span></p>
<p>finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span> <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{5}{{\sqrt {50} }}\,\,\,\left( { = \frac{1}{{\sqrt 2 }}} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>5</mn> <mrow> <msqrt> <mn>50</mn> </msqrt> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{4}"> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = {\text{cos}}\,\theta "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \sqrt {1 - \frac{{25}}{{50}}} "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <mn>25</mn> </mrow> <mrow> <mn>50</mn> </mrow> </mfrac> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \sqrt {1 - {\text{co}}{{\text{s}}^2}\,\theta } "> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \frac{{\sqrt 2 }}{2}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p>correct substitution of <strong>their</strong> values into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}ab\,{\text{sin}}\,{\text{C}}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>C</mtext> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6 \times \sqrt {50} "><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><mo>×</mo><msqrt><mn>50</mn></msqrt><mo>×</mo><msqrt><mn>1</mn><mo>-</mo><mfrac><mn>25</mn><mn>50</mn></mfrac></msqrt></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6 \times \sqrt {50} \times \frac{5}{{\sqrt {50} }}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>6</mn> <mo>×</mo> <msqrt> <mn>50</mn> </msqrt> <mo>×</mo> <mfrac> <mn>5</mn> <mrow> <msqrt> <mn>50</mn> </msqrt> </mrow> </mfrac> </math></span></p>
<p>area is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15"> <mn>15</mn> </math></span> <em><strong>A1</strong></em><em><strong> N3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A balloon in the shape of a sphere is filled with helium until the radius is 6 cm.</p>
</div>
<div class="specification">
<p>The volume of the balloon is increased by 40%.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the volume of the balloon.</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>Calculate the radius of the balloon following this increase.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>Units are required in parts (a) and (b).</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}\pi \times {6^3}">
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>6</mn>
<mn>3</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into volume of sphere formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 905{\text{ c}}{{\text{m}}^3}{\text{ }}(288\pi {\text{ c}}{{\text{m}}^3},{\text{ }}904.778 \ldots {\text{ c}}{{\text{m}}^3})">
<mo>=</mo>
<mn>905</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>288</mn>
<mi>π</mi>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>904.778</mn>
<mo>…</mo>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1) (C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Answers derived from the use of approximations of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> (3.14; 22/7) are awarded <strong><em>(A0)</em></strong>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Units are required in parts (a) and (b).</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{140}}{{100}} \times 904.778 \ldots = \frac{4}{3}\pi {r^3}">
<mfrac>
<mrow>
<mn>140</mn>
</mrow>
<mrow>
<mn>100</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mn>904.778</mn>
<mo>…</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{140}}{{100}} \times 288\pi = \frac{4}{3}\pi {r^3}">
<mfrac>
<mrow>
<mn>140</mn>
</mrow>
<mrow>
<mn>100</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mn>288</mn>
<mi>π</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1266.69 \ldots = \frac{4}{3}\pi {r^3}">
<mn>1266.69</mn>
<mo>…</mo>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for multiplying their part (a) by 1.4 or equivalent, <strong><em>(M1) </em></strong>for equating to the volume of a sphere formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^3} = \frac{{3 \times 1266.69 \ldots }}{{4\pi }}">
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mn>1266.69</mn>
<mo>…</mo>
</mrow>
<mrow>
<mn>4</mn>
<mi>π</mi>
</mrow>
</mfrac>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \sqrt[3]{{\frac{{3 \times 1266.69 \ldots }}{{4\pi }}}}">
<mi>r</mi>
<mo>=</mo>
<mroot>
<mrow>
<mfrac>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mn>1266.69</mn>
<mo>…</mo>
</mrow>
<mrow>
<mn>4</mn>
<mi>π</mi>
</mrow>
</mfrac>
</mrow>
<mn>3</mn>
</mroot>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \sqrt[3]{{(1.4) \times {6^3}}}">
<mi>r</mi>
<mo>=</mo>
<mroot>
<mrow>
<mo stretchy="false">(</mo>
<mn>1.4</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mrow>
<msup>
<mn>6</mn>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mroot>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^3} = 302.4">
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>302.4</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for isolating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(r = ){\text{ }}6.71{\text{ cm }}(6.71213 \ldots )">
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>6.71</mn>
<mrow>
<mtext> cm </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>6.71213</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft) <em>(C4)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A cylindrical container with a radius of 8 cm is placed on a flat surface. The container is filled with water to a height of 12 cm, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_06.17.11.png" alt="M17/5/MATSD/SP1/ENG/TZ2/12"></p>
</div>
<div class="specification">
<p>A heavy ball with a radius of 2.9 cm is dropped into the container. As a result, the height of the water increases to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> cm, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_06.18.54.png" alt="M17/5/MATSD/SP1/ENG/TZ2/12.b"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the volume of water in the container.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \times {8^2} \times 12">
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>8</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>12</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into the volume of a cylinder formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2410{\text{ c}}{{\text{m}}^3}{\text{ }}(2412.74 \ldots {\text{ c}}{{\text{m}}^3},{\text{ }}768\pi {\text{ c}}{{\text{m}}^3})">
<mn>2410</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>2412.74</mn>
<mo>…</mo>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>768</mn>
<mi>π</mi>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}\pi \times {2.9^3} + 768\pi = \pi \times {8^2}h">
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>2.9</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>768</mn>
<mi>π</mi>
<mo>=</mo>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>8</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>h</mi>
</math></span> <strong><em>(M1)(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into the volume of a sphere formula (this may be implied by seeing 102.160…), <strong><em>(M1) </em></strong>for adding their volume of the ball to their part (a), <strong><em>(M1) </em></strong>for equating <strong>a </strong>volume to the volume of a cylinder with a height of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}\pi \times {2.9^3} = \pi \times {8^2}(h - 12)">
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>2.9</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>8</mn>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo>−</mo>
<mn>12</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into the volume of a sphere formula (this may be implied by seeing 102.160…), <strong><em>(M1) </em></strong>for equating to the volume of a cylinder, <strong><em>(M1) </em></strong>for the height of the water level increase, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h - 12">
<mi>h</mi>
<mo>−</mo>
<mn>12</mn>
</math></span>. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h - 12">
<mi>h</mi>
<mo>−</mo>
<mn>12</mn>
</math></span> if adding 12 is implied by their answer.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(h = ){\text{ }}12.5{\text{ (cm) }}\left( {12.5081 \ldots {\text{ (cm)}}} \right)">
<mo stretchy="false">(</mo>
<mi>h</mi>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>12.5</mn>
<mrow>
<mtext> (cm) </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>12.5081</mn>
<mo>…</mo>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C4)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> If 3 sf answer used, answer is 12.5 (12.4944…). Follow through from part (a) if first method is used.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Emily’s kite ABCD is hanging in a tree. The plane ABCDE is vertical.</p>
<p>Emily stands at point E at some distance from the tree, such that EAD is a straight line and angle BED = 7°. Emily knows BD = 1.2 metres and angle BDA = 53°, as shown in the diagram</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_18.18.28.png" alt="N17/5/MATSD/SP1/ENG/TZ0/10"></p>
</div>
<div class="specification">
<p>T is a point at the base of the tree. ET is a horizontal line. The angle of elevation of A from E is 41°.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the length of EB.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the angle of elevation of B from E.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vertical height of B above the ground.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>Units are required in parts (a) and (c).</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{EB}}}}{{\sin 53{\rm{^\circ }}}} = \frac{{1.2}}{{\sin 7{\rm{^\circ }}}}">
<mfrac>
<mrow>
<mrow>
<mtext>EB</mtext>
</mrow>
</mrow>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>53</mn>
<mrow>
<mrow>
<msup>
<mi></mi>
<mo>∘</mo>
</msup>
</mrow>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1.2</mn>
</mrow>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>7</mn>
<mrow>
<mrow>
<msup>
<mi></mi>
<mo>∘</mo>
</msup>
</mrow>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into sine formula, <strong><em>(A1) </em></strong>for correct substitution.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({\text{EB}} = ){\text{ }}7.86{\text{ m}}">
<mo stretchy="false">(</mo>
<mrow>
<mtext>EB</mtext>
</mrow>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>7.86</mn>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="786{\text{ cm }}(7.86385 \ldots {\text{ m}}">
<mn>786</mn>
<mrow>
<mtext> cm </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>7.86385</mn>
<mo>…</mo>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="786.385 \ldots {\text{ cm}})">
<mn>786.385</mn>
<mo>…</mo>
<mrow>
<mtext> cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1) (C3)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>34° <strong><em>(A1) (C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Units are required in parts (a) and (c).</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 34^\circ = \frac{{{\text{height}}}}{{7.86385 \ldots }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>34</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>height</mtext>
</mrow>
</mrow>
<mrow>
<mn>7.86385</mn>
<mo>…</mo>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into a trigonometric ratio.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({\text{height}} = ){\text{ }}4.40{\text{ m}}">
<mo stretchy="false">(</mo>
<mrow>
<mtext>height</mtext>
</mrow>
<mo>=</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>4.40</mn>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="440{\text{ cm }}(4.39741 \ldots {\text{ m}}">
<mn>440</mn>
<mrow>
<mtext> cm </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>4.39741</mn>
<mo>…</mo>
<mrow>
<mtext> m</mtext>
</mrow>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="439.741 \ldots {\text{ cm}})">
<mn>439.741</mn>
<mo>…</mo>
<mrow>
<mtext> cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept “BT” used for height. Follow through from parts (a) and (b). Use of 7.86 gives an answer of 4.39525….</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a right triangle ABC. Point D lies on AB such that CD bisects AĈB.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p style="text-align: center;">AĈD = <em>θ</em> and AC = 14 cm</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \frac{3}{5}"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>θ</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}}"> <mrow> <mtext>BC</mtext> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> labelled sides on separate triangle, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^2}\,x + {\text{co}}{{\text{s}}^2}\,x = 1"> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> missing side is 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {1 - {{\left( {\frac{3}{5}} \right)}^2}} "> <msqrt> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2\theta "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>θ</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {\frac{{16}}{{25}}} \right) - 1"> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>16</mn> </mrow> <mrow> <mn>25</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\left( {\frac{3}{5}} \right)^2}"> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{16}}{{25}} - \frac{9}{{25}}"> <mfrac> <mrow> <mn>16</mn> </mrow> <mrow> <mn>25</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>9</mn> <mrow> <mn>25</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2\theta = \frac{7}{{25}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>7</mn> <mrow> <mn>25</mn> </mrow> </mfrac> </math></span> <em><strong>A1 N2</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>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{7}{{25}} = \frac{{14}}{{{\text{BC}}}}"> <mfrac> <mn>7</mn> <mrow> <mn>25</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>14</mn> </mrow> <mrow> <mrow> <mtext>BC</mtext> </mrow> </mrow> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = \frac{{14 \times 25}}{7}"> <mrow> <mtext>BC</mtext> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>14</mn> <mo>×</mo> <mn>25</mn> </mrow> <mn>7</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = 50"> <mrow> <mtext>BC</mtext> </mrow> <mo>=</mo> <mn>50</mn> </math></span> (cm) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a cuboid with a square base of length 6 cm, as shown in the diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The height of the cuboid, <em>x </em>cm, is equal to the height of the hemisphere.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <em>x</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the volume of the paperweight.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>1 cm<sup>3</sup> of glass has a mass of 2.56 grams.</p>
<p>Calculate the mass, in grams, of the paperweight.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>3 (cm) <em><strong>(A1) (C1)</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>units are required in part (a)(ii)</strong></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \frac{{4\pi \times {{\left( 3 \right)}^3}}}{3} + 3 \times {\left( 6 \right)^2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mn>3</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <strong>their</strong> correct substitution in volume of sphere formula divided by 2, <em><strong>(M1)</strong></em> for adding <strong>their</strong> correctly substituted volume of the cuboid.</p>
<p> </p>
<p>= 165 cm<sup>3 </sup>(164.548…) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<p><strong>Note:</strong> The answer is 165 cm<sup>3</sup>; the units are required. Follow through from part (a)(i).</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>their 164.548… × 2.56 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for multiplying their part (a)(ii) by 2.56.</p>
<p> </p>
<p>= 421 (g) (421.244…(g)) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a)(ii).</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A type of candy is packaged in a right circular cone that has volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{100 c}}{{\text{m}}^{\text{3}}}">
<mrow>
<mtext>100 c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<mtext>3</mtext>
</mrow>
</msup>
</mrow>
</math></span> and vertical height 8 cm.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_11.14.55.png" alt="M17/5/MATSD/SP1/ENG/TZ1/09"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the radius, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, of the circular base of the cone.</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 slant height, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l">
<mi>l</mi>
</math></span>, of the cone.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the curved surface area of the cone.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="100 = \frac{1}{3}\pi {r^2}(8)">
<mn>100</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>8</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into volume of cone formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 3.45{\text{ (cm) }}\left( {3.45494 \ldots {\text{ (cm)}}} \right)">
<mi>r</mi>
<mo>=</mo>
<mn>3.45</mn>
<mrow>
<mtext> (cm) </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3.45494</mn>
<mo>…</mo>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{l^2} = {8^2} + {(3.45494 \ldots )^2}">
<mrow>
<msup>
<mi>l</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>8</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>3.45494</mn>
<mo>…</mo>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into Pythagoras’ theorem.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l = 8.71{\text{ (cm) }}\left( {8.71416 \ldots {\text{ (cm)}}} \right)">
<mi>l</mi>
<mo>=</mo>
<mn>8.71</mn>
<mrow>
<mtext> (cm) </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>8.71416</mn>
<mo>…</mo>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em>(ft)</strong><em> </em><strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \times 3.45494 \ldots \times 8.71416 \ldots ">
<mi>π</mi>
<mo>×</mo>
<mn>3.45494</mn>
<mo>…</mo>
<mo>×</mo>
<mn>8.71416</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for their correct substitutions into curved surface area of a cone formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 94.6{\text{ c}}{{\text{m}}^2}{\text{ }}(94.5836 \ldots {\text{ c}}{{\text{m}}^2})">
<mo>=</mo>
<mn>94.6</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>94.5836</mn>
<mo>…</mo>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from parts (a) and (b). Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="94.4{\text{ c}}{{\text{m}}^2}">
<mn>94.4</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> from use of 3 sf values.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm.<br>A smaller right circular cone has a height of 12 cm and a slant height of 15 cm, and is removed from the top of the larger cone, as shown in the diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the radius of the base of the cone which has been removed.</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>Calculate the curved surface area of the cone which has been removed.</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>Calculate the curved surface area of the remaining solid.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{15}^2} - {{12}^2}} ">
<msqrt>
<mrow>
<msup>
<mrow>
<mn>15</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mn>12</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution into Pythagoras theorem.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{radius}}}}{{21}} = \frac{{15}}{{35}}">
<mfrac>
<mrow>
<mrow>
<mtext>radius</mtext>
</mrow>
</mrow>
<mrow>
<mn>21</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>15</mn>
</mrow>
<mrow>
<mn>35</mn>
</mrow>
</mfrac>
</math></span> <em><strong> (M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for a correct equation.</p>
<p>= 9 (cm) <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \times 9 \times 15">
<mi>π</mi>
<mo>×</mo>
<mn>9</mn>
<mo>×</mo>
<mn>15</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution into curved surface area of a cone formula.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 424\,\,{\text{c}}{{\text{m}}^2}\,\,\,\,\,\left( {135\pi ,\,\,424.115...{\text{c}}{{\text{m}}^2}} \right)">
<mo>=</mo>
<mn>424</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>135</mn>
<mi>π</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>424.115...</mn>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note</strong>: Follow through from part (a).</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="\pi \times 21 \times 35 - 424.115...">
<mi>π</mi>
<mo>×</mo>
<mn>21</mn>
<mo>×</mo>
<mn>35</mn>
<mo>−</mo>
<mn>424.115...</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution into curved surface area of a cone formula and for subtracting their part (b).</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1880\,\,{\text{c}}{{\text{m}}^2}\,\,\,\,\,\left( {600\pi ,\,\,1884.95...{\text{c}}{{\text{m}}^2}} \right)">
<mo>=</mo>
<mn>1880</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>600</mn>
<mi>π</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1884.95...</mn>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (b).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows triangle ABC, with <em>AB</em> = 6 and <em>AC</em> = 8.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\hat A = \frac{5}{6}">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mover>
<mi>A</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>6</mn>
</mfrac>
</math></span> find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\hat A">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mover>
<mi>A</mi>
<mo stretchy="false">^</mo>
</mover>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle ABC.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach using Pythagorean identity <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^2}\,A + {\left( {\frac{5}{6}} \right)^2} = 1">
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>A</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> (or equivalent) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,A = \frac{{\sqrt {11} }}{6}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>A</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>11</mn>
</msqrt>
</mrow>
<mn>6</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 8 \times 6 \times \frac{{\sqrt {11} }}{6}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>8</mn>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>11</mn>
</msqrt>
</mrow>
<mn>6</mn>
</mfrac>
</math></span> (or equivalent) <em><strong>(A1)</strong></em></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\sqrt {11} ">
<mo>=</mo>
<mn>4</mn>
<msqrt>
<mn>11</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span> passes through points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}( - 3,{\text{ }}4,{\text{ }}2)">
<mrow>
<mtext>A</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>4</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}( - 1,{\text{ }}3,{\text{ }}3)">
<mrow>
<mtext>B</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span> also passes through the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}(3,{\text{ }}1,{\text{ }}p)">
<mrow>
<mtext>C</mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>p</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The point D has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({q^2},{\text{ }}0,{\text{ }}q)"> <mo stretchy="false">(</mo> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>q</mi> <mo stretchy="false">)</mo> </math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DC}}} "> <mover> <mrow> <mtext>DC</mtext> </mrow> <mo>→</mo> </mover> </math></span> is perpendicular to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct approach <strong><em>A1</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1} \\ 3 \\ 3 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} { - 3} \\ 4 \\ 2 \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} 3 \\ { - 4} \\ { - 2} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} { - 1} \\ 3 \\ 3 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>AG N0</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>any correct equation in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = a + tb"> <mi>r</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>t</mi> <mi>b</mi> </math></span> (any parameter for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span>)</p>
<p> </p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3} \\ 4 \\ 2 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1} \\ 3 \\ 3 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> is a scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A2 N2</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \left( {\begin{array}{*{20}{c}} { - 3} \\ 4 \\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right),{\text{ }}(x,{\text{ }}y,{\text{ }}z) = ( - 1,{\text{ }}3,{\text{ }}3) + s( - 2,{\text{ }}1,{\text{ }} - 1),{\text{ }}r = \left( {\begin{array}{*{20}{c}} { - 3 + 2t} \\ {4 - t} \\ {2 + t} \end{array}} \right)"> <mi>r</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>y</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>r</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>4</mn> <mo>−</mo> <mi>t</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mo>+</mo> <mi>t</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + tb"> <mi>a</mi> <mo>+</mo> <mi>t</mi> <mi>b</mi> </math></span>, <strong>A1</strong> for the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = a + tb"> <mi>L</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>t</mi> <mi>b</mi> </math></span>, <strong>A0</strong> for the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = b + ta"> <mi>r</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>t</mi> <mi>a</mi> </math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 – finding value of parameter</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3} \\ 4 \\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ p \end{array}} \right),{\text{ }}( - 1,{\text{ }}3,{\text{ }}3) + s( - 2,{\text{ }}1,{\text{ }} - 1) = (3,{\text{ }}1,{\text{ }}p)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </math></span></p>
<p> </p>
<p>one correct equation (not involving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 2t = 3,{\text{ }} - 1 - 2s = 3,{\text{ }}4 - t = 1,{\text{ }}3 + s = 1"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>s</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>4</mn> <mo>−</mo> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>+</mo> <mi>s</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct parameter from their equation (may be seen in substitution) <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 3,{\text{ }}s = - 2"> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span></p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3} \\ 4 \\ 2 \end{array}} \right) + 3\left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ p \end{array}} \right),{\text{ }}3 - ( - 2)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 5\,\,\,\,\,\left( {{\text{accept }}\left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ 5 \end{array}} \right)} \right)"> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>accept </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1 N2</em></strong></p>
<p> </p>
<p><strong>METHOD 2 – eliminating parameter</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3} \\ 4 \\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ p \end{array}} \right),{\text{ }}( - 1,{\text{ }}3,{\text{ }}3) + s( - 2,{\text{ }}1,{\text{ }} - 1) = (3,{\text{ }}1,{\text{ }}p)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>p</mi> <mo stretchy="false">)</mo> </math></span></p>
<p> </p>
<p>one correct equation (not involving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 2t = 3,{\text{ }} - 1 - 2s = 3,{\text{ }}4 - t = 1,{\text{ }}3 + s = 1"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>s</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>4</mn> <mo>−</mo> <mi>t</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>+</mo> <mi>s</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct equation (with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>) <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + t = p,{\text{ }}3 - s = p"> <mn>2</mn> <mo>+</mo> <mi>t</mi> <mo>=</mo> <mi>p</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>−</mo> <mi>s</mi> <mo>=</mo> <mi>p</mi> </math></span></p>
<p>correct working to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="7 = 2p - 3,{\text{ }}6 = 1 + p"> <mn>7</mn> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6</mn> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>p</mi> </math></span></p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 5\,\,\,\,\,\left( {{\text{accept }}\left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ 5 \end{array}} \right)} \right)"> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>accept </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1 N2</em></strong></p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DC}}} "> <mover> <mrow> <mtext>DC</mtext> </mrow> <mo>→</mo> </mover> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CD}}} "> <mover> <mrow> <mtext>CD</mtext> </mrow> <mo>→</mo> </mover> </math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ 5 \end{array}} \right) - \left( {\begin{array}{*{20}{c}} {{q^2}} \\ 0 \\ q \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} {{q^2}} \\ 0 \\ q \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ 5 \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} {{q^2}} \\ 0 \\ q \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 3 \\ 1 \\ p \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>q</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p>correct vector for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DC}}} "> <mover> <mrow> <mtext>DC</mtext> </mrow> <mo>→</mo> </mover> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CD}}} "> <mover> <mrow> <mtext>CD</mtext> </mrow> <mo>→</mo> </mover> </math></span> (may be seen in scalar product) <strong><em>A1</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {3 - {q^2}} \\ 1 \\ {5 - q} \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} {{q^2} - 3} \\ { - 1} \\ {q - 5} \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} {3 - {q^2}} \\ 1 \\ {p - q} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>3</mn> <mo>−</mo> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>5</mn> <mo>−</mo> <mi>q</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>q</mi> <mo>−</mo> <mn>5</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>3</mn> <mo>−</mo> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mo>−</mo> <mi>q</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p>recognizing scalar product of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{DC}}} "> <mover> <mrow> <mtext>DC</mtext> </mrow> <mo>→</mo> </mover> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{CD}}} "> <mover> <mrow> <mtext>CD</mtext> </mrow> <mo>→</mo> </mover> </math></span> with direction vector of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span> is zero (seen anywhere) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {3 - {q^2}} \\ 1 \\ {p - q} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right) = 0,{\text{ }}\overrightarrow {{\text{DC}}} \bullet \overrightarrow {{\text{AC}}} = 0,{\text{ }}\left( {\begin{array}{*{20}{c}} {3 - {q^2}} \\ 1 \\ {5 - q} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 1 \end{array}} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>3</mn> <mo>−</mo> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>p</mi> <mo>−</mo> <mi>q</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>∙</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mover> <mrow> <mtext>DC</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>3</mn> <mo>−</mo> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>5</mn> <mo>−</mo> <mi>q</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>∙</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p> </p>
<p>correct scalar product in terms of only <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 - 2{q^2} - 1 + 5 - q,{\text{ }}2{q^2} + q - 10 = 0,{\text{ }}2(3 - {q^2}) - 1 + 5 - q"> <mn>6</mn> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>−</mo> <mi>q</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>5</mn> <mo>−</mo> <mi>q</mi> </math></span></p>
<p>correct working to solve quadratic <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(2q + 5)(q - 2),{\text{ }}\frac{{ - 1 \pm \sqrt {1 - 4(2)( - 10)} }}{{2(2)}}"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>±</mo> <msqrt> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>10</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = - \frac{5}{2},{\text{ }}2"> <mi>q</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> </math></span> <strong><em>A1A1 N3</em></strong></p>
<p> </p>
<p><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The position vectors of points P and Q are <strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> 2 <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−<!-- − --></mo>
</math></span> <strong><em>k </em></strong>and 7<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> 3<strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−<!-- − --></mo>
</math></span> 4<strong><em>k </em></strong>respectively.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation of the line that passes through P and Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line through P and Q is perpendicular to the vector 2<strong><em>i </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <em>n</em><strong><em>k</em></strong>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>valid attempt to find direction vector <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PQ}}} ,{\text{ }}\overrightarrow {{\text{QP}}} ">
<mover>
<mrow>
<mtext>PQ</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mover>
<mrow>
<mtext>QP</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span></p>
<p>correct direction vector (or multiple of) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>6<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 3<strong><em>k</em></strong></p>
<p><strong>any </strong>correct equation in the form <strong><em>r</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <em>t</em><strong><em>b </em></strong>(any parameter for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>) <strong><em>A2 N3</em></strong></p>
<p>where <strong><em>a </em></strong>is <strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> 2<strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> <strong><em>k</em></strong> or 7<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> 3<strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 4<strong><em>k </em></strong>, and <strong><em>b </em></strong>is a scalar multiple of 6<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 3<strong><em>k</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong><em>r</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> 7<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> 3<strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 4<strong><em>k</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <em>t</em>(6<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 3<strong><em>k</em></strong>), <strong><em>r</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {1 + 6s} \\ {2 + 1s} \\ { - 1 - 3s} \end{array}} \right),{\text{ }}r = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 6} \\ { - 1} \\ 3 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>6</mn>
<mi>s</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>1</mn>
<mi>s</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>−</mo>
<mn>3</mn>
<mi>s</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>r</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>A1 </em></strong>for the form <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <em>t</em><strong><em>b</em></strong>, <strong><em>A1 </em></strong>for the form <strong><em>L</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <em>t</em><strong><em>b</em></strong>, <strong><em>A0 </em></strong>for the form <strong><em>r</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <em>t</em><strong><em>a</em></strong>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct expression for scalar product <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 \times 2 + 1 \times 0 + ( - 3) \times n,{\text{ }} - 3n + 12">
<mn>6</mn>
<mo>×</mo>
<mn>2</mn>
<mo>+</mo>
<mn>1</mn>
<mo>×</mo>
<mn>0</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>n</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>12</mn>
</math></span></p>
<p>setting scalar product equal to zero (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong><em>u</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet ">
<mo>∙</mo>
</math></span> <strong><em>v</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0,{\text{ }} - 3n + 12 = 0">
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>12</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 4">
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>1</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>Hence or otherwise, solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>θ</mi><mo>≤</mo><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi>θ</mi><mo>≠</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to write all LHS terms with a common denominator of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>-</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to use algebraic division on RHS <em><strong>(M1)</strong></em></p>
<p>correctly obtains quotient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></math> and remainder <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> as required. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></p>
<p><strong><br>EITHER</strong></p>
<p>attempt to factorise in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>+</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>+</mo><mi>b</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Accept any variable in place of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><strong><br></strong><strong>OR</strong></p>
<p>attempt to substitute into quadratic formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><msqrt><mn>49</mn></msqrt></mrow><mn>4</mn></mfrac></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> only.</p>
<p><br>one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>11</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>330</mn></math>) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>12</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>11</mn><mtext>π</mtext></mrow><mn>12</mn></mfrac></math> (must be in radians) <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A0</strong></em> if additional answers given.</p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A lampshade, in the shape of a cone, has a wireframe consisting of a circular ring and four straight pieces of equal length, attached to the ring at points A, B, C and D.</p>
<p>The ring has its centre at point O and its radius is 20 centimetres. The straight pieces meet at point V, which is vertically above O, and the angle they make with the base of the lampshade is 60°.</p>
<p>This information is shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_17.16.13.png" alt="M17/5/MATSD/SP1/ENG/TZ2/03"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the length of one of the straight pieces in the wireframe.</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 total length of wire needed to construct this wireframe. Give your answer in centimetres correct to the nearest millimetre.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 60^\circ = \frac{{20}}{b}"> <mi>cos</mi> <mo></mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mi>b</mi> </mfrac> </math></span><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><strong>OR</strong><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{20}}{{\cos 60^\circ }}"> <mi>b</mi> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mrow> <mi>cos</mi> <mo></mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </mfrac> </math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into a correct trig. ratio.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(b = ){\text{ 40 (cm)}}"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">)</mo> <mrow> <mtext> 40 (cm)</mtext> </mrow> </math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 \times 40 + 2\pi (20)"> <mn>4</mn> <mo>×</mo> <mn>40</mn> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>20</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution in the circumference of the circle formula, <strong><em>(M1) </em></strong>for adding 4 times their answer to part (a) to their circumference of the circle.</p>
<p> </p>
<p>285.6637… <strong><em>(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from part (a). This <strong><em>(A1) </em></strong>may be implied by a correct rounded answer.</p>
<p> </p>
<p>285.7 (cm) <strong><em>(A1)</em>(ft)</strong> <strong><em>(C4)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>(A1)</em>(ft) </strong>for rounding their answer (consistent with their method) to the nearest millimetre, irrespective of unrounded answer seen.</p>
<p>The final <strong><em>(A1)(</em>ft) </strong>is not dependent on any of the previous <strong><em>M </em></strong>marks. It is for rounding their unrounded answer correctly.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced></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>Hence or otherwise, solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mn>2</mn><mi mathvariant="normal">π</mi></math>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> for proofs which work from both sides to find a common expression other than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></math>.</p>
<p> </p>
<p><strong>METHOD 1 (LHS to RHS)</strong></p>
<p>attempt to use double angle formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> <em><strong> M1</strong></em></p>
<p>LHS <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></math> OR</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></math> OR</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>=</mo></math> RHS <em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2 (RHS to LHS)</strong></p>
<p>RHS <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></math></p>
<p> attempt to use double angle formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> <em><strong> M1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong> A1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo></math> LHS <em><strong> AG</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>attempt to factorise <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong> A1</strong></em></p>
<p>recognition of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>⇒</mo><mfrac><mrow><mi>sin</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>tan</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p>one correct reference angle seen anywhere, accept degrees <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math>)</p>
<p> </p>
<p><strong>Note:</strong> This <em><strong>(M1)(A1)</strong></em> is independent of the previous <em><strong>M1A1</strong></em>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac><mo>,</mo><mfrac><mrow><mn>11</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math> <em><strong> A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for any two correct (radian) answers.<br>Award <em><strong>A1A0</strong></em> if additional values given with the four correct (radian) answers.<br>Award <em><strong>A1A0</strong></em> for four correct answers given in degrees.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Three airport runways intersect to form a triangle, ABC. The length of AB is 3.1 km, AC is 2.6 km, and BC is 2.4 km.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>A company is hired to cut the grass that grows in triangle ABC, but they need to know the area.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the size, in degrees, of angle BÂC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area, in km<sup>2</sup>, of triangle ABC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\cos \,A = } \right)\,\,\frac{{{{2.6}^2} + {{3.1}^2} - {{2.4}^2}}}{{2\left( {2.6} \right)\left( {3.1} \right)}}">
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mspace width="thinmathspace"></mspace>
<mi>A</mi>
<mo>=</mo>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mn>2.6</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>3.1</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mn>2.4</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2.6</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3.1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituted cosine rule formula, <em><strong>(A1)</strong></em> for correct substitutions.</p>
<p>48.8° (48.8381…°) <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 2.6 \times 3.1 \times {\text{sin}}\left( {48.8381 \ldots ^\circ } \right)">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>2.6</mn>
<mo>×</mo>
<mn>3.1</mn>
<mo>×</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>48.8381</mn>
<msup>
<mo>…</mo>
<mo>∘</mo>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em><strong>(ft)</strong></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituted area of a triangle formula, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p>3.03 (km<sup>2</sup>) (3.033997…(km<sup>2</sup>)) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows a circle with centre O and radius<em> r</em> cm.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The points A and B lie on the circumference of the circle, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\mathop {\text{O}}\limits^ \wedge {\text{B}}">
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>O</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> = <em>θ</em>. The area of the shaded sector AOB is 12 cm<sup>2</sup> and the length of arc AB is 6 cm.</p>
<p>Find the value of <em>r</em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>evidence of correctly substituting into circle formula (may be seen later) <em><strong>A1A1</strong></em><br><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\theta {r^2} = 12,\,\,r\theta = 6">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>θ</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
<mi>θ</mi>
<mo>=</mo>
<mn>6</mn>
</math></span></p>
<p>attempt to eliminate one variable <em><strong>(M1)</strong></em><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{6}{\theta },\,\,\theta = \frac{1}{r},\,\,\frac{{\frac{1}{2}\theta {r^2}}}{{r\theta }} = \frac{{12}}{6}">
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mn>6</mn>
<mi>θ</mi>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>r</mi>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>θ</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>r</mi>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>12</mn>
</mrow>
<mn>6</mn>
</mfrac>
</math></span></p>
<p>correct elimination <em><strong>(A1)</strong></em><br><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \frac{6}{r} \times {r^2} = 12,\,\,\frac{1}{2}\theta \times {\left( {\frac{6}{\theta }} \right)^2} = 12,\,\,A = \frac{1}{2} \times {r^2} \times \frac{l}{r},\,\,\frac{{{r^2}}}{{2r}} = 2">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>6</mn>
<mi>r</mi>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>θ</mi>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>6</mn>
<mi>θ</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>A</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mfrac>
<mi>l</mi>
<mi>r</mi>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mi>r</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p>correct equation <em><strong>(A1)</strong></em><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6r = 12,\,\,\frac{1}{2} \times \frac{{36}}{\theta } = 12,\,\,12 = \frac{1}{2} \times {r^2} \times \frac{6}{r}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mi>r</mi>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>36</mn>
</mrow>
<mi>θ</mi>
</mfrac>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>12</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mfrac>
<mn>6</mn>
<mi>r</mi>
</mfrac>
</math></span></p>
<p>correct working <em><strong>(A1)</strong></em><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3r = 12,\,\,\frac{{18}}{\theta } = 12,\,\,\frac{r}{2} = 2,\,\,24 = 6r">
<mn>3</mn>
<mi>r</mi>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mn>18</mn>
</mrow>
<mi>θ</mi>
</mfrac>
<mo>=</mo>
<mn>12</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>r</mi>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>24</mn>
<mo>=</mo>
<mn>6</mn>
<mi>r</mi>
</math></span></p>
<p><em>r</em> = 4 (cm) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve <span class="mjpage"><math alttext="{\log _2}(2\sin x) + {\log _2}(\cos x) = - 1" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>log</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msub> <mi>log</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>, for <span class="mjpage"><math alttext="2\pi < x < \frac{{5\pi }}{2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mn>2</mn> <mi>π</mi> <mo><</mo> <mi>x</mi> <mo><</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>correct application of <span class="mjpage"><math alttext="\log a + \log b = \log ab" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>log</mi> <mo></mo> <mi>a</mi> <mo>+</mo> <mi>log</mi> <mo></mo> <mi>b</mi> <mo>=</mo> <mi>log</mi> <mo></mo> <mi>a</mi> <mi>b</mi> </math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math alttext="\,\,\,\,\," xmlns="http://www.w3.org/1998/Math/MathML"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math alttext="{\log _2}(2\sin x\cos x),{\text{ }}\log 2 + \log (\sin x) + \log (\cos x)" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msub> <mi>log</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>log</mi> <mo></mo> <mn>2</mn> <mo>+</mo> <mi>log</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>log</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span></p>
<p>correct equation without logs <strong><em>A1</em></strong></p>
<p><em>eg</em><math alttext="\,\,\,\,\," xmlns="http://www.w3.org/1998/Math/MathML"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math alttext="2\sin x\cos x = {2^{ - 1}},{\text{ }}\sin x\cos x = \frac{1}{4},{\text{ }}\sin 2x = \frac{1}{2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span></p>
<p>recognizing double-angle identity (seen anywhere) <strong><em>A1</em></strong></p>
<p><em>eg</em><math alttext="\,\,\,\,\," xmlns="http://www.w3.org/1998/Math/MathML"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math alttext="\log (\sin 2x),{\text{ }}2\sin x\cos x = \sin 2x,{\text{ }}\sin 2x = \frac{1}{2}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>log</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span></p>
<p>evaluating <span class="mjpage"><math alttext="{\sin ^{ - 1}}\left( {\frac{1}{2}} \right) = \frac{\pi }{6}{\text{ }}(30^\circ )" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>sin</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mn>30</mn> <mo>∘</mo> </msup> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1)</em></strong></p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg</em><math alttext="\,\,\,\,\," xmlns="http://www.w3.org/1998/Math/MathML"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math alttext="x = \frac{\pi }{{12}} + 2\pi ,{\text{ }}2x = \frac{{25\pi }}{6},{\text{ }}\frac{{29\pi }}{6},{\text{ }}750^\circ ,{\text{ }}870^\circ ,{\text{ }}x = \frac{\pi }{{12}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>25</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>29</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msup> <mn>750</mn> <mo>∘</mo> </msup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msup> <mn>870</mn> <mo>∘</mo> </msup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span><strong>and</strong> <span class="mjpage"><math alttext="x = \frac{{5\pi }}{{12}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span>, one correct final answer</p>
<p><span class="mjpage"><math alttext="x = \frac{{25\pi }}{{12}},{\text{ }}\frac{{29\pi }}{{12}}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>25</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>29</mn> <mi>π</mi> </mrow> <mrow> <mn>12</mn> </mrow> </mfrac> </math></span> (do not accept additional values) <strong><em>A2</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a ball attached to the end of a spring, which is suspended from a ceiling.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The height, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres, of the ball above the ground at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds after being released can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the height of the ball above the ground when it is released.</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 minimum height of the ball above the ground.</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 the ball takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> seconds to return to its initial height above the ground for the first time.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the first 2 seconds of its motion, determine the amount of time that the ball is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math> metres above the ground.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the rate of change of the ball’s height above the ground when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi mathvariant="normal">π</mi><msqrt><mi>q</mi></msqrt><mo> </mo><msup><mi>ms</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mn>0</mn></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn><mfenced><mrow><mo>=</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn></math> (m) (above the ground) <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>uses the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced></math> which is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> (m)</p>
<p> </p>
<p><strong>OR</strong></p>
<p>the amplitude of motion is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn></math> (m) and the mean position is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn></math> (m) <strong>M1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>finds <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi mathvariant="normal">π</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced></math>, attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and determines that the minimum height above the ground occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mo>…</mo></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn></math> (m)</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>4</mn></math> (m) (above the ground) <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>the ball is released from its maximum height and returns there a period later <strong>R1</strong></p>
<p>the period is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi mathvariant="normal">π</mi></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn></mrow></mfenced><mo> </mo><mfenced><mi mathvariant="normal">s</mi></mfenced></math> <strong> A1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>2</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <strong> M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>…</mo></math> <strong> A1</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>so it takes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> seconds for the ball to return to its initial position for the first time <strong>AG</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>8</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math> (M1)</strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac></math> A1</strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mi>t</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></math> (A1)</strong></p>
<p> </p>
<p><strong>Note: </strong>Accept extra correct positive solutions for <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mi>t</mi></math></strong>.</p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn></mrow></mfenced></math> A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong>A1</strong> if solutions outside <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn></math> are also stated.</p>
<p>the ball is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math> metres above the ground for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></math>(s)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo></math>(s) <strong>A1</strong></p>
<p> </p>
<p><strong>[5 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER<br></strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced></math> <strong> (M1)</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>recognizes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced></math> is required <strong> (M1)</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi mathvariant="normal">π</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi mathvariant="normal">π</mi><mi>t</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p>attempts to evaluate their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>'</mo><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi mathvariant="normal">π</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi mathvariant="normal">π</mi><msqrt><mn>3</mn></msqrt><mo> </mo><mfenced><msup><mi>ms</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent correct answer forms where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>. For example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi mathvariant="normal">π</mi><msqrt><mn>3</mn></msqrt></math>.</p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Iron in the asteroid <em>16 Psyche</em> is said to be valued at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8973</mn></math> quadrillion euros <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtext>EUR</mtext></mfenced></math>, where one quadrillion <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>10</mn><mn>15</mn></msup></math>.</p>
</div>
<div class="specification">
<p>James believes the asteroid is approximately spherical with radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>113</mn><mo> </mo><mtext>km</mtext></math>. He uses this information to estimate its volume.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of the iron in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>k</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn><mo> </mo><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate James’s estimate of its volume, in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>km</mtext><mn>3</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The actual volume of the asteroid is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup><mo> </mo><msup><mtext>km</mtext><mn>3</mn></msup></math>.</p>
<p>Find the percentage error in James’s estimate of the volume.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. It appeared in a paper that permitted the use of a calculator, and so might not be suitable for all forms of practice.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>97</mn><mo>×</mo><msup><mn>10</mn><mn>18</mn></msup><mo> </mo><mo> </mo><mfenced><mtext>EUR</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>8</mn><mo>.</mo><mn>973</mn><mo>×</mo><msup><mn>10</mn><mn>18</mn></msup></mrow></mfenced></math> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>97</mn><mo> </mo><mo>(</mo><mn>8</mn><mo>.</mo><mn>973</mn><mo>)</mo></math>, <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><msup><mn>10</mn><mn>18</mn></msup></math>. Award <em><strong>(A1)(A0)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>97</mn><mtext>E</mtext><mn>18</mn></math>.<br>Award <em><strong>(A0)(A0)</strong></em> for answers of the type <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8973</mn><mo>×</mo><msup><mn>10</mn><mn>15</mn></msup></math>.</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><mrow><mn>4</mn><mo>×</mo><mi mathvariant="normal">π</mi><mo>×</mo><msup><mn>113</mn><mn>3</mn></msup></mrow><mn>3</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution in volume of sphere formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mn>040</mn><mo> </mo><mn>000</mn><mo> </mo><mfenced><msup><mtext>km</mtext><mn>3</mn></msup></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>04</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup><mo>,</mo><mo> </mo><mfrac><mrow><mn>5771588</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>6</mn><mo> </mo><mn>043</mn><mo> </mo><mn>992</mn><mo>.</mo><mn>82</mn></mrow></mfenced></math> <em><strong>(A1) (C2) </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfrac><mrow><mn>6</mn><mo> </mo><mn>043</mn><mo> </mo><mn>992</mn><mo>.</mo><mn>82</mn><mo>-</mo><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow><mrow><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfrac></mfenced><mo>×</mo><mn>100</mn></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution into the percentage error formula (accept a consistent absence of “<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></math>” from all terms).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>494</mn><mo> </mo><mfenced><mo>%</mo></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>494026</mn><mo>…</mo><mfenced><mo>%</mo></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2) </strong></em></p>
<p><strong><em><br></em></strong><strong>Note:</strong> Follow through from their answer to part (b). If the final answer is negative, award at most <em><strong>(M1)(A0)</strong></em>.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the vectors <em><strong>a</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ 3 \\ p \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <em><strong>b</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0 \\ 6 \\ {18} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> for which <em><strong>a</strong></em> and <em><strong>b</strong></em> are</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>parallel.</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>perpendicular.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg <strong>b</strong></em> = 2<em><strong>a</strong></em>, a = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span><em><strong>b</strong></em>, cos <em>θ </em>= 1, <em><strong>a</strong></em>•<em><strong>b</strong></em> = −|<em><strong>a</strong></em>||<em><strong>b</strong></em>|, 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 18</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 9 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of scalar product<em> <strong>(M1)</strong></em></p>
<p><em>eg </em><em><strong>a</strong></em>•<em><strong>b</strong></em>, (0)(0) + (3)(6) + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>(18)</p>
<p>recognizing <em><strong>a</strong></em>•<em><strong>b</strong></em> = 0 (seen anywhere) <em><strong>(M1)</strong></em></p>
<p>correct working<strong> (A1)</strong></p>
<p><em>eg</em> 18 + 18<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = 0, 18<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = −18 <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = −1 <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A cylinder with radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and height <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The sum of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> for this cylinder is 12 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an equation for the area, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, of the <strong>curved</strong> surface in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}A}}{{{\text{d}}r}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>A</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> when the area of the curved surface is maximized.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2\pi r\left( {12 - r} \right)">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 24\pi r - 2\pi {r^2}">
<mi>A</mi>
<mo>=</mo>
<mn>24</mn>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)(M1) (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r + h = 12">
<mi>r</mi>
<mo>+</mo>
<mi>h</mi>
<mo>=</mo>
<mn>12</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 12 - r">
<mi>h</mi>
<mo>=</mo>
<mn>12</mn>
<mo>−</mo>
<mi>r</mi>
</math></span> seen. Award <em><strong>(M1)</strong></em> for correctly substituting into curved surface area of a cylinder. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2\pi r\left( {12 - r} \right)">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong>OR </strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 24\pi r - 2\pi {r^2}">
<mi>A</mi>
<mo>=</mo>
<mn>24</mn>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi - 4\pi r">
<mn>24</mn>
<mi>π</mi>
<mo>−</mo>
<mn>4</mn>
<mi>π</mi>
<mi>r</mi>
</math></span> <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em>(ft)</strong><em><strong> (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em><strong>(ft)</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi">
<mn>24</mn>
<mi>π</mi>
</math></span> and <em><strong>(A1)</strong></em><strong>(ft)</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4\pi r">
<mo>−</mo>
<mn>4</mn>
<mi>π</mi>
<mi>r</mi>
</math></span> . Follow through from part (a). Award at most <em><strong>(A1)</strong></em><strong>(ft)<em>(A0)</em></strong> if additional terms are seen.</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align: left;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="24\pi - 4\pi r = 0">
<mn>24</mn>
<mi>π</mi>
<mo>−</mo>
<mn>4</mn>
<mi>π</mi>
<mi>r</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p style="text-align: left;"><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for setting <em>their</em> part (b) equal to zero.</p>
<p style="text-align: left;">6 (cm) <strong><em>(A1)</em>(ft)</strong><em><strong> (C2)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Follow through from part (b).</p>
<p style="text-align: left;"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> be an <strong>obtuse</strong> angle such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\theta = \frac{3}{5}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>5</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^x}\,{\text{sin}}\,x - \frac{{3x}}{4}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta "> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span> passes through the origin and has a gradient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta "> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>. Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </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>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 3. Line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M"> <mi>M</mi> </math></span> is a tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> at point P.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M"> <mi>M</mi> </math></span> is parallel to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L"> <mi>L</mi> </math></span>, find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of P.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>evidence of valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> sketch of triangle with sides 3 and 5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^2}\,\theta = 1 - {\text{si}}{{\text{n}}^2}\,\theta "> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> missing side is 4 (may be seen in sketch), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = - \frac{4}{5}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = - \frac{3}{4}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span> <em><strong>A2 N4</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution of either gradient <strong>or</strong> origin into equation of line <em><strong>(A1)</strong></em></p>
<p>(do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx + b"> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> </math></span>)</p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x\,{\text{tan}}\,\theta "> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = m\left( {x - 0} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mi>m</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = mx"> <mi>y</mi> <mo>=</mo> <mi>m</mi> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - \frac{3}{4}x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mi>x</mi> </math></span> <em><strong>A2 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = - \frac{3}{4}x"> <mi>L</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mi>x</mi> </math></span>.</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>valid approach to equate <strong>their</strong> gradients <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = {\text{tan}}\,\theta "> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f' = - \frac{3}{4}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,{\text{cos}}\,x + {{\text{e}}^x}\,{\text{sin}}\,x - \frac{3}{4} = - \frac{3}{4}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}\,\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right) - \frac{3}{4} = - \frac{3}{4}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span></p>
<p>correct equation without <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x = - {\text{cos}}\,x"> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x + {\text{sin}}\,x = 0"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - {\text{sin}}\,x}}{{{\text{cos}}\,x}} = 1"> <mfrac> <mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = - 1"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 135^\circ "> <mi>x</mi> <mo>=</mo> <msup> <mn>135</mn> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> (do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="135^\circ "> <msup> <mn>135</mn> <mo>∘</mo> </msup> </math></span>) <em><strong>A1 N1</strong></em></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> if additional answers are given.</p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>4</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is obtained by two transformations of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe these two transformations.</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>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>)</mo></math>.</p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>≥</mo><mn>7</mn></math>, find the smallest value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>translation (shift) by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></math> to the right/positive horizontal direction <em><strong>A1</strong></em></p>
<p>translation (shift) by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> upwards/positive vertical direction <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> accept translation by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mtd></mtr><mtr><mtd><mi>q</mi></mtd></mtr></mtable></mfenced></math></p>
<p><strong>Do not accept</strong> ‘move’ for translation/shift.</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><strong>METHOD 1</strong></p>
<p>minimum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math> (may be seen in sketch) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi><mo>≥</mo><mn>7</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>≥</mo><mn>8</mn><mo>.</mo><mn>5</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>8</mn><mo>.</mo><mn>5</mn></math>) <em><strong>A1</strong></em></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>8</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>r</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>4</mn><mo> </mo><mi>sin</mi><mfenced><mfrac><mrow><mo>-</mo><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mn>8</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mn>8</mn><mo>.</mo><mn>5</mn></math> <em><strong>(A1)</strong></em></p>
<p>smallest value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> to find an expression (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>) in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>g</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>r</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>4</mn><mo> </mo><mi>sin</mi><mfenced><mfrac><mrow><mo>-</mo><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>r</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi></math> <em><strong>A1</strong></em></p>
<p>minimum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi><mo>≥</mo><mn>7</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>6</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>≥</mo><mn>7</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math>) <em><strong>(A1)</strong></em></p>
<p>smallest value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi><mo>=</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>+</mo><mi>q</mi></math> is a maximum <em><strong>(M1)</strong></em></p>
<p>amplitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> <em><strong>(A1)</strong></em></p>
<p>attempt to find least maximum <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn><mo>×</mo><mn>4</mn><mo>+</mo><mn>7</mn></math></p>
<p>smallest value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Candidates knew aspects of the transformations performed but some were unable to correctly describe them fully, e.g., omitting direction (right/up/positive) or using 'move' instead of translate/shift. Each description requires three parts: transformation type, size and direction. e.g., translation of q units up. For part (b) few candidates were able to fully navigate the reasoning required in this question. A common error was to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mfrac><mrow><mo>-</mo><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></mfenced><mo>=</mo><mo>-</mo><mn>1</mn></math>, instead of 1. Those who used sketches to assist in their thinking were typically more successful.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand and simplify <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable substitution for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is a positive real constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">attempt to use binomial expansion <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><mprescripts></mprescripts><mn>3</mn></mmultiscripts><mo>×</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>3</mn></mmultiscripts><mo>×</mo><mn>1</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mn>3</mn></msup></math></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>a</mi><mo>+</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">attempt to substitute any double angle rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> Allow working RHS to LHS.</p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">recognizing to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>×</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>EITHER</strong></p>
<p style="text-align:left;">applies integration by inspection <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo>∫</mo><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>×</mo><msup><mfenced><mrow><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mn>6</mn></msup></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mi>m</mi></msubsup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>cos</mi><mo> </mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mn>32</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mn>32</mn><msup><mi>u</mi><mn>6</mn></msup><mo> </mo><mo>d</mo><mi>u</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>u</mi><mn>7</mn></msup><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mi>m</mi></msubsup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>u</mi><mn>7</mn></msup></mrow></mfenced><mn>0</mn><mrow><mi>sin</mi><mo> </mo><mi>m</mi></mrow></msubsup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mfenced><mrow><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup></mrow></mfenced><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></mrow></mfenced><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>+</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>128</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates successfully expanded the binomial, with the most common error being to omit the negative sign with a. The connection between (a)(i) and (ii) was often noted but not fully utilised with candidates embarking on unnecessary complex algebraic expansions of expressions involving double angle rules. Candidates often struggled to apply inspection or substitution when integrating. As a 'show that' question, b(i) provided a useful result to be utilised in (ii). So even without successfully completing (i) candidates could apply it in part (ii). Not many managed to do so.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>AC is a vertical communications tower with its base at C.</p>
<p>The tower has an observation deck, D, three quarters of the way to the top of the tower, A.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-03-06_om_10.32.25.png" alt="N16/5/MATSD/SP1/ENG/TZ0/11"></p>
<p>From a point B, on horizontal ground 250 m from C, the angle of elevation of D is 48°.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate CD, the height of the observation deck above the ground.</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>Calculate the angle of depression from A to B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan 48^\circ = \frac{{{\text{CD}}}}{{250}}"> <mi>tan</mi> <mo></mo> <msup> <mn>48</mn> <mo>∘</mo> </msup> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>CD</mtext> </mrow> </mrow> <mrow> <mn>250</mn> </mrow> </mfrac> </math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into the tangent ratio.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({\text{CD}} = ){\text{ }}278{\text{ }}({\text{m}}){\text{ }}(277.653 \ldots )"> <mo stretchy="false">(</mo> <mrow> <mtext>CD</mtext> </mrow> <mo>=</mo> <mo stretchy="false">)</mo> <mrow> <mtext> </mtext> </mrow> <mn>278</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mrow> <mtext>m</mtext> </mrow> <mo stretchy="false">)</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>277.653</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1) (C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan {\text{ABC (or equivalent)}} = \frac{{\frac{4}{3} \times 277.653 \ldots }}{{250}}"> <mi>tan</mi> <mo></mo> <mrow> <mtext>ABC (or equivalent)</mtext> </mrow> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mo>×</mo> <mn>277.653</mn> <mo>…</mo> </mrow> <mrow> <mn>250</mn> </mrow> </mfrac> </math></span> <strong><em>(M1)(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span> multiplying their part (a), <strong><em>(M1) </em></strong>for substitution into the tangent ratio, <strong><em>(M1) </em></strong>for correct substitution.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="90 - {\tan ^{ - 1}}\left( {\frac{{250}}{{\frac{4}{3} \times 277.653 \ldots }}} \right)"> <mn>90</mn> <mo>−</mo> <mrow> <msup> <mi>tan</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>250</mn> </mrow> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mo>×</mo> <mn>277.653</mn> <mo>…</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span> multiplying their part (a), <strong><em>(M1) </em></strong>for substitution into the tangent ratio, <strong><em>(M1) </em></strong>for subtracting from 90 and for correct substitution.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{(angle of depression}} = {\text{) }}56.0^\circ {\text{ }}(55.9687 \ldots )"> <mrow> <mtext>(angle of depression</mtext> </mrow> <mo>=</mo> <mrow> <mtext>) </mtext> </mrow> <msup> <mn>56.0</mn> <mo>∘</mo> </msup> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>55.9687</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1)</em>(ft) <em>(C4)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> have coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>9</mn><mo>,</mo><mo> </mo><mi>m</mi><mo>,</mo><mo> </mo><mo>-</mo><mn>6</mn></mrow></mfenced></math> respectively.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>, which passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>19</mn></mtd></mtr><mtr><mtd><mn>24</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>s</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider a unit vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">u</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">u</mi><mo>=</mo><mi>p</mi><mi mathvariant="bold-italic">i</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi mathvariant="bold-italic">j</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi mathvariant="bold-italic">k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>0</mn></math>.</p>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> is such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>BC</mtext><mo>→</mo></mover><mo>=</mo><mn>9</mn><mi mathvariant="bold-italic">u</mi></math>.</p>
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>valid approach to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>OB</mtext><mo>→</mo></mover><mo>-</mo><mover><mtext>OA</mtext><mo>→</mo></mover><mo> </mo><mo> </mo><mo>,</mo><mo> </mo><mo> </mo><mtext>A</mtext><mo>-</mo><mtext>B</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mi>m</mi><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>8</mn></mtd></mtr></mtable></mfenced></math> <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo>,</mo><mo> </mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>19</mn></mtd></mtr><mtr><mtd><mn>24</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>s</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math></p>
<p>one correct equation <em><strong> (A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>+</mo><mn>2</mn><mi>s</mi><mo>=</mo><mn>9</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>6</mn><mo>=</mo><mn>24</mn><mo>-</mo><mn>5</mn><mi>s</mi></math></p>
<p>correct value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> <em><strong>A1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>6</mn></math></p>
<p>substituting <strong>their</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> value into their expression/equation to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> <em><strong>(M1)</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>19</mn><mo>+</mo><mn>6</mn><mo>×</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>5</mn></math> <em><strong> A1 N3</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>BC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>9</mn><mi>p</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>C</mi><mo>=</mo><mn>9</mn><mi mathvariant="bold-italic">u</mi><mo>+</mo><mi>B</mi><mo> </mo><mo>,</mo><mo> </mo><mover><mtext>BC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mi>x</mi><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mi>y</mi><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mi>z</mi><mo>+</mo><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>correct working to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> <em><strong> (A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>OC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>9</mn><mi>p</mi><mo>+</mo><mn>9</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mtext>C</mtext><mo>=</mo><mn>9</mn><mfenced><mtable><mtr><mtd><mi>p</mi></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mo>-</mo><mn>3</mn></math></p>
<p>correct approach to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">u</mi></mfenced></math> (seen anywhere) <em><strong>A1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo> </mo><mo>,</mo><mo> </mo><msqrt><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>4</mn><mn>9</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>9</mn></mfrac></msqrt></math></p>
<p>recognizing unit vector has magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">u</mi></mfenced><mo>=</mo><mn>1</mn><mo> </mo><mo>,</mo><mo> </mo><msqrt><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><mn>1</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>9</mn></mfrac><mo>=</mo><mn>1</mn></math></p>
<p>correct working <em><strong> (A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>4</mn><mn>9</mn></mfrac><mo> </mo><mo>,</mo><mo> </mo><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>substituting <strong>their</strong> value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi><mo>-</mo><mn>9</mn></mtd></mtr><mtr><mtd><mi>y</mi><mo>-</mo><mn>5</mn></mtd></mtr><mtr><mtd><mi>z</mi><mo>+</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mtext>C</mtext><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mtext>C</mtext><mo>=</mo><mn>9</mn><mfenced><mtable><mtr><mtd><mfrac><mn>2</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mn>3</mn></mfrac></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>9</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>-</mo><mn>9</mn><mo>=</mo><mn>6</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mfenced><mrow><mn>15</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>3</mn></mrow></mfenced></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>15</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math>) <em><strong> A1 N4</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The marks for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> are independent of the first two marks.<br>For example, it is possible to award marks such as <em><strong>(M0)(A0)A1(M1)(A1)A1 (M0)A0</strong></em> or <em><strong>(M0)(A0)A1(M1)(A0)A0 (M1)A0</strong></em>.</p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>+</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>4</mn></math> may be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>+</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn><mi mathvariant="normal">π</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>correct substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>+</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>correct substitution using double-angle identities <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempting to factorise <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempting to use the quadratic formula <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><msqrt><msup><mn>5</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>2</mn></msqrt></mrow><mn>4</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><mn>3</mn></mrow><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext><mo>=</mo><mn>10</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext><mo>=</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AC</mtext><mo>=</mo><mn>2</mn><mi>x</mi></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mover><mtext>C</mtext><mo>^</mo></mover><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math>, find the area of the triangle.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><msqrt><mi>q</mi></msqrt></mrow><mn>2</mn></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>attempt to use the cosine rule to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mi>x</mi></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mfrac><mn>3</mn><mn>4</mn></mfrac></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>100</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>50</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>50</mn></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mover><mi>C</mi><mo>^</mo></mover></math> (seen anywhere) <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mover><mi>C</mi><mo>^</mo></mover><mo>+</mo><msup><mfenced><mfrac><mn>3</mn><mn>4</mn></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup></math> or right triangle with side <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> and hypotenuse <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mover><mi>C</mi><mo>^</mo></mover><mo>=</mo><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac></math> <em><strong> (A1)</strong></em></p>
<p><strong><br>Note:</strong> The marks for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mover><mi>C</mi><mo>^</mo></mover></math> may be awarded independently of the first three marks for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<p><br>correct substitution into the area formula using their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math>) and their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mover><mi>C</mi><mo>^</mo></mover></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>5</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><mn>10</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><msqrt><mn>50</mn></msqrt><mo>×</mo><mn>2</mn><msqrt><mn>50</mn></msqrt><mo>×</mo><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><mn>25</mn><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to find the height, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, of the triangle in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>10</mn><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p>equating their expressions for either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>h</mi><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>10</mn><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>x</mi></mrow></mfenced><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>100</mn><mo>-</mo><mfrac><mn>25</mn><mn>16</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac><mi>x</mi></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>50</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mn>50</mn></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>correct substitution into the area formula using their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math>) <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>2</mn><msqrt><mn>50</mn></msqrt><mo>×</mo><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac><msqrt><mn>50</mn></msqrt></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>×</mo><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced><mfenced><mrow><mfrac><msqrt><mn>7</mn></msqrt><mn>4</mn></mfrac><mn>5</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><mn>25</mn><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Find the least positive value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>determines <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math>) as the first quadrant (reference) angle <em><strong>(</strong><strong>A1)</strong></em></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>,</mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mfenced><mrow><mo>,</mo><mo>…</mo></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>⇒</mo><mi>x</mi><mo><</mo><mn>0</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> is rejected <em><strong>(R1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>17</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> (must be in radians) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>This question proved to be a struggle for many candidates, and some candidates made no attempt here. While a good number of candidates recognized the reference angle of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math>, this led to a final answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math>, which many left as their final answer. In other cases, some candidates heeded the requirement that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> must be a positive value, however they gave an incorrect final answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>11</mn><mi>π</mi></mrow><mn>6</mn></mfrac></math>. Few candidates correctly rejected their initial reference angle of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>4</mn></mfrac></math> and correctly solved an equation using <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mn>4</mn></mfrac></math>.</p>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question, all lengths are in metres and time is in seconds.</strong></p>
<p>Consider two particles, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math>, which start to move at the same time.</p>
<p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> moves in a straight line such that its displacement from a fixed-point is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>10</mn><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Particle <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> also moves in a straight line. The position of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math>.</p>
<p>The speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> is greater than the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mi>q</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>recognizing velocity is derivative of displacement <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mtext>d</mtext><mi>s</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo> </mo><mo>,</mo><mo> </mo><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>10</mn><mo>-</mo><mfrac><mn>7</mn><mn>4</mn></mfrac><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p>velocity<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>14</mn><mn>4</mn></mfrac><mi>t</mi><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced></math> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced><mo> </mo><mo>,</mo><mo> </mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math> , velocity<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p>correct speed <em><strong>(A1)</strong></em></p>
<p><em>eg </em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>recognizing relationship between speed and velocity (may be seen in inequality/equation) <em><strong>R1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced></math> , speed = | velocity | , graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> speed , <img src="data:image/png;base64,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"> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>1</mn></msub></math> speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>P</mi><mn>2</mn></msub></math> velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>5</mn></math></p>
<p>correct inequality or equation that compares speed or velocity (accept any variable for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>) <em><strong>A1</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>t</mi></mrow></mfenced><mo>></mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo><</mo><mo>-</mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>></mo><mn>5</mn><mo> </mo><mo>,</mo><mo> </mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mi>q</mi><mo>=</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math> (seconds) (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math> , do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>10</mn><mn>7</mn></mfrac></math>) <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the last two <em><strong>A1</strong></em> marks without the <em><strong>R1</strong></em>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \theta = \frac{{\sqrt 5 }}{3}">
<mi>sin</mi>
<mo><!-- --></mo>
<mi>θ<!-- θ --></mi>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> is acute.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta ">
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 2\theta ">
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>evidence of valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>right triangle, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\cos ^2}\theta = 1 - {\sin ^2}\theta ">
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span>missing side is 2, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {1 - {{\left( {\frac{{\sqrt 5 }}{3}} \right)}^2}} ">
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta = \frac{2}{3}">
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 2\theta ">
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times {\left( {\frac{2}{3}} \right)^2} - 1,{\text{ }}1 - 2{\left( {\frac{{\sqrt 5 }}{3}} \right)^2},{\text{ }}{\left( {\frac{2}{3}} \right)^2} - {\left( {\frac{{\sqrt 5 }}{3}} \right)^2}">
<mn>2</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 2\theta = - \frac{1}{9}">
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A triangular postage stamp, ABC, is shown in the diagram below, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AB}} = 5{\text{ cm}},{\rm{ B\hat AC}} = 34^\circ ,{\rm{ A\hat BC}} = 26^\circ ">
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>=</mo>
<mn>5</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
<mo>,</mo>
<mrow>
<mrow>
<mi mathvariant="normal">B</mi>
<mrow>
<mover>
<mi mathvariant="normal">A</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mn>34</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
<mo>,</mo>
<mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mrow>
<mover>
<mi mathvariant="normal">B</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mn>26</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{A\hat CB}} = 120^\circ ">
<mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mrow>
<mover>
<mi mathvariant="normal">C</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
<mo>=</mo>
<msup>
<mn>120</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_11.34.31.png" alt="M17/5/MATSD/SP1/ENG/TZ1/13"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the length of BC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the postage stamp.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{BC}}}}{{\sin 34^\circ }} = \frac{5}{{\sin 120^\circ }}">
<mfrac>
<mrow>
<mrow>
<mtext>BC</mtext>
</mrow>
</mrow>
<mrow>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>34</mn>
<mo>∘</mo>
</msup>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mrow>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>120</mn>
<mo>∘</mo>
</msup>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substituted sine rule formula, <strong><em>(A1) </em></strong>for correct substitutions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = 3.23{\text{ (cm) }}\left( {3.22850 \ldots {\text{ (cm)}}} \right)">
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>=</mo>
<mn>3.23</mn>
<mrow>
<mtext> (cm) </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3.22850</mn>
<mo>…</mo>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C3)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}(5)(3.22850)\sin 26^\circ ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>5</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>3.22850</mn>
<mo stretchy="false">)</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>26</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(M1)(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substituted area of a triangle formula, <strong><em>(A1) </em></strong>for correct substitutions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3.54{\text{ }}({\text{c}}{{\text{m}}^2}){\text{ }}\left( {3.53820 \ldots {\text{ }}({\text{c}}{{\text{m}}^2})} \right)">
<mo>=</mo>
<mn>3.54</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3.53820</mn>
<mo>…</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows triangle ABC, with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AB}} = 3{\text{ cm}}">
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = 8{\text{ cm}}">
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{A\hat BC = }}\frac{\pi }{3}">
<mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mrow>
<mover>
<mi mathvariant="normal">B</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
<mo>=</mo>
</mrow>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.17.57.png" alt="N17/5/MATME/SP1/ENG/TZ0/04"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = 7{\text{ cm}}">
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>=</mo>
<mn>7</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.</p>
<p><img src="images/Schermafbeelding_2018-02-11_om_10.50.00.png" alt="N17/5/MATME/SP1/ENG/TZ0/04.b"></p>
<p>Find the exact perimeter of this shape.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>evidence of choosing the cosine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{c^2} = {a^2} + {b^2} - ab\cos C">
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>a</mi>
<mi>b</mi>
<mi>cos</mi>
<mo></mo>
<mi>C</mi>
</math></span></p>
<p>correct substitution into RHS of cosine rule <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{3^2} + {8^2} - 2 \times 3 \times 8 \times \cos \frac{\pi }{3}">
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>8</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mo>×</mo>
<mn>8</mn>
<mo>×</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span></p>
<p>evidence of correct value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \frac{\pi }{3}">
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span> (may be seen anywhere, including in cosine rule) <strong><em>A1</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \frac{\pi }{3} = \frac{1}{2},{\text{ A}}{{\text{C}}^2} = 9 + 64 - \left( {48 \times \frac{1}{2}} \right),{\text{ }}9 + 64 - 24">
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> A</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>C</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo>+</mo>
<mn>64</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>48</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mo>+</mo>
<mn>64</mn>
<mo>−</mo>
<mn>24</mn>
</math></span></p>
<p>correct working clearly leading to answer <strong><em>A1</em></strong></p>
<p>e<em>g</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}{{\text{C}}^2} = 49,{\text{ }}b = \sqrt {49} ">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>C</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>49</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<msqrt>
<mn>49</mn>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = 7{\text{ (cm)}}">
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>=</mo>
<mn>7</mn>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</math></span> <strong><em>AG N0</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award no marks if the only working seen is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}{{\text{C}}^2} = 49">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>C</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>49</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = \sqrt {49} ">
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>=</mo>
<msqrt>
<mn>49</mn>
</msqrt>
</math></span> (or similar).</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution for semicircle <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{semicircle}} = \frac{1}{2}(2\pi \times 3.5),{\text{ }}\frac{1}{2} \times \pi \times 7,{\text{ }}3.5\pi ">
<mrow>
<mtext>semicircle</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>π</mi>
<mo>×</mo>
<mn>3.5</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mi>π</mi>
<mo>×</mo>
<mn>7</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3.5</mn>
<mi>π</mi>
</math></span></p>
<p>valid approach (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{perimeter}} = {\text{AB}} + {\text{BC}} + {\text{semicircle, }}3 + 8 + \left( {\frac{1}{2} \times 2 \times \pi \times \frac{7}{2}} \right),{\text{ }}8 + 3 + 3.5\pi ">
<mrow>
<mtext>perimeter</mtext>
</mrow>
<mo>=</mo>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>+</mo>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>+</mo>
<mrow>
<mtext>semicircle, </mtext>
</mrow>
<mn>3</mn>
<mo>+</mo>
<mn>8</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<mi>π</mi>
<mo>×</mo>
<mfrac>
<mn>7</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>8</mn>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mn>3.5</mn>
<mi>π</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11 + \frac{7}{2}\pi {\text{ }}( = 3.5\pi + 11){\text{ (cm)}}">
<mn>11</mn>
<mo>+</mo>
<mfrac>
<mn>7</mn>
<mn>2</mn>
</mfrac>
<mi>π</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>3.5</mn>
<mi>π</mi>
<mo>+</mo>
<mn>11</mn>
<mo stretchy="false">)</mo>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.<br>This is shown in the following diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The vectors <em><strong>p</strong></em> , <em><strong>q</strong></em> and <em><strong>r</strong></em> are shown on the diagram.</p>
<p>Find <em><strong>p</strong></em>•(<em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r</strong></em>).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1 </strong>(using |<em><strong>p</strong></em>| |2<em><strong>q</strong></em>| cos<em>θ</em>)</p>
<p>finding <em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r (A1)</strong></em></p>
<p><em>eg </em> 2<em><strong>q</strong></em>, <img src="data:image/png;base64,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"></p>
<p>| <em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r </strong></em>| = 2 × 3 (= 6) (seen anywhere) <em><strong>A1</strong></em></p>
<p>correct angle between <em><strong>p</strong></em> and <em><strong>q</strong></em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{3}"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> (accept 60°)</p>
<p>substitution of <strong>their</strong> values <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 3 × 6 × cos<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{\pi }{3}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct value for cos<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{\pi }{3}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>×</mo> <mn>6</mn> <mo>×</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span></p>
<p><em><strong>p</strong></em>•(<em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r</strong></em>) = 9 <em><strong> A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong> (scalar product using distributive law)</p>
<p>correct expression for scalar distribution <em><strong>(A1)</strong></em></p>
<p>eg <em><strong>p</strong></em>• <em><strong>p</strong></em> + <em><strong>p</strong></em>•<em><strong>q</strong></em> + <em><strong>p</strong></em>•<em><strong>r</strong></em></p>
<p>three correct angles between the vector pairs (seen anywhere) <em><strong>(A2)</strong></em></p>
<p><em>eg </em> 0° between <em><strong>p</strong></em> and <em><strong>p</strong></em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{3}"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> between <em><strong>p</strong></em> and <em><strong>q</strong></em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> between <em><strong>p</strong></em> and <em><strong>r</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for only two correct angles.</p>
<p>substitution of <strong>their</strong> values <em><strong>(M1)</strong></em></p>
<p><em>eg </em> 3.3.cos0 +3.3.cos<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{3}"> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> + 3.3.cos120</p>
<p>one correct value for cos0, cos<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{\pi }{3}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> or cos<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{2\pi }{3}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> (seen anywhere) <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2},\,\,\,3 \times 6 \times \frac{1}{2}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>×</mo> <mn>6</mn> <mo>×</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span></p>
<p><em><strong>p</strong></em>•(<em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r</strong></em>) = 9 <em><strong> A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong> (scalar product using relative position vectors)</p>
<p>valid attempt to find one component of <em><strong>p</strong></em> or <em><strong>r</strong></em> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> sin 60 = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{3}"> <mfrac> <mi>x</mi> <mn>3</mn> </mfrac> </math></span>, cos 60 = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{3}"> <mfrac> <mi>x</mi> <mn>3</mn> </mfrac> </math></span>, one correct value <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{2},\,\,\frac{{3\sqrt 3 }}{2},\,\,\frac{{ - 3\sqrt 3 }}{2}"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p>one correct vector (two or three dimensions) (seen anywhere) <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \left( \begin{gathered} \,\,\,\frac{3}{2} \hfill \\ \frac{{3\sqrt 3 }}{2} \hfill \\ \end{gathered} \right),\,\,q = \left( \begin{gathered} 3 \hfill \\ 0 \hfill \\ \end{gathered} \right),\,\,r = \left( \begin{gathered} \,\,\,\,\frac{3}{2} \hfill \\ - \frac{{3\sqrt 3 }}{2} \hfill \\ \,\,\,\,0 \hfill \\ \end{gathered} \right)"> <mi>p</mi> <mo>=</mo> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>q</mi> <mo>=</mo> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mo>=</mo> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mtd> </mtr> <mtr> <mtd> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </math></span></p>
<p>three correct vectors <em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r </strong></em>= 2<em><strong>q</strong></em> <em><strong>(A1)</strong></em></p>
<p><em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r </strong></em>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ \end{gathered} \right)"> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 6 \hfill \\ 0 \hfill \\ 0 \hfill \\ \end{gathered} \right)"> <mrow> <mo>(</mo> <mtable displaystyle="true" columnspacing="1em" rowspacing="3pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </math></span> (seen anywhere, including scalar product) <em><strong>(A1)</strong></em></p>
<p>correct working <em><strong>(A1)</strong></em><br><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{3}{2} \times 6} \right) + \left( {\frac{{3\sqrt 3 }}{2} \times 0} \right),\,\,9 + 0 + 0"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>×</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>9</mn> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> </math></span></p>
<p><em><strong>p</strong></em>•(<em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r</strong></em>) = 9 <em><strong> A1 N3</strong></em></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Point A has coordinates (−4, −12, 1) and point B has coordinates (2, −4, −4).</p>
</div>
<div class="specification">
<p>The line <em>L</em> passes through A and B.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation for <em>L</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Point <em>C</em> (<em>k</em> , 12 , −<em>k</em>) is on <em>L</em>. Show that <em>k</em> = 14.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mo>∙</mo>
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span>.</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>Write down the value of angle OBA.</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>Point D is also on <em>L</em> and has coordinates (8, 4, −9).</p>
<p>Find the area of triangle OCD.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>correct approach <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AO}}}\limits^ \to \,\, + \,\,\mathop {{\text{OB}}}\limits^ \to ,\,\,\,{\text{B}} - {\text{A}}\,{\text{, }}\,\left( \begin{gathered} \,\,2 \hfill \\ - 4 \hfill \\ - 4 \hfill \\ \end{gathered} \right) - \left( \begin{gathered} \, - 4 \hfill \\ - 12 \hfill \\ \,\,\,1 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>AO</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>+</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>B</mtext>
</mrow>
<mo>−</mo>
<mrow>
<mtext>A</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>, </mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>AG N0</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>any correct equation in the form <em><strong>r</strong></em> = <em><strong>a</strong></em> + <em>t<strong>b</strong></em> (any parameter for <em>t</em>) <em><strong>A2 N2</strong></em></p>
<p>where <strong><em>a</em></strong> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \,\,2 \hfill \\ - 4 \hfill \\ - 4 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \, - 4 \hfill \\ - 12 \hfill \\ \,\,\,1 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> and <em><strong>b</strong></em> is a scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p><em>eg</em> <em><strong>r</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \, - 4 \hfill \\ - 12 \hfill \\ \,\,\,1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right),\,\,\left( {x,\,\,y,\,\,z} \right) = \left( {2,\,\, - 4,\,\, - 4} \right) + t\left( {6,\,\,8,\,\, - 5} \right),">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
</math></span> <em><strong>r </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \, - 4 + 6t \hfill \\ - 12 + 8t \hfill \\ \,\,\,1 - 5t \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
<mo>+</mo>
<mn>6</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
<mo>+</mo>
<mn>8</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
<mo>−</mo>
<mn>5</mn>
<mi>t</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for the form <em><strong>a</strong></em> + <em>t<strong>b</strong></em>, <em><strong>A1</strong></em> for the form <em><strong>L</strong></em> = <em><strong>a</strong></em> + <em>t<strong>b</strong></em>, <em><strong>A0</strong></em> for the form <em><strong>r</strong></em> = <em><strong>b</strong></em> + <em>t<strong>a</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong> (solving for <em>t</em>)</p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \,k \hfill \\ 12 \hfill \\ - k \hfill \\ \end{gathered} \right) = \left( \begin{gathered} \,\,2 \hfill \\ - 4 \hfill \\ - 4 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right),\,\,\left( \begin{gathered} \,k \hfill \\ 12 \hfill \\ - k \hfill \\ \end{gathered} \right) = \left( \begin{gathered} \, - 4 \hfill \\ - 12 \hfill \\ \,\,\,1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p>one correct equation <em><strong>A1</strong></em></p>
<p>eg −4 + 8<em>t</em> = 12, −12 + 8<em>t</em> = 12</p>
<p>correct value for <em>t <strong>(A1)</strong></em></p>
<p><em>eg t</em> = 2 or 3</p>
<p>correct substitution <em><strong>A1</strong></em></p>
<p><em>eg </em> 2 + 6(2), −4 + 6(3), −[1 + 3(−5)]</p>
<p><em>k</em> = 14 <em><strong>AG N0</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong> (solving simultaneously)</p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \,k \hfill \\ 12 \hfill \\ - k \hfill \\ \end{gathered} \right) = \left( \begin{gathered} \,\,2 \hfill \\ - 4 \hfill \\ - 4 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right),\,\,\left( \begin{gathered} \,k \hfill \\ 12 \hfill \\ - k \hfill \\ \end{gathered} \right) = \left( \begin{gathered} \, - 4 \hfill \\ - 12 \hfill \\ \,\,\,1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p>two correct equations in <em><strong>A1</strong></em></p>
<p><em>eg k</em> = −4 + 6<em>t,</em> −<em>k</em> = 1 −5<em>t</em></p>
<p><strong>EITHER</strong> (eliminating <em>k</em>)</p>
<p>correct value for <em>t</em> <em><strong>(A1)</strong></em></p>
<p><em>eg t</em> = 2 or 3</p>
<p>correct substitution <em><strong>A1</strong></em></p>
<p><em>eg </em> 2 + 6(2), −4 + 6(3)</p>
<p><strong>OR</strong> (eliminating <em>t</em>)</p>
<p>correct equation(s) <em><strong>(A1)</strong></em></p>
<p><em>eg </em> 5<em>k</em> + 20 = 30<em>t</em> <strong>and </strong>−6<em>k</em> − 6 = 30<em>t</em>, −<em>k</em> = 1 − 5<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{k + 4}}{6}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>4</mn>
</mrow>
<mn>6</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct working clearly leading to <em>k</em> = 14 <em><strong>A1</strong></em></p>
<p><em>eg </em>−<em>k</em> + 14 = 0, −6<em>k</em> = 6 −5<em>k</em> − 20, 5<em>k</em> = −20 + 6(1 + <em>k</em>)</p>
<p><strong>THEN </strong></p>
<p><em>k</em> = 14 <em><strong>AG N0</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into scalar product <em><strong>A1</strong></em></p>
<p><em>eg </em>(2)(6) − (4)(8) − (4)(−5), 12 − 32 + 20</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OB}}}\limits^ \to \, \bullet \mathop {{\text{AB}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mspace width="thinmathspace"></mspace>
<mo>∙</mo>
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span> = 0 <em><strong>A1 N0</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{O}}\mathop {\text{B}}\limits^ \wedge {\text{A}} = \frac{\pi }{2},\,\,90^\circ \,\,\,\,\,\left( {{\text{accept}}\,\frac{{3\pi }}{2},\,\,270^\circ } \right)\,">
<mrow>
<mtext>O</mtext>
</mrow>
<mover>
<mrow>
<mtext>B</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>A</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msup>
<mn>90</mn>
<mo>∘</mo>
</msup>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<msup>
<mn>270</mn>
<mo>∘</mo>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
</math></span> <strong><em>A1 N1</em></strong></p>
<p><strong><em>[1 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> × height × CD)</p>
<p>recognizing that OB is altitude of triangle with base CD (seen anywhere) <em><strong> M1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \left| {\mathop {{\text{OB}}}\limits^ \to } \right| \times \left| {\mathop {{\text{CD}}}\limits^ \to } \right|,\,\,{\text{OB}} \bot {\text{CD}},">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>CD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>OB</mtext>
</mrow>
<mi mathvariant="normal">⊥</mi>
<mrow>
<mtext>CD</mtext>
</mrow>
<mo>,</mo>
</math></span> sketch showing right angle at B</p>
<p><img src="data:image/png;base64,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"></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{CD}}}\limits^ \to = \left( \begin{gathered} - 6 \hfill \\ - 8 \hfill \\ \,5 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>CD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{DC}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>DC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p>correct magnitudes (seen anywhere) <em><strong>(A1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\mathop {{\text{OB}}}\limits^ \to } \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 4} \right)}^2} + {{\left( { - 4} \right)}^2}} = \left( {\sqrt {36} } \right)">
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>36</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\mathop {{\text{CD}}}\limits^ \to } \right| = \sqrt {{{\left( { - 6} \right)}^2} + {{\left( { - 8} \right)}^2} + {{\left( 5 \right)}^2}} = \left( {\sqrt {125} } \right)">
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>CD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>125</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct substitution into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}bh">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>b</mi>
<mi>h</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6 \times \sqrt {125} ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<msqrt>
<mn>125</mn>
</msqrt>
</math></span> </p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\sqrt {125} ,\,\,15\sqrt 5 ">
<mo>=</mo>
<mn>3</mn>
<msqrt>
<mn>125</mn>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>15</mn>
<msqrt>
<mn>5</mn>
</msqrt>
</math></span> <em><strong>A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong> (subtracting triangles)</p>
<p>recognizing that OB is altitude of either ΔOBD or ΔOBC(seen anywhere) <em><strong>M1</strong></em></p>
<p>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \left| {\mathop {{\text{OB}}}\limits^ \to } \right| \times \left| {\mathop {{\text{BD}}}\limits^ \to } \right|,\,\,{\text{OB}} \bot {\text{BC}},">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>BD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>OB</mtext>
</mrow>
<mi mathvariant="normal">⊥</mi>
<mrow>
<mtext>BC</mtext>
</mrow>
<mo>,</mo>
</math></span> sketch of triangle showing right angle at B</p>
<p><img 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"></p>
<p>one correct vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{BD}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>BD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{DB}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>DB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{BC}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>BC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{CB}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>CB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p>eg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{BD}}}\limits^ \to = \left( \begin{gathered} \,6 \hfill \\ \,8 \hfill \\ - 5 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>BD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{CB}}}\limits^ \to = \left( \begin{gathered} - 12 \hfill \\ - 16 \hfill \\ \,10 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>CB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>12</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>16</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>10</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\mathop {{\text{OB}}}\limits^ \to } \right| = \sqrt {{{\left( 2 \right)}^2} + {{\left( { - 4} \right)}^2} + {{\left( { - 4} \right)}^2}} = \left( {\sqrt {36} } \right)">
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>36</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p>one correct magnitude of a base (seen anywhere)<em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\mathop {{\text{BD}}}\limits^ \to } \right| = \sqrt {{{\left( 6 \right)}^2} + {{\left( 8 \right)}^2} + {{\left( 5 \right)}^2}} = \left( {\sqrt {125} } \right),\,\,\left| {\mathop {{\text{BC}}}\limits^ \to } \right| = \sqrt {144 + 256 + 100} = \left( {\sqrt {500} } \right)">
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>BD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>125</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>BC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>144</mn>
<mo>+</mo>
<mn>256</mn>
<mo>+</mo>
<mn>100</mn>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>500</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6 \times \sqrt {500} - \frac{1}{2} \times 6 \times 5\sqrt 5 ,\,\,\frac{1}{2} \times 6 \times \sqrt {500} \times {\text{sin}}90 - \frac{1}{2} \times 6 \times 5\sqrt 5 \times {\text{sin}}90">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<msqrt>
<mn>500</mn>
</msqrt>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<mn>5</mn>
<msqrt>
<mn>5</mn>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<msqrt>
<mn>500</mn>
</msqrt>
<mo>×</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mn>90</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>6</mn>
<mo>×</mo>
<mn>5</mn>
<msqrt>
<mn>5</mn>
</msqrt>
<mo>×</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mn>90</mn>
</math></span></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\sqrt {125} ,\,\,15\sqrt 5 ">
<mo>=</mo>
<mn>3</mn>
<msqrt>
<mn>125</mn>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>15</mn>
<msqrt>
<mn>5</mn>
</msqrt>
</math></span> <em><strong>A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong> (using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span><em>ab</em> sin <em>C</em> with ΔOCD)</p>
<p>two correct side lengths (seen anywhere) <em><strong>(A1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\mathop {{\text{OD}}}\limits^ \to } \right| = \sqrt {{{\left( 8 \right)}^2} + {{\left( 4 \right)}^2} + {{\left( { - 9} \right)}^2}} = \left( {\sqrt {161} } \right),\,\,\left| {\mathop {{\text{CD}}}\limits^ \to } \right| = \sqrt {{{\left( { - 6} \right)}^2} + {{\left( { - 8} \right)}^2} + {{\left( 5 \right)}^2}} = \left( {\sqrt {125} } \right),\,">
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>9</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>161</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>CD</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>125</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\mathop {{\text{OC}}}\limits^ \to } \right| = \sqrt {{{\left( {14} \right)}^2} + {{\left( {12} \right)}^2} + {{\left( { - 14} \right)}^2}} = \left( {\sqrt {536} } \right)">
<mrow>
<mo>|</mo>
<mrow>
<mover>
<mrow>
<mrow>
<mtext>OC</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>14</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>14</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>536</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>attempt to find cosine ratio (seen anywhere) <em><strong>M1</strong></em><br><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{536 - 286}}{{ - 2\sqrt {161} \sqrt {125} }},\,\,\frac{{{\text{OD}} \bullet {\text{DC}}}}{{\left| {OD} \right|\left| {DC} \right|}}">
<mfrac>
<mrow>
<mn>536</mn>
<mo>−</mo>
<mn>286</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>161</mn>
</msqrt>
<msqrt>
<mn>125</mn>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mrow>
<mtext>OD</mtext>
</mrow>
<mo>∙</mo>
<mrow>
<mtext>DC</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>O</mi>
<mi>D</mi>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>D</mi>
<mi>C</mi>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>correct working for sine ratio <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\left( {125} \right)}^2}}}{{161 \times 125}} + {\text{si}}{{\text{n}}^2}\,D = 1">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>125</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>161</mn>
<mo>×</mo>
<mn>125</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>D</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p>correct substitution into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}ab\,\,{\text{sin}}\,C">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>a</mi>
<mi>b</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>C</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.5 \times \sqrt {161} \times \sqrt {125} \times \frac{6}{{\sqrt {161} }}">
<mn>0.5</mn>
<mo>×</mo>
<msqrt>
<mn>161</mn>
</msqrt>
<mo>×</mo>
<msqrt>
<mn>125</mn>
</msqrt>
<mo>×</mo>
<mfrac>
<mn>6</mn>
<mrow>
<msqrt>
<mn>161</mn>
</msqrt>
</mrow>
</mfrac>
</math></span></p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\sqrt {125} ,\,\,15\sqrt 5 ">
<mo>=</mo>
<mn>3</mn>
<msqrt>
<mn>125</mn>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>15</mn>
<msqrt>
<mn>5</mn>
</msqrt>
</math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>6</mn><mo>+</mo><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn><mi>π</mi></math>.</p>
<p>The following diagram shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> touches the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, as shown. The shaded region is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, between the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
</div>
<div class="specification">
<p>The right cone in the following diagram has a total surface area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi mathvariant="normal">π</mi></math>, equal to the shaded area in the previous diagram.</p>
<p>The cone has a base radius of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>, height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, and slant height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area of the shaded region is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi mathvariant="normal">π</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the volume of the cone.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>+</mo><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> (or setting their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math>) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mi mathvariant="normal">π</mi><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow></munderover><mfenced><mrow><mn>6</mn><mo>+</mo><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mi mathvariant="normal">π</mi><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow></msubsup></math> <em><strong>A1A1</strong></em></p>
<p>substitute their limits into their integrated expression and subtract <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>18</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>3</mn><mi mathvariant="normal">π</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>6</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo> </mo><mi mathvariant="normal">π</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>6</mn><mfenced><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow></mfenced><mo>+</mo><mn>0</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>6</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>0</mn></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>18</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>6</mn><mi mathvariant="normal">π</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>area<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>=</mo><mn>12</mn><mi mathvariant="normal">π</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute into formula for surface area (including base) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><msup><mn>2</mn><mn>2</mn></msup></mfenced><mo>+</mo><mi mathvariant="normal">π</mi><mfenced><mn>2</mn></mfenced><mfenced><mi>l</mi></mfenced><mo>=</mo><mn>12</mn><mi mathvariant="normal">π</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn><mtext>π</mtext><mi>l</mi><mo>=</mo><mn>12</mn><mi mathvariant="normal">π</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mtext>π</mtext><mi>l</mi><mo>=</mo><mn>8</mn><mi mathvariant="normal">π</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find the height of the cone <em><strong>(M1)</strong></em></p>
<p>e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mtext>their</mtext><mo> </mo><mi>l</mi></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msqrt><mn>12</mn></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup><mi>h</mi></math> with their values substituted <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><mfenced><msup><mn>2</mn><mn>2</mn></msup></mfenced><mfenced><msqrt><mn>12</mn></msqrt></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>volume</mtext><mo>=</mo><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi><msqrt><mn>12</mn></msqrt></mrow><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi><msqrt><mn>3</mn></msqrt></mrow><mn>3</mn></mfrac><mo>=</mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi></mrow><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A calculator fits into a cuboid case with height 29 mm, width 98 mm and length 186 mm.</p>
</div>
<div class="question">
<p>Find the volume, in cm<sup>3</sup>, of this calculator case.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>evidence of 10 mm = 1 cm <strong><em>(A1)</em></strong></p>
<p><strong>Note:</strong> Award <strong><em>(A1)</em></strong> for dividing their volume from part (a) or part (b) by 1000.</p>
<p>529 (cm<sup>3</sup>) (528.612 (cm<sup>3</sup>)) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from parts (a) or (b). Accept answers written in scientific notation.</p>
<p><em><strong>[2 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a circle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> lie on the circumference of the circle, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mover><mtext>O</mtext><mo>^</mo></mover><mtext>B</mtext><mo>=</mo><mn>1</mn><mo> </mo><mtext>radian</mtext></math>.</p>
<p style="text-align: left;">The perimeter of the shaded region is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact area of the <strong>non-shaded</strong> region.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>minor arc <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext></math> has length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> <em><strong>(A1)</strong></em></p>
<p>recognition that perimeter of shaded sector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>r</mi></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>r</mi><mo>=</mo><mn>12</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>-</mo><mi mathvariant="normal">A</mi><mover><mi mathvariant="normal">O</mi><mo>^</mo></mover><mi mathvariant="normal">B</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong> (M1)</strong></em></p>
<p>Area of non-shaded region <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><msup><mn>4</mn><mn>2</mn></msup></mfenced></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>area of circle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo></math> area of shaded sector <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mi mathvariant="normal">π</mi><mo>-</mo><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>1</mn><mo>×</mo><msup><mn>4</mn><mn>2</mn></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>16</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>8</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>8</mn><mfenced><mrow><mn>2</mn><mi mathvariant="normal">π</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempting to use the change of base rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = \frac{{{\text{lo}}{{\text{g}}_3}\left( {{\text{cos}}\,2x + 2} \right)}}{{{\text{lo}}{{\text{g}}_3}9}}">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mn>9</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{lo}}{{\text{g}}_3}\left( {{\text{cos}}\,2x + 2} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{2}}\,{\text{sin}}\,x = \sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{4}}\,{\text{si}}{{\text{n}}^2}\,x = {\text{cos}}\,2x + 2">
<mrow>
<mtext>4</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2x = 1 - 2\,{\text{si}}{{\text{n}}^2}\,x">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6\,{\text{si}}{{\text{n}}^2}\,x = 3">
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x = \left( \pm \right)\frac{1}{{\sqrt 2 }}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mo>±</mo>
<mo>)</mo>
</mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: Award <em><strong>A0</strong></em> if solutions other than <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> are included.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x = \frac{1}{3}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,4x">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
<mi>x</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>correct substitution into formula for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {2x} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\left( {2x} \right)">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\left( {\frac{1}{3}} \right)^2}">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\left( {\frac{{\sqrt 8 }}{3}} \right)^2} - 1">
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {\frac{1}{3}} \right)\left( {\frac{{\sqrt 8 }}{3}} \right)">
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{{\sqrt 8 }}{3}} \right)^2} - {\left( {\frac{1}{3}} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {2x} \right) = \frac{7}{9}">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>7</mn>
<mn>9</mn>
</mfrac>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\left( {2x} \right) = \frac{{2\sqrt 8 }}{9}\,\,\,\,\left( { = \frac{{\sqrt {32} }}{9} = \frac{{4\sqrt 2 }}{9}} \right)">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>9</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>32</mn>
</msqrt>
</mrow>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (may be seen in substitution) <em><strong>A2</strong></em></p>
<p>recognizing 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is double angle of 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {2\left( {2x} \right)} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{co}}{{\text{s}}^2}\,\left( {2\theta } \right) - 1">
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2\,{\text{si}}{{\text{n}}^2}\,\left( {2\theta } \right)">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^2}\,\left( {2\theta } \right) - {\text{si}}{{\text{n}}^2}\,\left( {2\theta } \right)">
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct substitution of <strong>their</strong> value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {2x} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,\left( {2x} \right)">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> into formula for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {4x} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\left( {\frac{7}{9}} \right)^2} - 1">
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>7</mn>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{98}}{{81}} - 1">
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\left( {\frac{{2\sqrt 8 }}{9}} \right)^2}">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - \frac{{64}}{{81}}">
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>64</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{7}{9}} \right)^2} - {\left( {\frac{{2\sqrt 8 }}{9}} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>7</mn>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{49}}{{81}} - \frac{{32}}{{81}}">
<mfrac>
<mrow>
<mn>49</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>32</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {4x} \right) = \frac{{17}}{{81}}">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>17</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>recognizing 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is double angle of 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {2\left( {2x} \right)} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>double angle identity for 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{co}}{{\text{s}}^2}\,\left( {2\theta } \right) - 1">
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2\,{\text{si}}{{\text{n}}^2}\left( {2x} \right)">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^2}\,\left( {2\theta } \right) - {\text{si}}{{\text{n}}^2}\,\left( {2\theta } \right)">
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>correct expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {4x} \right)">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\left( {1 - 2\,{\text{si}}{{\text{n}}^2}\,\theta } \right)^2} - 1">
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\left( {2\,{\text{sin}}\,x\,{\text{cos}}\,x} \right)^2}">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - 2\,{\text{si}}{{\text{n}}^2}\,\theta } \right)^2} - {\left( {2\,{\text{sin}}\,\theta \,{\text{cos}}\,\theta } \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>correct substitution for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,x">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\left( {1 - 2{{\left( {\frac{1}{3}} \right)}^2}} \right)^2} - 1">
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {1 - 4{{\left( {\frac{1}{3}} \right)}^2} + 4{{\left( {\frac{1}{3}} \right)}^4}} \right) - 1">
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\left( {2 \times \frac{1}{3} \times \frac{{\sqrt 8 }}{3}} \right)^2}">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>8</mn>
</msqrt>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {\frac{{49}}{{81}}} \right) - 1">
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>49</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2\left( {\frac{{32}}{{81}}} \right)">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>32</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{49}}{{81}} - \frac{{32}}{{81}}">
<mfrac>
<mrow>
<mn>49</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>32</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\left( {4x} \right) = \frac{{17}}{{81}}">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>17</mn>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><strong>Note: In this question, distance is in metres and time is in seconds.</strong></p>
<p>Two particles <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_1}">
<mrow>
<msub>
<mi>P</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_2}">
<mrow>
<msub>
<mi>P</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> start moving from a point A at the same time, along different straight lines.</p>
<p>After <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds, the position of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_1}">
<mrow>
<msub>
<mi>P</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is given by <strong><em>r</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ { - 1} \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Two seconds after leaving A, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_1}">
<mrow>
<msub>
<mi>P</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is at point B.</p>
</div>
<div class="specification">
<p>Two seconds after leaving A, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_2}">
<mrow>
<msub>
<mi>P</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> is at point C, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 4 \end{array}} \right)">
<mover>
<mrow>
<mtext>AC</mtext>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of A.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} "> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </math></span>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\overrightarrow {{\text{AB}}} } \right|"> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos {\rm{B\hat AC}}"> <mi>cos</mi> <mo></mo> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mrow> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mi mathvariant="normal">C</mi> </mrow> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the distance between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_1}"> <mrow> <msub> <mi>P</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_2}"> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> </math></span> two seconds after they leave A.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </math></span> at A <strong><em>(M1)</em></strong></p>
<p>A is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(4,{\text{ }} - 1,{\text{ }}3)"> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ { - 1} \\ 3 \end{array}} \right) + 2\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 2} \end{array}} \right),{\text{ }}(6,{\text{ }}3,{\text{ }} - 1)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span></p>
<p>correct approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} "> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </math></span> <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AO}} + {\text{OB}},{\text{ B}} - {\text{A, }}\left( {\begin{array}{*{20}{c}} 6 \\ 3 \\ { - 1} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 4 \\ { - 1} \\ 3 \end{array}} \right)"> <mrow> <mtext>AO</mtext> </mrow> <mo>+</mo> <mrow> <mtext>OB</mtext> </mrow> <mo>,</mo> <mrow> <mtext> B</mtext> </mrow> <mo>−</mo> <mrow> <mtext>A, </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ 4 \\ { - 4} \end{array}} \right)"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} "> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </math></span> is two times the direction vector <strong><em>(M1)</em></strong></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = 2\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 2} \end{array}} \right)"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 2 \\ 4 \\ { - 4} \end{array}} \right)"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\overrightarrow {{\text{AB}}} } \right| = \sqrt {{2^2} + {4^2} + {4^2}} ,{\text{ }}\sqrt {4 + 16 + 16} ,{\text{ }}\sqrt {36} "> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msqrt> <mn>4</mn> <mo>+</mo> <mn>16</mn> <mo>+</mo> <mn>16</mn> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msqrt> <mn>36</mn> </msqrt> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\overrightarrow {{\text{AB}}} } \right| = 6"> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>6</mn> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (vector approach)</strong></p>
<p>valid approach involving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} "> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AC}}} "> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> </math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} \bullet \overrightarrow {{\text{AC}}} ,{\text{ }}\frac{{\overrightarrow {{\text{BA}}} \bullet \overrightarrow {{\text{AC}}} }}{{{\text{AB}} \times {\text{AC}}}}"> <mover> <mrow> <mtext>AB</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mover> <mrow> <mtext>BA</mtext> </mrow> <mo>→</mo> </mover> <mo>∙</mo> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mrow> <mtext>AB</mtext> </mrow> <mo>×</mo> <mrow> <mtext>AC</mtext> </mrow> </mrow> </mfrac> </math></span></p>
<p>finding scalar product and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\overrightarrow {{\text{AC}}} } \right|"> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </math></span> <strong><em>(A1)(A1)</em></strong></p>
<p>scalar product <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(3) + 4(0) - 4(4){\text{ }}( = - 10)"> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mo>−</mo> <mn>10</mn> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\overrightarrow {{\text{AC}}} } \right| = \sqrt {{3^2} + {0^2} + {4^2}} {\text{ }}( = 5)"> <mrow> <mo>|</mo> <mrow> <mover> <mrow> <mtext>AC</mtext> </mrow> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>0</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>5</mn> <mo stretchy="false">)</mo> </math></span></p>
<p>substitution of <strong>their </strong>scalar product and magnitudes into cosine formula <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos {\rm{B\hat AC = }}\frac{{6 + 0 - 16}}{{6\sqrt {{3^2} + {4^2}} }}"> <mi>cos</mi> <mo></mo> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mrow> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mi mathvariant="normal">C</mi> <mo>=</mo> </mrow> </mrow> <mfrac> <mrow> <mn>6</mn> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mn>16</mn> </mrow> <mrow> <mn>6</mn> <msqrt> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,B\hat AC = - \frac{{10}}{{30}}\left( { = - \frac{1}{3}} \right)"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mrow> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mi>C</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>10</mn> </mrow> <mrow> <mn>30</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p> </p>
<p><strong>METHOD 2 (triangle approach)</strong></p>
<p>valid approach involving cosine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos {\rm{B\hat AC = }}\frac{{{\text{A}}{{\text{B}}^2} + {\text{A}}{{\text{C}}^2} - {\text{B}}{{\text{C}}^2}}}{{2 \times {\text{AB}} \times {\text{AC}}}}"> <mi>cos</mi> <mo></mo> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mrow> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mi mathvariant="normal">C</mi> <mo>=</mo> </mrow> </mrow> <mfrac> <mrow> <mrow> <mtext>A</mtext> </mrow> <mrow> <msup> <mrow> <mtext>B</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <mtext>A</mtext> </mrow> <mrow> <msup> <mrow> <mtext>C</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <mtext>B</mtext> </mrow> <mrow> <msup> <mrow> <mtext>C</mtext> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mrow> <mtext>AB</mtext> </mrow> <mo>×</mo> <mrow> <mtext>AC</mtext> </mrow> </mrow> </mfrac> </math></span></p>
<p>finding lengths AC and BC <strong><em>(A1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = 5,{\text{ BC}} = 9"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mrow> <mtext> BC</mtext> </mrow> <mo>=</mo> <mn>9</mn> </math></span></p>
<p>substitution of <strong>their </strong>lengths into cosine formula <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos {\rm{B\hat AC}} = \frac{{{5^2} + {6^2} - {9^2}}}{{2 \times 5 \times 6}}"> <mi>cos</mi> <mo></mo> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mrow> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>5</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>6</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mn>9</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>6</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos {\rm{B\hat AC}} = - \frac{{20}}{{60}}{\text{ }}\left( { = - \frac{1}{3}} \right)"> <mi>cos</mi> <mo></mo> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mrow> <mover> <mi mathvariant="normal">A</mi> <mo stretchy="false">^</mo> </mover> </mrow> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mrow> <mn>60</mn> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Award relevant marks for working seen to find BC in part (c) (if cosine rule used in part (c)).</p>
<p> </p>
<p><strong>METHOD 1 (using cosine rule)</strong></p>
<p>recognizing need to find BC <strong><em>(M1)</em></strong></p>
<p>choosing cosine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{c^2} = {a^2} + {b^2} - 2ab\cos {\text{C}}"> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mi>cos</mi> <mo></mo> <mrow> <mtext>C</mtext> </mrow> </math></span></p>
<p>correct substitution into RHS <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}{{\text{C}}^2} = {(6)^2} + {(5)^2} - 2(6)(5)\left( { - \frac{1}{3}} \right),{\text{ }}36 + 25 + 20"> <mrow> <mtext>B</mtext> </mrow> <mrow> <msup> <mrow> <mtext>C</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>36</mn> <mo>+</mo> <mn>25</mn> <mo>+</mo> <mn>20</mn> </math></span></p>
<p>distance is 9 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p> </p>
<p><strong>METHOD 2 (finding magnitude of </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {BC} "> <mover> <mrow> <mi>B</mi> <mi>C</mi> </mrow> <mo>→</mo> </mover> </math></span><strong>) </strong></p>
<p>recognizing need to find BC <strong><em>(M1)</em></strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OB}}} "> <mover> <mrow> <mrow> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>→</mo> </mover> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OC}}} "> <mover> <mrow> <mrow> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mo>→</mo> </mover> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OB}}} = \left( {\begin{array}{*{20}{c}} 6 \\ 3 \\ { - 1} \end{array}} \right)"> <mover> <mrow> <mrow> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OC}}} = \left( {\begin{array}{*{20}{c}} 7 \\ { - 1} \\ 7 \end{array}} \right),{\text{ }}\overrightarrow {{\rm{BA}}} + \overrightarrow {{\rm{AC}}} "> <mover> <mrow> <mrow> <mi mathvariant="normal">O</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mover> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">A</mi> </mrow> </mrow> <mo>→</mo> </mover> <mo>+</mo> <mover> <mrow> <mrow> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mo>→</mo> </mover> </math></span></p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{BC}}} = \left( {\begin{array}{*{20}{c}} 1 \\ { - 4} \\ 8 \end{array}} \right),{\text{ }}\overrightarrow {{\rm{CB}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \\ { - 8} \end{array}} \right),{\text{ }}\sqrt {{1^2} + {4^2} + {8^2}} = \sqrt {81} "> <mover> <mrow> <mrow> <mi mathvariant="normal">B</mi> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mover> <mrow> <mrow> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msqrt> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>8</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mo>=</mo> <msqrt> <mn>81</mn> </msqrt> </math></span></p>
<p>distance is 9 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p> </p>
<p><strong>METHOD 3 (finding coordinates and using distance formula)</strong></p>
<p>recognizing need to find BC <strong><em>(M1)</em></strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math>attempt to find coordinates of B or C, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}(6,{\text{ }}3,{\text{ }} - 1)"> <mrow> <mtext>B</mtext> </mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>3</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}}(7,{\text{ }} - 1,{\text{ }}7)"> <mrow> <mtext>C</mtext> </mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>7</mn> <mo stretchy="false">)</mo> </math></span></p>
<p>correct substitution into distance formula <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BC}} = \sqrt {{{(6 - 7)}^2} + {{\left( {3 - ( - 1)} \right)}^2} + {{( - 1 - 7)}^2}} ,{\text{ }}\sqrt {{1^2} + {4^2} + {8^2}} = \sqrt {81} "> <mrow> <mtext>BC</mtext> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <msqrt> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>8</mn> <mn>2</mn> </msup> </mrow> </msqrt> <mo>=</mo> <msqrt> <mn>81</mn> </msqrt> </math></span></p>
<p>distance is 9 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A ladder on a fire truck has its base at point B which is 4 metres above the ground. The ladder is extended and its other end rests on a vertical wall at point C, 16 metres above the ground. The horizontal distance between B and C is 9 metres.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle of elevation from B to C.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A second truck arrives whose ladder, when fully extended, is 30 metres long. The base of this ladder is also 4 metres above the ground. For safety reasons, the maximum angle of elevation that the ladder can make is 70º.</p>
<p>Find the maximum height on the wall that can be reached by the ladder on the second truck.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,B = \frac{{12}}{9}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mfrac> <mrow> <mn>12</mn> </mrow> <mn>9</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em><em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 12 seen, <em><strong>(M1)</strong></em> for correct substitution into tan (or equivalent). Accept equivalent methods, such as Pythagoras, to find BC and correct substitution into other trig ratios. If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^{ - 1}}\left( {\frac{{16}}{9}} \right)"> <mrow> <mtext>ta</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>16</mn> </mrow> <mn>9</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> seen award <em><strong>(A0)(M1)(A0)</strong></em>.</p>
<p>53.1° (53.1301…°) <em><strong>(A1) (C3)</strong></em></p>
<p><strong>Note:</strong> If radians are used the answer is 0.927295…; award at most <em><strong>(A1)(M1)(A0)</strong></em>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="30\,{\text{sin}}\,70^\circ + 4">
<mn>30</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<msup>
<mn>70</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mn>4</mn>
</math></span> <em><strong>(M1)</strong></em><em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,70^\circ = \frac{x}{{30}}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<msup>
<mn>70</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>30</mn>
</mrow>
</mfrac>
</math></span> (or equivalent) and <em><strong>(M1)</strong></em> for adding 4.</p>
<p>32.2 (32.1907…) (m) <em><strong>(A1) (C3)</strong></em></p>
<p><strong>Note:</strong> If radians are used the answer is 27.2167…; award at most <em><strong>(M1)(M1)(A0)</strong></em>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{OA}}}\limits^ \to = \left( \begin{gathered} 2 \hfill \\ 1 \hfill \\ 3 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>OA</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to = \left( \begin{gathered} 1 \hfill \\ 3 \hfill \\ 1 \hfill \\ \end{gathered} \right)">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>, where O is the origin. <em>L</em><sub>1</sub> is the line that passes through A and B.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector equation for <em>L</em><sub>1</sub>.</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>The vector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 2 \hfill \\ p \hfill \\ 0 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> is perpendicular to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{AB}}}\limits^ \to ">
<mover>
<mrow>
<mrow>
<mtext>AB</mtext>
</mrow>
</mrow>
<mo stretchy="false">→</mo>
</mover>
</math></span>. Find the value of <em>p</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>any correct equation in the form <em><strong>r</strong> = <strong>a</strong> + t<strong>b</strong></em> (accept any parameter for <em>t</em>)</p>
<p>where <strong><em>a</em></strong> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 2 \hfill \\ 1 \hfill \\ 3 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>, and <strong><em>b</em></strong> is a scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 1 \hfill \\ 3 \hfill \\ 1 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N2</strong></em></p>
<p>eg <em><strong>r</strong> = </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 2 \hfill \\ 1 \hfill \\ 3 \hfill \\ \end{gathered} \right) = t\left( \begin{gathered} 1 \hfill \\ 3 \hfill \\ 1 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>r</strong> = 2<strong>i</strong> + <strong>j</strong> + 3<strong>k</strong> + s</em>(<em><strong>i</strong> + 3<strong>j</strong> + <strong>k</strong></em>)</p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> for the form<em> <strong>a</strong> + t<strong>b</strong>, <strong>A1 </strong></em>for the form<em> L = <strong>a</strong> + t<strong>b</strong></em>, A0 for the form <em><strong>r</strong> = <strong>b</strong> + t<strong>a</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>correct scalar product <em><strong>(A1)</strong></em></p>
<p><em>eg </em> (1 × 2) + (3 × <em>p</em>) + (1 × 0), 2 + 3<em>p</em></p>
<p>evidence of equating <strong>their</strong> scalar product to zero <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em><strong>a•b</strong></em> = 0, 2 + 3p = 0, 3<em>p</em> = −2</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = - \frac{2}{3}">
<mi>p</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1 N3</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>valid attempt to find angle between vectors <em><strong>(M1)</strong></em></p>
<p>correct substitution into numerator and/or angle <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,\theta = \frac{{\left( {1 \times 2} \right) + \left( {3 \times p} \right) + \left( {1 \times 0} \right)}}{{\left| a \right|\left| b \right|}},\,\,{\text{cos}}\,\theta = 0">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>×</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mi>p</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>×</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>|</mo>
<mi>a</mi>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mi>b</mi>
<mo>|</mo>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = - \frac{2}{3}">
<mi>p</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A park in the form of a triangle, ABC, is shown in the following diagram. AB is 79 km and BC is 62 km. Angle A<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\text{B}}\limits^ \wedge ">
<mover>
<mrow>
<mtext>B</mtext>
</mrow>
<mo>∧<!-- ∧ --></mo>
</mover>
</math></span>C is 52°.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the length of side AC in km.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of the park.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(AC<sup>2</sup> =) 62<sup>2</sup> + 79<sup>2</sup> − 2 × 62 × 79 × cos(52°) <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting in the cosine rule formula, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p>63.7 (63.6708…) (km) <em><strong> (A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> × 62 × 79 × sin(52°) <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting in the area of triangle formula, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p>1930 km<sup>2</sup> (1929.83…km<sup>2</sup>) <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The vectors <strong><em>a</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>b</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {k + 3} \\ k \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> are perpendicular to each other.</p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <strong><em>c</em></strong> = <strong><em>a</em></strong> + 2<strong><em>b</em></strong>, find <strong><em>c</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>evidence of scalar product <strong><em>M1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet "> <mo>∙</mo> </math></span> <strong><em>b</em></strong>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4(k + 3) + 2k"> <mn>4</mn> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>k</mi> </math></span></p>
<p>recognizing scalar product must be zero <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet "> <mo>∙</mo> </math></span> <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0,{\text{ }}4k + 12 + 2k = 0"> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>4</mn> <mi>k</mi> <mo>+</mo> <mn>12</mn> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p>correct working (must involve combining terms) <strong><em>(A1)</em></strong></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6k + 12,\,\,\,6k = - 12"> <mn>6</mn> <mi>k</mi> <mo>+</mo> <mn>12</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>6</mn> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mn>12</mn> </math></span></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = - 2"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <strong>their </strong>value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><strong><em>b</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2 + 3} \\ { - 2} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, 2<strong><em>b</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ { - 4} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 2 \\ { - 4} \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} {4 + 2k + 6} \\ {2 + 2k} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>6</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mi>k</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong><em>c</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ { - 2} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two fixed points, A and B, are 40 m apart on horizontal ground. Two straight ropes, AP and BP, are attached to the same point, P, on the base of a hot air balloon which is vertically above the line AB. The length of BP is 30 m and angle BAP is 48°.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Angle APB is acute.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the diagram, draw and label with an <em>x</em> the angle of depression of B from P.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the size of angle APB.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the size of the angle of depression of B from P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img 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RbFJUV48+03KUKbIuAZBJQUPbPU3pvoE/l5zJYmRGXi8xi8Tm1KND0L2Lu7AtdcN0KIgwhFzJZNqliQrEgkn372KSw/6xkyr6kl3pskEmqgomnVuaWiqH2lpLGmIjf/sB85UUNs6sBDEjbyjfZIAibtGmLukpiIBpatclxTrZRHZP+WBVbiYL1G7hGy8Rz3CHfuYNJvMrSbVo59UJ4h4R+f8xPs3btXQlL4jwXZo6RMasAS9uEgNSUDjz/6mIZpCIr64hUElBS9stIem2dpWRneeONdpCS7ibqNBieOKRKOwOg9oFffnkiIixNzZlPiIFwkLDaSGAmlcNcuWI6bKJzyWAFDGjWtIGAnhgnFdpDo68IIeyE8EpHR+nikLGMa5TWSIc+ZRkLkeTZeS0pLCWuKrr4nGh33QyX1m+vtKh6lJE3yMmzEd+mCj//ykfRv5DQlWsq/8sorUbuvXipuMDuOVOOgSRgh2AFb6kLaiYl47PHHzND0qAhEPQJKilG/xN6c4KuLX8GJXRLDHppucu5wDAbENcUKorSoDLnDhwkhGdIyxMjPxqRJwvp74f+6ehfZkA4vNF+GyVD0MdpL2Ui4IQudT0yMECHprqlZluTEZsiW7805Y+pknzzHcfj9ieIVy7hCSe7NB9gpq1/wLvbN7kV7dMeVmhCLbbt2RgjXzMfMk/LPPutscQiyHZ+QKbVOyiWpOr6QEHFKXAJWrlwRqe0oA9cXRSCKEVBSjOLF9erU6BAz93fzEJecIFzBvTYm0pb9Nkk54xJIVk4mfvgvPxSYDGkZE2ZTLZFEsuWTzUiMTxESElNs2NTIQH3u6Ykna5iryFfUPk2jTPINiYik25Rs+ZmtqWbI/kiIPMdn/H4bnWLC+U7JhdRKqdk5Idghn6R5k6QB7J/XaEK1bVSWlgqpNiV6M0/2Ge/3I6t7BmC5GXJIsBKcQe61TekrID4xEY8+qdqiWU89RjcCSorRvb6enN2ri17GD75/ihCEUxfed3PoiOJ6VlLZC4WCOOvsQREtsSlQJA4SF8mJpMSfiooqxMa4MYGMCzRaIk2YNFlSyyIh8Vp9bTUy0jPlOcolKfHHaGnmyGuGhA1h8hzfG0LkvfG+eDTUhwDKt5jJhuOn9ytNuCF3T5FjYryi8L2bSGBvoA6N9fXSB+U21UINGQ8fdj7Kq6pghT1RI7GL4dJXNMsmJSVj1burdG+RIGqLegSUFKN+ib03wbdXvO3GElKb8ocQrAPi423XGSYculBZW40+p7l5SYkQSZCkQRIyhEhyNOSx56svhHRCLPRLzdC2xIGHv0BiSRWd1DWfUts66eQTBXhDgCQ/08w7I9v0w3sMeTYlyU6dOsHX2XXaEdMmvV7D+UutMOmLbKZ+oycqx+I4qKqoR4C6X5P9Sd7Hz4aMe+R0A02kUnUjTPBSScNy09/RtEqzrT/ehzVrV5sp6FERiFoElBSjdmm9ObE169Zhb00NGLdHgiCJ2HZIqtbT7EnNx3YsxPv8yMzMEpBIUoYoDBkZbY3kEQCr3rtamuW3Jb0aNUWmSeO+Xii8vyiamu1IdYyMjPTIApj9Qso0JGn6403UCmlENdfMGNg337PFxcRKSInEV4b3LSV3KcMoSP7ieONmtqGXqqSwcwLYu6cmoilSPsdi+qJsy7bxvVO+72q+9GDlvKglWraYZpnKjtl5kuOS8fLrr0fmpG8UgWhFQEkxWlfWo/N6d+kSpKdluGWc6sJJry0bvng3hIJ/7INWCD869VRByJCTu7Pngma0qAiJhUKorK1wzaPixuoG7nM/j6ZTmjXF8YZkFLKwNxhAakqaCKMWaPYL2ZchOfbBz2w8R+rj0TjX8HrTlhjf1SUumlx9bugFvV+ptDosPOy4GXokNISJyi3A749HaTizjenbmGWNdKdTJ2RkZaLh6wbRLkmwtuXuU0o8JstTMT+s38bHf9qIioqKpsPS94pA1CGgpBh1S+rdCbHs0dbCAglepwmQGg5/qEVxD040OwD1VbU4q+9ZQkKGDA0BGlIyJk2eD4RCCNDzk40en+GSU2I/laoY4VJMPO+z0cWfGFkEQ0IkJfZl5PMGQ75Ge+M5cz/PsRkyi0mUro2RVmIRZV9TYjBcb1g3xpAeNAzCZ0koB8XFJREtmJqimachXY7pzNNPR92evWISFoLn/is9ULmHSS2UY7UcfO+UTKxcvSo8Mj0oAtGJgJJidK6rJ2f15Zf/h71f1YiWxD/ofGOcX6ywp6aYB+EgPTNdzJUkCaOxGa2O50hOhjArSsuQaNtikuTzjBMUUyzJg5nRwo43JJBgTQhpackR/CmzqVm0KekZkmp6zozHeKga4vRZCd+QFD1M6XUqVT4YQ2G7HqicM4ksHMjv8ydhz+7d+5lLOTCOic0QbnZODgJ13AllFCfzuboVOIQQyc7URoNAQlIyPvno48jc9I0iEI0IKClG46p6dE7L3nwbXRMTYYfcChIkMP6Bly0/bpfZQcn4khAbh86JiaI1ESpDPAY28gDJyRDUnr17JDsNaYMOKUK0THoajg8kQQqBUFP0O/DFdxFRJB1DeDxhSIjyeZ7UZIjRnOM9ZjzmyGf3oRE+9s++jNdp2KNWcswxb6mkegv/SoeJem+9WzSYc2EfbOybzZD+voYG5HTLFDI18ZfSB0mWDkXUugmlD1j3wXp5Vl8UgWhFQEkxWlfWg/P65ONPkJyWhFDYvOl6YoYQdBypBCEmQVKDbSEpKSliyjRkZQjMeJ+SREhMQRIgSzRRA5NNRB5chxvxRCX5hrUzkla83xZCJekYYm26HKYfc90QIu9hf2xmz49HyojrFAvEW3ACISmKzLHYQsp0JCJdhqQUlJSB4m819zfpsBo6QZ4XbTUcFmK018jYLAvJGZlhz1WXOmWaRgMWb1uXkOt216CiUvcVm66nvo8uBJQUo2s9PT2bbTu2y96hmPsC5ClH0pVJXJ+YU0NCIPHx3Gz8Jkm2IaIISYRRNORI9mRRprCPTZhsXO/WAJ1caEIlD5GYLAsxCQkiwZAP5RsNkBdMP+ZoSJKfXUr6xkPUjC3khBDkxijztoZDMqixsj96i1LDk+1FkrNodu7eZ2OwNmIepiz2wf6MCdXIP7lzV3FAItlKuEfQNcnyfZBlMzjJkIP4rolYtUL3FcNfET1EIQJKilG4qF6cUlVVFRIT4+UPt4QoUKkz5s6wWZM0yD/wnRM7R4jJEBfJyOz9ET9qaEaD5DZbkIxDQqLDTQCw/eFYxXAgPT1aDaHY+76JdzQanzFVUmbTZojKnLfC2hzJi2PjeY6LjYH1VsCNt2Rf1HglREPCTGx3T5CeoqLXuqTG54xGyvdmTuzHfKYLa7zBLvysYwXdzDnsM5w+jlybkZqOotICeVZfFIFoREBJMRpX1YNzKi4uQiDAisGu1kYNisVzxXmE1k25wvg9C52oOobJwmhp5mhIiGTVlL78LCbssAKFm26NNRO5V8mK9ywq7BYctuDUOfAn+YV0DcnyaLRCym16/ttkzOuGSKmp8rNp5GUmI4iYaqm9kgR9toxDzMNMEm5Kg4RNukYr5NyMvKbjYKB/QpKr3co+qXjUugnT2afJ1iMhID6gYLtbwNiMS4+KQDQhoKQYTavp4bnU1QUk75mYEx1HQhYklIDmQFoU+U2nxsO9uNjYCFLGZMmjIatvnyOjOlQ/6XnKEAWWauIxnFWGJMS9PGl07An/Vhk6M9qoISRznv2QjNkMOfK9IWOOp2mjwwvHL2zMvjm3eOZ0ZbA9azq69RytoHtPOIhE+qB8I43k+O1G7VaceIyNmDcw+4/kQ3W1UvbPvKt795Z++3H9rAhEDQJKilGzlDqRrl06R/bFmjrVuBqdu1fG5NdsTbU3s3doyIvXDVnxHM2tdYGgmEzF89QQB4Pkqa011UqZ4cZx5ZN6SEBGCzX9Gi2NREVi5A/HQLLk/fzMxlczTjkR1vw4BhIY2VcImvuK9B+ikES3Yob8g4Dy4Ii8pmMw/TedL+WT9IxsIWCfLdq2m+HGvSbhJ9aBpCrj0xdFIAoQUFKMgkXUKQB7a/YIDJZoUm66MhM/yFNOKOiaGm2fmBdJPMacyCMJgpxizpsjyYSmRXHkpMmUVTZINuRWn2u+pJbIPtg3E3RTWzXERgIy7w3hGTKkbJIez7PxXkNYvMc8a66TB0UNlcG42Xp88QxNdJOdS4WLMHFyLBxXbHxyhFiNHPZFwjVEac6L2ZXT8rvxnTIvOg/RPBsO5Oezfn9nGa++KALRiICSYjSuqgfnVF5eKaWUxKQp4Ymu9yWhoHOMRYcRSZYdAuobhChIOmykJFIdCaqpmZHnSWgnnnQi9obqIo4t4mTDQP2Aa6D0mbRoJEvuO3JPj8/u2ydHymUz/ZGM2Ng75fO8ISYeeY73mHOGHEUGHX3E8SVcEcPh3qY7Nyqw0jctvZZbVor7nZRHWaYfvjdzk/MNDahCg5vFhl6sNMMyb6z8a8L9E0HZEthPEzJVb22KQJQi4P6zN0onp9PyDgJZmZkyWWpp5CQm/pbQBSkI7H7NWS2Q+22NEnf4DTYkp5gmZGRIBOF9RidQg0SJ62A2G8slQ9uBX+yVrG9IKnW1OEsSkbuOPKxu0cDBNCU9EnETkuJjTQlLyCqcdcYQNe/5OmyydJMEcCaARY1OkpyHM9GQ2sOERjeZIAKwO/mE7M1ceGRz6dolavnHwJ6AkKAgFe/uJ1IcS0qJEZaerk5QEoiHaKvVpghEKQKqKUbpwnptWj6fS0QOvTFJhPwvnJGFRCG1hcOmxT17agQe/mknIRhNjJ9JiDznGPJobES87UdcUoJoTwzpECtmyCcmUxEZ3mM0cYokD8qgLBNiQbJjP0ZLNFqb6d8cTf+GEPlZftDoZuaRIEzqbAzPcPcRRaOTBAJuBh/eyHFRazzp5K6obayV8Zg9RPZl5m7obW/dXljMryrabnj/kCZYH5Gk/Zletj75BwD3V7UpAtGKgJJitK6sx+ZFpY1FdeXvt1SNsCXlGnOB0vWUQfbcj+N+WShUh4bGhgP21EhYhiRIStTgSCCIi0OiL1G8WCWLDN1LfXRJZXiEGzQvZttw4Du1NENuJD82fmbjK+Wy8ZVExWau8xrf82hIjO9F++Xo3FSnbrkoEl+dIcKwmZM30BEXIexrOAGZmRlIiEkQmZwfZVI+58ejIenq6mrXacfNayBj4kaqMZnyBDViv9/HVOHudX1VBKIQASXFKFxUL04pLiEZTl0QUmJXzIriF+qWdBLVzg1XCIUcMX8GQ8EI+TQlLpIEyaKp5kbyYpFdIUA6nIS9NN1UNuGsOWbP0rGwp6omoik2JTmuC/viD+UbcjJam7nOI/sUU2b4GRZ0ZGo5UVOpuIVJ3mFNR94TzsUq+Vl9FuyghbrQXqR3des6Nu2PY+Jn8w8AjuOrL3a7ic5NQnE/EKxzs/hQEaY5WpIiAOia/T3pU18UgWhEQEkxGlfVg3P6wSnfh53oF22OZlPxEg3QE9QlM4v2RDqROA6qa2tRt6dOUHJ1NteMagiMZOHqbyboH+jctYukT+Oeouzj0dmFTi8Mi2CsYFg2hdqxNhobGoR0DAHySPlsph9DToaIDTkasjJExvu+qqkR82mEkNkvPUOZ4Ybxg3RN5X5m0D0vZbOCFrqkdRXyJfGZ/jkWfjb9doqNRWFRkWiYMp+woxD3TOmw49ZtdM219Q21+NEp3WQe+qIIRCMCSorRuKoenFNcQhwcOoCE4/fEtMkE2mEnGNF2Aq45MLGzH6VlJRFtjSREkjCNGhrpiyRiiCM1OQ2hYKOYK+l8abHiPTU0ki37JBFTkXOAhuoq2X/ks2xGsiFGQ8SGnNg3x2D6Ms8Z0yrvo8WSe4QmPpEmTPbtBu+HFUjOl4WHZUPRQnJSEixW9QiPw/TPMfGcIUoWIqZHKesUG6KXuookWUnzFtaOfRa+/OdX6PbDH8m89EURiEYElBSjcVU9OqfBgwa51SHoaEKWCTvaSD5QcjwAACAASURBVBkkU3DYdkCv0O07dgopGDIiZCQrfqaGRiIzpMXPPXp1R3VjwOU/Zq+h1ycJRzIDuJU36IJDEq6rC4HalyEdQ4Lso6nMpiZMOuTwfp4z+34kR37mT21VrRAuSZeaoRQ9FkXVDeKXJQ8727geqg7iO8fDb8VLn7xOImT/lNf0uPXTT9G5c1fJ+kM+5T8oaC4185P4SAqg6TlYjZ8PHybd6YsiEI0IKClG46p6dE5nnHEWyisrgSDLKbkgiLmRG3HcK6tztSgSRWGBm9SaxGMcWkhKRlsjaZjGdz2790DN7r2ypSdhH1LoN6yVhStIyL6e30ZNcK94a/J5Q4IkJJKR0doM8ZnE3CROXud5Dt2855gY7ygaHy2kTBQQcB2GmJwgZGIlwwH3Yv5kxhuGjLAaSNgUSnlN++DYzAxLS0uQlJQgWqhwIv2IGILCf1SQEUmS4eD9bj1PQ6wB1wCkR0UgihBQUoyixfT6VM4bdh6YA5WmTbGk0iTIP+a0dLIwb7ybLJsVIApLihFw3BRoJCEShCEigyM/N9WuTjklxyVbqUThhj1wj5JNtFG+AKgur5KjIUK5Ht7HMxoa72wq25CkIUSSoSHMukAdbMQJ0dEblFlsOC8SJIPsRYNjdadwphuaWWn2TEtNjxAx5XGOTUlXiN+y8OX/fekG/VPTZFFhhmIw9IT5XUn+9DcNY3lGz14yN31RBKIVASXFaF1ZD84rJS0FOVmZCNYxFRlLKLkkRQ8V+aLTOUZi7C10sTujYNeOiAem0eiMyZJmVBKJMakSzsyskyQhtphMqUDxae7Dubwo5MQ9uZo6Nw7SyGxKjnxPuU0JsSkBmuskZCEtAIE6qmwN4lRDrY3zE2WNe4rhMcgeKnOx2m6dx/p99cjKypS+KMfIM/MzYwgGAyguKZOxiwbMjN9Mas7QFSY+9ztSpJlEWVpahqHDh3rwm6VT9hICSopeWm0PzPUn5/wEtQ1VjGCIxNOJc0qTbzpzhcYlx2DHzp0RUyXJiI3aVFOS4vswtQrJSAhE2Dwpnp9NvFrFdEmTpgXUNjRE0CYpGVOlITpDVJTPZj7zSEJmM8dAoA7MIkNNl+ZMEpXrT0Rm9rmEb7nJycUxxgdxiOnVu5eQoSF4I4/aoiHJv/zlI/h8bmHiULjqh/TNWE8SPquKMOWbDQSCdThn4GB3cPqqCEQpAk3+VETpDHVankJg1Oib8WXxV/D53DhFMSkydIJaVXhfjBokf7Z/sl0C+wmQISWTycbsv5EsDal0z+mO2opaUQ35fIgB/DQryr6bm0mHe3idExPx1RfF8hzJx8g3xGuOPG/e80iCZF9mX9NodfW1tWLeJEFJ/lYqjuGE3xHqZBIBGYsje6qJXRORmR7WFMP7mcaZx5A8+9+4aSOSYpPdEBbjUSP5TV2HGyFS2w1jue6GG3Q/UVZTX6IZASXFaF5dD84tNSUVA4f+WPYUJX6P1kCGKjjhShLcA2R+UB/EHPjl7i8lppDkRVIiEbFFNMbwOV7P7paDPTV7YIfzoDK3qpAt9+LIIeFE3bYTj5KyL0XDpEwSntHM+N4QLjU2krE5RxmGsAwRcxzcU4yLSZAxS1gG87oyq06YJJnBxuwFSoULXwjZWdmoD+d4ZX8cB/ti41jYd10ggLKS3QBNpAxdIS7h/UNJZhdOmUeLak0wgEsvvlCe1xdFIJoRUFKM5tX16NyuuvpalJWXimZITU7CM8L7fvyjz0YCS09Lw9rVaxETDp8wRESSMgTC9yQUkkhsTCx69e7tJgAPVxIWfZTv6aDCIHorhOSUGJSVlLtaKE2PYScbQ4yUR3oyBGj6knE1IWeOh9f2Mler7RIX9xLpGUoS43vRDs1vse0mK3fqHNlbden9mxhMzoWNc2HfBYUFgK9RtiMllyoda6ygJE03ssU2G3KQnpKK7t17yPP6oghEMwLm1yma56hz8xgCuUOGyj4Yp809NmpAkqKsSUyhqy3aKCjYFUHHeHuSiEhgbOY9SYRk1qNnD3FucZyQ7LPJSzg+UPoLSd0mMME24yFpTiW5UR6f53s2Q1iGqHjNkCavkzSN+bSsskKScZMMqfwxRZ3s8/HGsMmTzj7GY7RhXyOyc3KkH76Y/s1ceI59/enPHyK2Uxyo+IoF2BQvphJKb1bGYjoWSitKcdcd90Tk6RtFIJoRUFKM5tX18NymTJuGiqoq0bAIA2s9+CQ7i8/12CThWQ7Kyivxj3/8736ERMKgNsWjeU/y4vs+p/ZAdXW9Sxg0Nhp1j31wr4/7jH6gurJW0CchkdyMPN5utE/eQKIyxOgaN10SI2nyGbbA3r1iOg0G3BAMi041FjVH1/NVQieY8s1yaznu/upL9OzVO+Lcw/5JuqZffq6srMKWDZukeojstVpuWSwxy7JTU1PRCSGzayaGDFEHG1kMfYl6BJQUo36JvTnBn56TKywl+3w+wA75EGQmGjvkpkEjOTkWUpKS8MZbyyMEYojDcB3JhCRC8mI7sWuamFFpLiVxsCCvvKe8cCwf5QZq9ggRGUIyRxIrCc9ob+zPaHCmD3OvGUNFRTmkJBZj8fkbK4RFRnVjE6XYMff/fBCNMiMrBwlxcZExc+Tsz8jnPDZsWIe0zDTXe5WxjkwAFPbYFWceKT8VQklpBW66+0Y3HlLu0hdFILoRUFKM7vX17OxiEyxMmTwFpcXFosFRsxJNimZB7vOFA90tn43igiLsrayIaIskKqMhkkyMaZPHhOQUnHRS13A6OTKh5aZfC4cvGLLcE6iJmEq5CEYLpPZHOYbwmhKVIUpe5xjMc8Wlu2HR9EvNMGi8TC0xe5LI4uNtSKhIiGQcwNln9N5v3dkHiZaN8+JYPvrLJ/D5/ELkkvRbnIQcMFxFgv8tS/4hkZGRhguHX7yfPP2gCEQzAkqK0by6Hp/bsOHDkdM9x9XgEIJTZ9FfJRJGYYjRtm188sknEfIjiRjiIpmQrIz5c19DA07rfTpqqxtEUyOhsDnGAzQcyV9bWQ8rFIo8R5kkOpIhicmQbVMipBwSFq8bsvTt+xpfflkkgfRiJvWHk5CHzZu2HZJakVY4XpIFhXv27hWRY+RHxt/YiB1bP0NJWbFgQRMyn3VxcbPZyH5lyEJhSREeeOB+j3+LdPpeQ0BJ0Wsr7rH5TntwOgp2FkioAdO/sYqEmwaN1lU3AwwrbLz35ttCWoY8CJNLd66XKMmR19j+X+5PUReqc4kqTEZgAHw4NIOaaDAUwO6aGtHQjOZHoiPh8bNx6iFJ8jMb++A97ic5hd1798DvjwfJj6ZaK2i75lrbQqCOibtt+HzhNGwhC3Gd4pCZmSVyKNeMm0fTz4o1K5GRkgGH5afqGJ5iIxAK106kX5IVQnV9PQaffRb69x/gDkRfFQGPIKCk6JGF9uo0e5zaA9fccAVqa2sRZBA/fWEc291ftILyniZPx/bhL+vXR0iM9zUlEn42Wh5D/3OycmjLFIKSfctwkV/eR8eV+MR4FOzatZ+myGuG8AxBknipQfLHkFbTPceS0hIk+hLDBYbdrDXizxpywHqHZl/T4p5iyEHXk7vIfiL7omxDjDzy8xdffoHS4lKGOLqJvul56jhgfGOI2jTNy7Cxt24PHsrPoxhtioCnEFBS9NRye3Oyd9x2F4J7a8JhDG4ib5II9wOD4aD+1JRkrFm7NmJ2NCTVlLDMOWp7mZkZrpMKK0iQTCxbQiKEIOMdxMbFYseOHUKkJCQ+y0YyNO+peZJ4eYXn2UwfhpC/+vJL+JL8bsq6cFFjSS/HyA869vCHD9pA7b569Ot7lhAhCZAy2AwR8/OyN99GanICfBbzptJ7lhVEHLfAMPcVfUBFSSluHjkKTISgTRHwGgJKil5bcQ/ONzklBTOefQolZSWkL9cxhoHvrgOnaEZBJ4jSikrs2LJFSMWQF4+GsFyKcQHMHZaL8vIyMcX6LB+sAOCPp9CQ7C/S27WyqlzqKvIJI8PsF/JoiM8sCQmR9/GaMeNWVVcj1hcr9RMZTyiDjw9X/2CJLIaZcEMwBNTU1OEn4dAJI7upzJKSYhQXFUKS14RF0ZQsG4rxbo5TarnZPXrg5ptHm2HpURHwFAJKip5abu9OdkC//hg0aDCqq+rFXMhwBjqU8CcQcIPhM1JS8MbyFWJmNORFUqHW1ZSoSFjpaelISUsUD04GzYf8ITfTDTNnc38RQZSUVErwPp81Mqi1GW3QaJA8mv64QrzH7DXWMJsNTaV+n5AZNVyJKwxnn+H91E5pAs7JyYIVEyuLbLRDQ7LsY/WadUiISwIdi+hgI6nvRGP2uQnHbQe7CguQnzfdu18UnbnnEVBS9PxXwDsAzJo5E3E0G0ribFazCMnemltnwhatj9rdtm37e6KSoNgMURkt7MdnDRRCFS9WFjBmonF6c7KYhW0jNiYkRY9JSCYZN0mqqcZJuSRM3sMjG6+TJPlTXl7q7lvyqozbNZmG6NQjQfxUHkOor6pF7959I5pmU82UJP5/xf+HDR+sF/OoY4m+LH1x7JbPjd0sKy4DMcrKynIHoq+KgAcRUFL04KJ7dcr0Cp01bw4KCgrdAHjxNgFCPgbhByV+MSUlFUuXLBUzK4mKzWh2PLLxLAnrzL6no6q+SjQ3uzM9Tl1yIdmSHOurgZo9e+WZpkRIuUYWyYsky2aOJDHew352765zvVrJgRwvc6ySfFlCivbfEHO7WthdsxuDBg0SOUZLNOOm3GUr3kFGVoaYdmnu5XMSpM9UbkGgsrYKA4cMxPDhw0WGvigCXkVASdGrK+/Reffs0ROP5uWjvKxUEnYLsVALZF5R7q8xp6ll44P1H0RMnkbrMhocP5N4sv81B53tRBdJVrsPp1mTpOMBIC4hFkVFBfuRalNtkA8aLZEka/YRjZcr+ystLZKCv+IVyj3FoCNkSEIjR4qWaoXQ94y+iI9PjJhhKY/j5PGvn32Gndt3uulwGDkSDIm2zP5Fq7WAlKR4PPbQTI9+K3TaisA3CCgpfoOFvvMIAheffz5yzxuOqopyN2aRKeBko841STJu8a3X30AowGSmbjPkZbQwQ5T9hvQTMuXjxhNU9uv8kDCHwpKSCDnxGUN4TQnQaHS8RvkkMmqKRUVF8MUngmnchATpKcsixjSjGnJzLDQ2NqJ/n/6yf2m0WzPOhsYGvPrGW8jITBUzq5AotU3uTYqiGURleTmeeeY5MAuQNkXA6wjob4HXvwEenf/UKVORkZmJ+sZ6hFguSdSusANKEOhyUhe8/PqrQlKEiCZIo32RxAxJDhkwGJVV1bLhSBkkLeHXsPb52aZPIyRHOZRh5JHAKMc945pp+d4QG0s7Jcb7XKcYsMIhE4FLIh0gEBLnG5pB6SjU+6y+QoqU13ScH374F1RXVokzDh1yxCmHplNx1AmhpLgEefkzdB9RVkVfFIHw75cCoQh4DQFqXi+++BICdQEJWJf5kyyY8szvoNPXMdi6eTP2lJUJyZh9Pt5Hjc/s/6WmpiEjI4VnXVNmOEE4iZFmyt01X6GyuipiiiXhUYujPDbK4TkSLeWa9yQ27n3GxsSJQ42rGrqFko23KbVbkuRpPXtKID+fMc5AlFdXWYHXlr6GhKQEIVaTl5UOsrS9VpVW4r4H7sfgwe5epAxIXxQBjyOgmqLHvwBenj7NpEuWLkUFCxKHNxfp1UnCJIGkpaZh9qy5QlQkL6OFkXyM+ZMk1qvvGWiorxWiMcQjRwBdu5yIgu07DiA/o3ny+abaotHyuC61VZUyFu5VSgtrodQOKZ9DJuEOO/fcCHGTDCmDGueKVSuRktTZNesy+Q7tpTS/wkFZWSkuuOwCXH35lV7+CujcFYEDEFBSPAASPeElBLKzsvDsCy+guLDY9exk9fpQyPUoZZaYYDX++Kc/iXZn9vxIZEYjI1bnDR+Ompqg7C36eIGNZaVsR1KuFRQXhU9+E8RPGWbfz2iLJDOeJ7E1NjRgT11NZO9Q9gKZGIAmUNIoydv2IT05FSd/7+QIcZu9zs+2fop1azfAsjtJtQshQzoShUKoLa/GmQMHYPz4CZFx6RtFQBFwEVBS1G+C5xE4s8/peOqpp1BSXOru2YVNqNzFS4pLEm1yb3l5ZK+O5MVmNMfYmFj06t1TSDAQDMEOM6Nkm7EslBQVYd++fZH7zXOGwAw5GpMqpQeCQdTtYekNJmsFQnzLMAy/I/KpzVZVVqPv2X2lviPHZLTX+vpavL58OTIy06UIssPCxyRUhonsa0Bmj254ZtZTrmx9VQQUgf0QUFLcDw794FUEhgwZgilTJkkSb2IgYRVhh5T0lBQsfW1ZZB/QkJjR8Hh//379UFVRBZ/ti4Q8MIUa9/wKi0rRqVMn0S75rHmuKTlSO6SSySO1xbpAnSQxl2QAHI9kySFbkt/c/ctgKIjzh58v9zfVXpcvX4Ha2ioJt2BKNw6C86mtrUdSQjKef/oZry6zzlsR+E4ElBS/EyK9wSsIXHDBBcjPz0dxSZEQG02WDIewO/uxc+d2fPjhnwUKanKGvMzxjL59JX1aKOSmjOONNoPswXRqQez6267IXp/RCJuSo5FptMfCXQVITIyXkBE67Zh9RCQykbmDUMDB6f36yHv2xXGw/eOLf2Ljhk2RShncR2QxjYrqSqSnpeGVl14G91K1KQKKwMERUFI8OC561qMInH/++Zg8ZSrKystk/06cXEIOUpPT8eabb6Iu4Jo0DXnxaMypQ4YOkfJUEt0hdS2kDpNUm/jgw40RjY5kaPYPCbMhR5fWXILbsXMXYjrFSMJv3kNzqRUMJwL3QZKbD88dJnIoi+NgTcT58xchJYWkZ0spKMRbqCyvRnZGFl5askQJ0aPfa532kSOgpHjkWOmdHkHgkosuwr333IuCwkLJUsNp0xTK/cUnZj8hTjBGMzN7ebynX7/+qK2ulfAHamgSC0j7KYCykkIhMFozaeo02qLRNM2R95IcS8qKZe+Qj9usdch8peFCxhzLWWecJcWEm8p6d/kKfN1Y65I5s9843HesQlp6Cl5YvAixVBm1KQKKwGERUFI8LDx60asIXH7l5Xj26Wew42+fiyeqz+cIOVWWVeHDDz8U4jJEZjS/k088EWkZGbKXx9AO2ftjGASAiopa1FZXSWJwEqmhJ6NpGs2TpMksNMWFTATuiBk3FHILIxsTanVFFX4+5OdCrhwDZW3bvhMbN65DbFyC1MSi2beqtgrZmVl46cWXlRC9+kXWeTcbASXFZkOmD3gFAQa1v7h4CYqKi6SeIQPeU9JSsGjxYvzP3/8WyURDrY8aG8nugvOGoryqCvDTgkli5KYeJElAYamb8o33GicbHo3jDnElwe7+6iskJvrcsBCaTS1HUru5iUpDSEpMwfe7/0D6o1ZJs+nixQuRkpoqmW7Yd1VlJc4880wsXLRITaZe+cLqPFsEASXFFoFRhUQrAgP698crr76C6opy1O+rFweXjPQMvPzSy6hvqJdpk8iMWbR37zPEycUK2m6CcTF5AslpSfjzHz8UIuNDZh+RRxIbn2cjQZYWlUQ8TBneIY429CAN2qiqrcYV114Reb6xvh7z581FXAxZGIDflueHDR+Gp2bOchMRuFf0VRFQBI4AASXFIwBJb/E2AqedehrefP991NUF0dDIPKcWamsbsOi5xREvVJIbG7XF3r16I+KFSjMqQw0dG59u3Sr3GLOrOfKkCcegGbWgpARJSQkIhVPFsT8xncY7ElLRo0f3SEziHz/8C4qKS2HHdxJP2eLiYowZOwaTJ0+WvvRFEVAEmoeAkmLz8NK7PYoAwxl+/957SMvIRFVFJeJiY7Fz13Z8+MH6CDEakrvgogtQVlnupn1j0WFJrRaSkI2ior9HzK5mH9EE3Zv9xZJwBhybBYDDYR3MV1pWUYlhw3IlWJ8k/Ne/fY7XlixFSlyCOAKVlBRLkeCRI0d5dJV02orAsSOgpHjsGKoEjyCQlJSEFxcuRO6wXJSUliA5OQVLl76G0v/9u2iIhtTifX70O/Ms8RiV6H3uKzoWfPE2/va3/404yJBE2aglmn1Gft62bRtM0m43G42bei4jJQ0/Peff5Jnq6mr87vkXkZGRjvqvG1BXWY/Xly0DkxBoUwQUgaNHQEnx6LHTJz2IAMMaaJq8576xKCotEueWZ+e/gMrqaiFG43Dzk38bgOrKahchmj+5j3hCLD7+2DWhkg6NpkiTK3+o/f39f/4HPn+87CMyd6oVrnvIUI/zzj8PISck3qlz58yT+6vKS5GSnoE33lmGH/Xo4cEV0SkrAi2LwAlff/311y0rUqUpAt5A4K87d+KmEaNg+33w+xOQ/a/Z6CQkFsQJ+2z895/Wolv37uHk3awAZaFgVyEGDRrEisTYt68enTrF4euvGxA6IRb2Pgf/90UJQqEALL8t2iX3EpkAfFdBIX72b7k4oZOFqupylJaVYe/er3DFJSMxbvwd6lDjja+czrINEFBSbAOQtYvoRaCstAz/fvEvkZWaxogNUCd0xPYJiaCQWEMpR+WmfLMcS/b/6DzDWPpQnSPhG6JLhks7OQG3wga1RCYD55FhGaaxUkZpSTmuHnEl7rrjDnNaj4qAItACCKj5tAVAVBHeRSA9Ix1Pz5yFgvIScYoJMXTCcqR2oWShCZIcyZbuvmJIkpG63qQM8LfjLddUGmTMvXue1SwkITlhZakonoaFIHVGZqmprcWZA3sqIXr3a6czb0UElBRbEVwV7Q0EBgwciHtuGyvON25VC1u0RokvJAQ2q1Qwd7dLjKL02S5ZMmVbpIgwyzuBpGoLh/I+0RAdt/QTibKxsR6dLODxR2d5A1ydpSLQxgiYbFNt3K12pwhEFwIjRo5CUXEJ1qxdh6RkVzH00T7KrDZBsYtKiD5JztUIZVsRYAUMnyNHkp4YSemt6qOSaMEJMPcpwgH8Dvbs3oO333kb9ITVpggoAi2PgO4ptjymKtGjCNTX1uOK665BY1UtYuJiv9kPJB0aT1LHcfcaw0Qnyb7NHiI1Q4lLdBOQM1bDYk3GcFLxopIiPDXrWQwZMtCjCOu0FYHWR0BJsfUx1h48hACJ8Wfn/hwpyfEMwpDcp/u+3oeY2E5obGxEDGMTnUY07ouFs68BdfUBJMbHIyaGGmUsLP83DjWNDeHnGvahrmYvbr55NEaMGOEhNHWqikDbI6Dm07bHXHuMYgRYwHfktaNw3gXDkZGeLqES3EuU/cTwvFkb0Zz7/cr3ceFFFyMUcjPe8HiwNm1aHq669tqDXdJzioAi0IIIqKNNC4KpohQBInDVNZfh3eXvRmIHSYK2bUd+zOevLQvY52qGvM7W9D7zvrKyEr179dDyT/r1UgTaAAElxTYAWbvwFgJ0gvH5fftphwdDgNpiTGzCd963fds29Os/4GAi9JwioAi0MAJKii0MqIpTBIhAt5wc/PWzzw8LRoxloRG1EY3yUDdv/HgLsrOzDnVZzysCikALIqCk2IJgqihFwCBwzqBBeGP5W+bjQY/7eLY+9qDXzMldO3ch+5RM81GPioAi0MoIKCm2MsAq3psIMHE4tcWKyspDAkDzqRUr1HjIez7YsB6XXHr5Ia/rBUVAEWhZBJQUWxZPlaYIRBAYMHAQtm37OPL5YG86hStoHOwaz5E4SbDaFAFFoG0Q0N+2tsFZe/EgAt265WDenNnYvm3HIWbv4PPPPkdh4T8Oer2stBSDhgw66DU9qQgoAq2DgJJi6+CqUhUBQaCopATDLrgggoYdCkoVDaZwY1mo7j17yTWeh88PJxSS9G7MZLN9xw5kZeVEntU3ioAi0PoIKCm2Psbag5cRCIWQlpICFh9mYzFhtqafTWFic5QbAPhPOCESx2jO6VERUARaFwHdU2xdfFW6xxFgFYwlixfhiy+/ECRIfPwJhEL4j1X/iYXPL0BhcWEEJV6rr6/FH/7we2za9jGcEyKX9I0ioAi0AQKqKbYByNqFdxGIj4/HZddcg1UrV2Hv3j1IT81AdW0tairLcN4vL0Ra+nD8/g+/x3//55+QnfU9VNbWoKKiHL/8919i9+69sA7vnOpdYHXmikArIaCk2ErAqlhFgAiI92hMLM4fPlxMpw2NDYiNcWMTaVClZnjuL84VsHgtITYBPELypR48D6oiqwgoAq2HgJJi62GrkhUBBOpCqK+ths/2CRo2LCFClhPmex4R3m+Uz1JQ2BEC3VtXB8ToDod+jRSBtkRASbEt0da+PIeAL97CG8vewNmnn42cnt0RGxMjpFhTU4P169ehuKQMA/v3R99+/eCQHBsbsbumDps3bcBft36KeFYY1qYIKAJthoCSYptBrR15EYGMrGycN/w87CrchZdffgk+2w/n6xA6J3VF7pBcJKUk49PPPsNLzy9A54REBBCEz4rH8GFD8NFHHyG7W7YXYdM5KwLHDQEtMnzcoNeOvYBAcXExnsx7HNffNFKma1u2azJ1HNEYGaLBfUUTqmEwqaqswAcbN2HSxInmlB4VAUWgDRDQDYs2AFm78C4CWVlZOOlfs7Fx0yYhv4aGenGiaQgToSFEHnmOrbqyCvMXLMSoUaO8C5zOXBE4TgiopnicgNduvYXAlClTYNk2hg4dioS4OCFIIuDExMBqbIwc//7Pf+C9d97D9GnTQELVpggoAm2LgJJi2+KtvXkYgQ0fbMT8BXMRl5CMoUOH4MSuJ4LB/bYDbNi0AZ98tA25uUNxzXXXgIWKtSkCikDbI6Ck2PaYa48eR6CooAgFBTux+ZPteP21JfD5bGTndEfvfj0RayW4IRpMBxc2p3ocLp1+GyBwy+jR+g+xMM7qfdoGXzjtQhFoigA9Sk86+SQsXbZUgvvLSnbDRgFKCgtgWZac4/0mdJ+/pNHyvq6mBp9s24Yf//hsWJaNaJpbR57LVVdfraQY/iVVTbHpXyt9rwi0EQKz58zG79//PX502qnYDG6g9QAAIABJREFUsnkzFixYiO7dugkhCjECiEYvuC1bNuGnP/kp/vmPr5CcltBGaGs3isCRIxCNv3dHPnu9UxE4Dgi8++4KvPbaa8ib/gQee/gRdM/phnF33o2KygrRFDmkaP3FtGDLPmrIHzgOyGuXisB3IxCtv3vfPXO9QxE4Dgh8vmMH8mfk4bYxY9Dn9J6IjYvD9Px81NZWgx6qDSFjKD0Og2uDLnfvrUHIByRaKW3Qm3ahCDQfASXF5mOmTygCR4UANcH7J05A7qBhuP7qayNaYXpaGvLzZ0kGm+efW3BUsjvKQ7XVlehsxyM2Qf/0dJQ189o49ZvptRXX+R4XBBqcejz84KPw+eJxz/33HDCGM/v1wejRN2PRgvlYs3rNAdej5cRXX+1BRqbGX0bLekbjPNT7NBpXVefU7hB4aeHLoJPJ0teWICnh4A4mI64ficKCIkydPhU9evZEVlZmu5vHsQ6orGw3srOjb17Hios+334QUE2x/ayFjiRKEVi3Zi2em/8c8vLykPkdWtJ9996H7Kxs3H33naisrog6RP75fwX4QU63qJuXTih6EFBSjJ611Jm0QwSKioswLW8aRo4cgYGDBn7nCOMSXMebyooKPPbwE995f0e74bPPPkXOD37Q0Yat4/UQAkqKHlpsnWrbIlBdXY3x48bj9NP748abRx9x51kZGZiWNx1r163B/Pnzj/i5jnDjR3/Zjh49uneEoeoYPYqAkqJHF16n3boIMLTi8SceRyAYxG/vvxexdvO27wf0H4CbR9+GBQsWYO369a072DaSXlRUjJqa3Tj1tD5t1KN2owg0HwElxeZjpk8oAt+JwFuvvYk1a9ZgxvQZSE1J/c77D3bDzaNGIndYLqZPmYLi4pKD3dKhzu0q2IGuXU5Cly5dOtS4dbDeQkDTvHlrvXW2bYDAho0bMe7O2/Hb++/HL3954X49btq8GWvWrMbyN95FyAkiO9sNT7B9Ns4+6xxcffXV+3md1tfW46rrroHf58fzz/+uVfJTNtQ62F1Tjrq6Opx04omRPmj+/d/Cf+Bfu/0LuNd5rC3v8SfwxpLF+NPmzc3WnI+1b31eEThSBFRTPFKk9D5F4AgQYID+Iw9NwQWXXHYAIfLx/v36YeKEiejVuztIhK+89CbefvsdzJg+C6tXrcTIEdejtPQbrZBk9OzTz6C0rASPPvrwEYyg+bcsX7EUvzj3Fzj1hz/EQw8/JAJo6vzxOeegT9/TMPfZZ5sv9CBP7PxsK3r27quEeBBs9FT7QUBJsf2shY6kgyNAzWrcuHH4l3/phjvGjD3sbBLiU6RChBVWwLK7ZWHEzaOwe88ezJkzb79nMzIzMG1qvgT1L1y4eL9rLfHhyiuvxYP3P4BgKIQ5c+airLgc856bjd/e/1vcNfYODBk2tCW6wX/96U8YMLB/i8hSIYpAayHQvN3/1hqFylUEOjgCdKx5bv58lJaUYtHCRd9pbtyHxgNmfPrpZ0qVjJKS4gOuDR48AKPHjMG8eXPQq3dP0TgPuOkYTgweMhiJiYmoqanBr++8Cc88/TukZ6RLOrpjEBt5tKCwEMWFRbjkkssi5/SNItAeEVBNsT2uio6pwyHwh5Ur8cbytzBt2uNCJt85gZAjtziOe+SHsrJSyYeanBx/0MdHjhgJkteEceNQVl5+0HuO9mRaapo49fB5n7/Lkc2hGZ0tW/oaevXqiYz09GY8pbcqAm2PgJJi22OuPUYZApu3fIIZM2dg7Jix6Nev95HNzuavni9yL02vz819Tj5fd93IyPmmb1hn8cEHHkRySjLuvP32Fq+ocd7550t3f/5gbdNuW+T9nz/8EwYPaRkzbIsMSIUoAodAQEnxEMDoaUXgSBBg5plHH5qMoYNzxXP0SJ4x99D79Jk5szBu/DhceunFiE/0S1mp/v0Pve+WlJSEGTNnoaS0BI9NzzOijvlIUv5402b4/H4U/eOfWPneSpHJjDzH2ioqK7Fu7QYMy809VlH6vCLQ6gjonmKrQ6wdRCsCDJe4/+GH4I+Pxz33HVj54rDzDjmwLR9+PfZOxPm+0RgP+0z4YrecHEwady8mTnkIPXt0w+VXXn0kjx1wz/p167BqzRoMGTIYWzZtxWP5+fh8+3b89x//iIkPTUbDCQ0IVAWQfW32Ac825wTjNS0riJ//7OfNeUzvVQSOCwJKiscFdu20oyPAvcAXX34Jf/v0MyxavOiQlS8OOU8xnwYPefm7Lgw7/5fYVfQPPDlnDrp174Ezzzzzux454Pr8ec/hlddfwcpVq/D+O29LfOKkyVPw53N/gW0ff4z/ev8/8ficWQc819wTL7/8EkaMHPWdzkfNlav3KwKtgYAG77cGqioz6hFYs3o1Jk+Zgvy8fAwaPKjZ871tzG345KNt+M/1a5utKZrO6PF699hx2FXwORYtXtxsJxbGVO7Yvh1nnXH2foS1a+cu1NbXok+fPpFCyKbP5h7pcfqvp/4I77//e+QOHdLcx/V+RaDNEdA9xTaHXDvs6AgwvIBloEaMHHlUhCjzF+/TIEINDUcNB/OpTp3+ABIS4jB50mTQnNucxvRzgwYN3o8Q+Xz3Ht1x+umnHzMhUtaKlSuQcXJX/PjHZzdnaHqvInDcEFBSPG7Qa8cdEQE6pEycMAF9zjwdo2++udlTKK8ox8aNG/Hxx9sQCIawctVqlJYcfXgFie3RvHxs2/4Jnpz7ZLPH05oPUJOdM38OfjViTPPNy605MJWtCBwGATWfHgYcvaQINEWAf+QfnDIJO7buwAuLXziqRN9VVVXYvfsrWLYNn+2DEwrBF28jLTWjaVfNfr9m1SqMnzhRQjYuvHj/fKvNFtZCD7z99ru44YZr8be//w8y0jQ+sYVgVTGtjIA62rQywCo+ehB4Y+kSbPhgAxbOX3xUhEgkkpOT5aelUckdPhw3fP4ZHnviMXTv1g2n9jmtpbtotrwXXpyPSy64QAmx2cjpA8cTATWfHk/0te8OgwArX8ydNx/jx49Htx7HFqLQWpO+7fY70atHL9w5YRxopj2ebcuWLVi1Yi3G3nXX8RyG9q0INBsBNZ82GzJ9wGsIFJeW4uaRIzF8WC7uvHtcu54+PUpvuOZ6fP8H2Zg1Z85xqUhBM/NFF/4SneMT8dqyZe0aLx2cIvBtBFRT/DYi+lkRaIIAPTqnjJ+InJwcUBNr742ON/kzH8fHH32CZ+fOPS7D3bZlK9auXoOx96qWeFwWQDs9JgSUFI8JPn042hGYM/tJlFSWYcrUqcdF6zoafHv17IXxE8djyeJFWLnSTdd2NHKO5hlqiUxbd8U1l2FQ/4FHI0KfUQSOKwLqaHNc4dfO2zMCby1fjuUrV+DZpxcgPS2tPQ/1gLFdctFF+PzzTzFt+jSJO+zerfsB97TGiXfffBMfffQRnpu/tTXEq0xFoNUR0D3FVodYO+iICGzdvg233zYao2+7DVdfeW2LBLK3NQ4NtQ5uunUEKisrsGTxEiSnpLTqEKpra3FG3744//wLMGvWzFbtS4UrAq2FgJJiayGrcjssAvTcvOXGW3DGWWdh8uRJHXYeHDjrLo64+npJtfb0U8eex/RwYNw97m6sePNtfPLZ5wdkyTncc3pNEWhPCOieYntaDR3LcUeAjjWTHpyMzp0Tce89zax8cdxHf+AAaPadlp+PLZs2Yubs2Qfe0EJnNm/airlz5+HRGTOVEFsIUxVzfBBQUjw+uGuv7RQBVr4o+HwnpudNj5o/7v3O7IO77rkLLy9ahPXr17U48kx9d9ONN+Cqa66SupAt3oEKVATaEAElxTYEW7tqvwiwFBQ9NZcsXIRpedORmZnVfgd7FCO79NLLcdFFl2DSxMnYVVBwFBIO/ghxe+zxR1FWWY78aS1X9PjgvelZRaD1EdA9xdbHWHvoAAgUFBRIgu+rr78Wo0aO6gAjbv4QaRq+8dZfIRhw8OILL0j9xOZL2f+JZW+/ieuuuQ7vvfcehuXm7n9RPykCHRABJcUOuGg65JZFgOa/X914I7pn90DejGktK7ydSSstK8P1V1+Nfv36IS8//5hG9/mOHfhZ7lCMvH6kmJuPSZg+rAi0EwSUFNvJQugwjg8CDDa/f9JEFO4oxPMvP4uUpNTjM5A27HX9hg8w7s67MWrUSIwefdtR9cx0cj/7+S8kfvP377/fIUNWjmri+lDUI6B7ilG/xDrBwyHw6uIl2LRxI/Jn5XuCEInF4IGDMGbsGCxYsBAffPDB4eA56LX6YBA33XArgnU1eOmll5QQD4qSnuyoCCgpdtSV03EfMwIfrP8A85+bj4kTJqJbTs4xy+tIAq6/9noMGZaLiZMmoqCwqFlDv3/iBKz7YA3ee+e9Dpfpp1kT1Zs9iYCaTz257DrpkpJijBp1c4eofNFaq0XHm1/d+CvYloXnn3/hO0NQaGp+dNo0PPHEE3j/vfcxeMjg1hqaylUEjhsCSorHDXrt+HghUFtdi5t+fSu6JqZi5pwZHSbRd2vgVVZahsuvvhwD+w/A9Px8HMp0REJ8Yf5zuPvOcVj29ms4//zzW2M4KlMROO4IHOp34LgPTAegCLQWAk8++QQqy8sxZdr9niZE4puekY68vHysXrMGCxcuOCTkv3tqHu6883Y8+9KzSoiHREkvRAMCSorRsIo6hyNG4K23lmH16jXIn52HtNSOVfniiCfZzBsHDhiAsWPvwPx587F5y+YDnp40aRLG3T8ev3vpBcmjesANekIRiCIElBSjaDF1KodH4NOdW5GfNxc3jxmNPj1OP/zNHrs6YsT1OOfsQbj77rtRWlois+ee42/u/A1mz56Nt95+VQnRY98Jr05X9xS9uvIemzcrX9x4wwic/ZOBmDyxY1e+aK2lYxKDG264DonxXTDzqRm48Vc3SsjG79/7AwYM6t9a3apcRaBdIaCk2K6WQwfTGghQ46GDSCBYh6eefqpF0pu1xjjbg8xdBbswatQoBPbWoSZQI2EX3bq3TYHi9jB/HYMioOZT/Q5EPQLzX/gddhXsQN6MfCXE71jt7t26Y+qUqYBtYcZjM6GE+B2A6eWoQ0BJMeqWVCfUFAFWvnj9tdcwdeo0DTRvCsxh3g8ZMgR33XUvpj46tUUrahymS72kCLQbBJQU281S6EBaGoGCwkLk5edh1MiRGDhoYEuLj2p5l1x+MXKHDMVdd4wF9xq1KQJeQUD3FL2y0h6bZ1VlJW685RZ0y+mB/PzornzRWksrGW9uuRGZqRmY8dTMQwb2t1b/KlcROB4IqKZ4PFDXPlsVgYZaB48+8RjgOLj/t/e0al/RLDwuIQ4z8vKxeetmzJ03DyworE0RiHYElBSjfYU9OL9X3liIDR9swMyZM5GckuJBBFpuyllZWZg6bQqWLlyE9R+sbznBKkkRaKcIKCm204XRYR0dAuvXrse8ufMxddJUZGdnH50QfWo/BIYMHorRY0dj8sTJ6nizHzL6IRoR0D3FaFxVj86ppLQEI0eMwIW/vBBjxo71KAqtM20pxjxuPApLivHiCy9oaEvrwKxS2wECSortYBF0CMeOQFVVFW699dc48aSumP3kbC18e+yQHiChob4e19xwA7pn5eCh/OmeT6Z+AEB6IioQUPNpVCyjtydBLWb+vLmorKzApEmTlRBb6esQGxeH/Px8bNi0ES8vXtRKvahYReD4IqCkeHzx195bAIHlb76JFStWYdbs2Rqg3wJ4Hk5Et5wcTJoyCQvmPYcNGzcc7la9pgh0SASUFDvksumgDQKbt3yC2XNm47axY/GjHj3MaT22IgLDcofh+pHXY9KEiSgsKmzFnlS0ItD2COieYttjrj22EAIVFRW49tprMXjoYEwcP7GFpKqYI0GAJuu77rwdleVVeGHJ7xBnJxzJY3qPItDuEVBSbPdLpAM8GALMtnL77b8BYOGpp54CA821tS0CbqmpG9Gjew6m5+XpXm7bwq+9tRICaj5tJWBVbOsiwMoXRcVFEliuhNi6WB9KelJSEvKmTce69RuxdKk63hwKJz3fsRBQUuxY6+X50TLV2MoVK9zKF5PzkJGR6XlMjicAPXp1x7TpkzH7yfnYuGnT8RyK9q0ItAgCSootAqMKaSsECouKpC7iqJGjMWBQv7bqVvs5DAK5Q3Nx2TWXYNKECSgtKzvMnXpJEWj/COieYvtfIx1hGIHKigr86qab0KtXT0ybOk33sNrRN0Mcb26/DbXVQSxYuAC2bbej0elQFIEjR0A1xSPHSu88jgjUB4N47PEn4ff7cN899yohHse1OFjXsbaNhx/JQ1n5HkydPg0kSW2KQEdEQEmxI66aB8f88qJF2LxpA6ZPz9PKF+10/VNTUjEz/xGsXrUKf1i5sp2OUoelCBweASXFw+OjV48zAnSsWbNmLRYueA4Txk8EM6poa78InNrnNKmmMW3aNGzdsrX9DlRHpggcAgElxUMAo6fbBwKlZaV48onpuPb6Ecgdnts+BqWjOCwCw88bjvPPvwATJk5AWXn5Ye/Vi4pAe0NAHW3a24roeCIIMDj81ltuwYknZSL/iTytyhBBpv2/oYZ/882jZKBPz5uPOJ+v/Q9aR6gISDoQhUERaIcI0FFjztw5qKquwsRJ9ykhtsM1OtyQLMvC9Lx8FBUWYfaMmSBJalMEOgICaj7tCKvkwTEuX/YaVq1YhfzHn9DKFx10/dPT0jBr1lN4a/lbWLFqVQedhQ7bawgoKXptxTvAfDdu2YzZc+di7B134dSePTvAiHWIh0LgtD6nYcLEicibro43h8JIz7cvBHRPsX2th+dHw4wot9w4Aj/9f0Mxbtx4z+MRDQDQdDp16lRs3vwJFi1+Hgzd0KYItFcElBTb68p4cFxS+eI3dyLkBPHMM89o5Yso+g5wj/iWkSORkOTHk4/PR2yCGqmiaHmjair6zYyq5ezYk5n3/FwUf1mE6XnTlBA79lIeMHpmvMnLz8Nnn+/E3OdmHXBdTygC7QUBJcX2shIeH8e7K97FG8vewuQHpmjliyj9LrCiyYwZM/DqK69i9erVUTpLnVZHR0BJsaOvYBSMf+eunfLHcvTNYzBwwIAomJFO4VAI9O/XH6Nvuw1Tpk4B112bItDeENA9xfa2Ih4bT0VlBW668Ub07tUXU6dN9djsvTldOt5MnjgJO3buxEsvvYSEpARvAqGzbpcIqKbYLpfFG4Oi88WUhx5BvD8e99xzlzcmrbOUCif3P/AAbL+Ne397r1bU0O9Eu0JASbFdLYe3BvPhhg34aMM6jBl7l1a+8NbSiyPVfffeg40bNoLfA22KQHtBQEmxvayEx8bB8Iu5c+bixwOG4qz+Z3ps9jpdItCrz5kYNmw4Zs9+EjSja1ME2gMCSortYRU8NgaaTR966EGE4GDqIw9qXlOPrb+ZLsM07r//t4Bl47FHHlMzqgFGj8cVASXF4wq/Nzt/delibNi0EY9Mm4akpCRvgqCzFgS4/jNn5GPLls14ceFCJUb9Xhx3BJQUj/sSeGsAJMO5c+Zh0sRJ+FGPHt6avM72oAhkZ+dg4sSJWLTgOfxl48aD3qMnFYG2QkBDMtoKae0HRUXFuPnmkbjwvAsx5s6xiogisB8C8597DsuWLsUzC57DD3O67XdNPygCbYWAkmJbIe3xfiT35ehRSLB8eHLePN1H9Pj34WDTb6h18Nv7x6GstAzPPv+smtYPBpKea3UE1Hza6hBrBwzWfmTqVJSVlOPh/DwlRP1KHBQBJgl/6JGHEQjV4bHH1PHmoCDpyVZHQEmx1SHWDt555x2sWrkCM/Oe1LJB+nU4LAJJCQl4ZHoe1q9bj5cWLQT/QaVNEWhLBJQU2xJtD/a1dctWPPb4Y5j02yk49XR1rPHgV6DZU6YD1qQpk7Bw4UJs3LSp2c/rA4rAsSCgpHgs6Omzh0WAAdkPTn4Qw88bjgsuveCw9+pFRaApAsNyh+HKy6/A5IkTUVhU2PSSvlcEWhUBdbRpVXi9K5yONXfdeTtqa6vw7FOLEJdkexcMnflRIcDv0J1jxyJQU4dnnn1Wa2weFYr6UHMRUE2xuYjp/UeEwJxZM/G3z/+KvOn5SohHhJje9G0EmPFm2vRp2LN3r5jgdX/x2wjp59ZAQEmxNVD1uMw169fi9VffwtS86Vow2OPfhWOdfmpKKu6fNgVr163FkteWHqs4fV4R+E4ElBS/EyK9oTkI7CoowOTxEzFmzG0Y2F8LBjcHO7334Aj0690XE8ZNwLy5c0HHLW2KQGsioHuKrYmux2Sz8sV1N1yDbpnd8PBMjUf02PK36nRpOp01cyZWrlyJhYsWIysrs1X7U+HeRUBJ0btr36Iz5x+t8XePQ2FpMV5+4SV1imhRdFUYEeA/un59529gO8DT8+drEgj9WrQKAmo+bRVYvSd0waIF2LR1M/KmTldC9N7yt8mM4xLiMCMvH6VlpXgsP08rarQJ6t7rREnRe2ve4jNm5YuF8xdh8tQp6N6je4vLV4GKgEEgNTUVU6ZOxep3V+IPK1eZ03pUBFoMASXFFoPSm4KKi0swecJEXHvVlcgdPNSbIOis2xSBfmf2w90TxiMvf7o63rQp8t7oTPcUvbHOrTJL7iOOuP56+Dsn4qn8uRqP2Cooq9CDIcDv3vS8adi4YSMWPLcQ6RnpB7tNzykCzUZASbHZkOkDRIDZRh6aMgU7tm3B8y8u1kTf+rVocwRYaurW34xEnD8JT86ZpY43bb4C0dmhmk+jc11bfVb/ufJ9rFm9Gg8/lKeE2OpoawcHQ4ClpqZPy8PfC/8HT8yY0WqON/R6zct/AoMHD8HQoYMxf/588YT99piKioswbtw4XHjphbj79rtRUKg5W7+NUUf4rKTYEVapnY1x8+bNmDr9UUyaNBmnnd6nnY1Oh+MlBDIyMzB1yhQsf+N1/Pe6dS0+dVpEHnn0YfQfcBaeeeYpDBt+npDiSy+/tF9fJMBRI0bi/PPOxztvvoPzLjkPY0aP1mTm+6HUMT6o+bRjrFO7GWV5RTlGjrgF/QeehckTJ7WbcelAvI3Aa8uWYe6cJ/H0nAXo07dni4FRUlKCQDCEbjnZIpMm2yt+dSlO7toV8+cviPRDDbGktBhLFi+FZVlSB/Lqq69Gt27dMX36tMh9+qb9I6CaYvtfo3YzwvpgEA8+MBGpKZ1x3/gJ7WZcOhBF4KKLL8agwUPw0NRJYMmylmqZmZkRQqRMmmxP7NwF/fsNinTBfyiuXbsWvbr1FULkBRLj4IHnYNWqlSguLY3cq2/aPwJKiu1/jdrFCOntN/fpWfjbXwvEFf7/t3c2UFGd577/X+4E8StqKDGUQ1keyrHERDmWEko9LpeLUuqyVo3Geg1FJIZQBBVREcdhGMZxHFFEggSRECSU+G0s5VKul0s4lBqLn9EYDtfF8rIoJYQYgh+EUHrXO0YU+ZwBZvae+e+1WDPz7vfjeX7PnnnY+33e9xEZDHiQgFQIiOsxcfsOODk5IikxcdTmF283NMDF2QWvh4V0q36+qsr43m2aa3eZePO97z/8XFdT26OcH6RNgE5R2vaRjHQiS8Gx3zPzhWQMQkF6ERgzdix0OgM+vf4pDqYf6HV+OAXin8I/ni5GVMRaiDvDvzc0dHf3t79/bnzv5t7TKU6ZNMVYXt/09+66fCN9AnSK0reR1SX89MYtqFQqrN8UxcwXVrcGBRiIgIeHOzRaHY4fP2F8pDlQXVPOfXb9U3zdfg+urq64fPkqoqI2dEegdrR3GLtyUjj16LKrq9M4t9ijkB8kT4BOUfImsq6AbW1tSFJvQYCvP1auWGVdYTg6CQyBQECAP8LXroVSpURNbc0QWgxe5cWZL2HlyuXIzs7B4qWL8be/3cafP/5PY8PvPfec8bW1rbVHR3futBnnFqdOerZHOT9ImwCdorTtY1XpxCOjpKREdMIByTt3dgcRWFUoDk4CQyAQFhqKOf7+SEhQ417bvSG0GHqVqKgoY+X7be3GV5/ZPsbXuv/38DHqo56avguwmTZ9+qMivsqAAJ2iDIxkLRGzc3Jw4Xw1DAYDM19Yywgc1ywCIvozKWknFIou7NixfUQDb5ynOMPJUYGf+L9ilM3b2xte3tNRU3uph6w3bn6CWf8+C57TpvUo5wdpE6BTlLZ9rCadyHyRl50DlU7NL7XVrMCBh0NApJrSaDS4dO0ajuRmm9WVWN5RfelS9/yheHqSk5uDZctWwnXq4/1WI0Ij8Un1DYhdbcRRd+sWbty4iajISLPGZSPrEeDifeuxl+zIDQ31WLlqFZYtXoyYDbGSlZOCkcBQCJRXlCMhPgEGnQFz5j1eXziUtkUlxdCpNXBzc8X8oGC0tdyBs6sLwsPCezXPz89HSUkJfuLvj79UViIsLBzBwUG96rFA2gToFKVtH4tL19nZidDQMDz77ARkZGRwHtHiFuCAo0EgKzMLBUcLkJeTi2menkMeQtwZivRoHV3tcFQ44vnnnh9wKqG5uRlf3f0K//L8vwxYb8gCsKLFCdApWhy5dAcUPwBKtQo3r97A4fcOc6Nv6ZqKkplIQOxhujk21rjbzeF3DtNhmcjPnqpzTtGerD2IrmfOnERZSRmSk5LoEAdhxdPyIiB2vNm5cyfu3r2PxKTR2/FGXlQobV8E6BT7omKHZSKYIMWQBqU2gZkv7ND+9qDyxIkTsc9gQGVVJT4ozOfCenswuhk60imaAc3WmrS0tECZkIDghUFYELjA1tSjPiTQTcDTywtatQbZmbk4f/5CdznfkMAjApxTfETCTl9F5ouY6Ch0dXYiKysbCm70badXgn2pnZa2H3/8YzGysrMwzYPrCO3L+gNryzvFgfnY9FkRWJN98G3cqqkxrueiQ7Rpc1O5Jwi8FbUOP/L8EbRqVfcaxCdO860dE6BTtGPjnys7h/ffPwN9Sgrc3NztmARVtzcCxlRTukR80fw1knclj+iON/bG0tZsdBvyAAAgAElEQVT0pVO0NYsOUZ/rN25CrVZj48a18PP1G2IrViMB2yEgtmtL3q1DZVklzpw+bTuKUZNhEaBTHBY+eTYWmS8S1SrM8QvAilWPk6XKUxtKTQLmE5g5YwY2JWxC2t49uFBdbX5HbGkzBBhoYzOmHJoiYhHz1thYNDY24EjhUYjHSDxIwN4J6PQ6lJeXIy8vv8eepvbOxR71552inVn9/bw8XLhUjZR9++kQ7cz2VLd/Apui4/HDH/4QKpWSgTf9Y7KLM3SKdmHmh0pWVlYhKysLGrUO7u4MrLEj01PVQQiMGe+AxO2JqL9dj9S0VAbeDMLLlk/TKdqydZ/QTaS0UauUCAsLRWDgvCfO8C0JkIAgMNV1KgwGPYqKi/Cn4mJCsVMCnFO0A8O3tbfjrYhwTJo4Can7D/CxqR3YnCqaT+DkyZPYu2cPDmRnwfflWeZ3xJayJMA7RVmabehCiwX6u7VqtH91F7uSd9IhDh0da9opgSVLliAoKBhJ8dvQeueOnVKwX7XpFG3c9mdOnsG50gokJ+/EpClTbFxbqkcCwyfg4OCAbSolXnjBFRu2xDHwZvhIZdUDnaKszGWasNVXr0K/Rw+lWokXZ75kWmPWJgE7JiCWKmm0GtTfqsOhdw8x8MaOrgU6RRs1tjHzxdatWLxoMRYuYOYLGzUz1RpFAq6ubjCkpODk8dP436UlozgSu5YSAQbaSMkaIyRLZ2cn1kauhUOnA97OPoCxivEj1DO7IQH7I5Cbn4/cnGwcyMiAz4yX7Q+AnWnMO0UbNHhKyj401jdCq9PQIdqgfamSZQmErlqF+QGB2KlUoeVOi2UH52gWJ0CnaHHkoztgSUkJTp06Aa1WC/H4hwcJkMDwCIjAm03bN2HcOCeoEpLxoPPe8Dpka0kToFOUtHlME+4zkRdRq8HGjRvh6+trWmPWJgES6JfAxPHjodWn4JObl3Ak5/1+6/GE/AnQKcrfhkYNROaLbfFbMM9vHlasWGEjWlENEpAOAXd3N+h1BuTl5qLs3DnpCEZJRpQAA21GFKd1OhM71ig3bcbfv/wCR/LzuUDfOmbgqHZCICs7GwUF+cjJyYGXp5edaG0/avJO0QZsfaygAH+9ehEGvZ4O0QbsSRWkTWBteDjm+M1FfFw8xBMaHrZFgE5R5vYsKy9DTnYu9AY9PDw8ZK4NxScB6RMw7nizYyugAJTxiXjQ0SF9oSnhkAnQKQ4ZlfQq3qq7DZ1Gi5DQlZgbMFd6AlIiErBRAhMnToTBIAJvqvF+fp6NammfanFOUaZ2v9d2D2+uexPOk6Zg//4DEP+98iABErAsgYqKSsTFxWKfPgVz5vMfU8vSH53R+Es6OlxHvdfk5CS0ttzBzp276BBHnTYHIIG+CcydOwdhoSFQa9Woq7/ddyWWyooAnaKszPVQ2MKjBSgrq8De3WkQj3F4kAAJWI/AmohI+Mz2Q/yWLQy8sZ4ZRmxkOsURQ2mZjqouVCM9NQ0avQbTZzAc3DLUOQoJ9E9AZNTYtn0z2ts7sHvPbmbU6B+VLM7QKcrCTA+FFJkvNColghctQnBgkIwkp6gkYNsEnKc4I0WXgoqKCnxQcNS2lbVx7RhoIxMDf9PZid+tDYfCwREHsjK5HlEmdqOY9kXgXGkpVCoV9Cl6zAmYy/l+GZqfd4oyMVpqSgrq6hug1mnpEGViM4ppfwQCg4Lwm9+8Bp02FQ0NjfYHwAY0plOUgREfZr44ZdyxxnXqVBlITBFJwH4JrIvZAM9/ewEJCfFobW21XxAy1ZxOUeKGu3bjhvFxzKbNm5n5QuK2ongkIAiINcPqHVo0NzcjIzOdgTcyuyzoFCVssAf3HmDH1gTMD5yP5a++KmFJKRoJkMCTBFxcXLAvNRUlJaU4dfTkk6f4XuIE6BQlaiARWBO3JQ6TnZ9F4vZETthL1E4UiwT6I/Citzc2rl+PjMxUVF+62l81lkuMAJ2ixAzySJwjuTm4ePUyNFotxo4f+6iYryRAAjIisGTJq1i4cCkSt29CY1OTjCS3X1HpFCVo+/KKCuRmF8AgMl+4M/OFBE1EkUhgyAQ2Rsfh+9+fhoT4eIg9i3lImwCdosTsU3vrFlRKJULXhjDzhcRsQ3FIwBwCY8Y7QKvXobGxEZmZGeZ0wTYWJECnaEHYgw0l/otUK5WY7eMDkciUBwmQgG0QmOriAo1Wj7Nni3DqzBl0dXXZhmI2qAWdokSMKr4kSclJuNt2H8m7mPlCImahGCQwYgR8Z89EVPRapKQY8FlNzYj1y45GlgCd4sjyNLu3o8dPo7y8HCnpBkwcP97sftiQBEhAmgTE+sUVK0KwcOFCxG7caFzHKE1J7VsqOkUJ2L/qwnmkp+6GVqOB17TpEpCIIpAACYwWgY3RGyHWMYo9UsVaZB7SIkCnaGV7iDBtjUqNRYuXIig42MrScHgSIIHRJiCWWBnEXsZ1dcjKYuDNaPM2tX9myTCV2AjWf9DWiXUx4WJfKGRn53CB/giyZVckIHUC5y9cQGzsBiQkJGBB8AJ+/yViMN4pWtEQe9N1aGhsgk5v4BfCinbg0CRgDQL+fn6ICI+EwWDAf9XWWkMEjtkHATrFPqBYoqiouBjFZ4ug1+shwrV5kAAJ2B+B0LAQzJs3D9u2xKO5pdn+AEhQYzpFKxhFZL7QqJRYv34TfHx8rCABhyQBEpAKgc2bNsPRSYGdybvwzQMG3ljbLnSKFrZA2717SExQIjB4AZavXGHh0TkcCZCA1AhMnDgR+/e/jevXruCdd9+Vmnh2Jw+dogVNLjJfbN26GeMmj8OObdtB+BaEz6FIQMIEXF2nQqPToSAvDyXFRRKW1PZF4++yBW38blY2rl+/Dp1Gz8wXFuTOoUhADgQC/P0RGRkBfco+1N5i4I21bEanaCHyZefKUJCfD71OBw8PdwuNymFIgATkRCAsLBwiKjVeBN40M/DGGrajU7QA9Vt1dVBr1QgJD0NAwBwLjMghSIAE5Epg29atxrXLe1NTuXG4FYxIpzjK0O+1diAhfgv8fHyxOixslEdj9yRAAnInMGnKFOPaxerz55GVnU3HaGGD0imOInARWJO0Kx6dnZ1ITE7CGIViFEdj1yRAArZCwHPaNMTHJyA/Lw/nq6psRS1Z6EGnOIpmOnP8JCrLq5CsN0CEXfMgARIggaESCAwKREhoKOKVCWhoqB9qM9YbJgE6xWEC7K+5yHyRlpoKtVaDF728+qvGchIgARLol8Ca19fCz88P62Li0NbW1m89nhg5AnSKI8eyu6f6xkZj5ouFyxYhKDCou5xvSIAESMAUAmPGO2D71m1w6OrErt27TGnKumYSYJYMM8H11+xe2z3ErFuHMU5jcTDrYH/VWE4CJEACQyZws+YmoiOjEBIWgt+sDGF8wpDJmV6Rd4qmMxuwxd70VDQ0NSJZlzxgPZ4kARIggaES8J7ujZjYDcjOyMblS9VDbcZ6ZhCgUzQDWn9NPiwuRtGZs9Dr9HB2du6vGstJgARIwGQCixYuwsqQEKiUSty+zcAbkwEOsQEfnw4R1GDVrl25grfeehPr4zdhxWJu9D0YL54nARIwncA397oQuy0S99s6cODAAUa1m45w0Ba8UxwU0eAVRFRYYmIS5gcGYfmi5YM3YA0SIAESMIOACLzRJOnQ3NyEvWl7zeiBTQYjQKc4GKFBzosF+rFxsRg3zgk71Go4OBDpIMh4mgRIYBgEnKc4w7BnL0pLSlFQUMAdb4bBsq+m/AXvi4oJZYcys1BTU4OUfSmMCDOBG6uSAAmYT+BFb2/jpuEZmRm4du2a+R2xZS8CdIq9kAy9oKy8HIUFedDr9XB1dRt6Q9YkARIggWESWLRoEZYuWoS42FiItdE8RoYAA23M5CgyX4SErEJoaBgi1q41sxc2IwESIAHzCYjAm3Uxa6FwUGDf/v3M02o+yu6WdIrdKIb+5l5HG9asegPubm4w7EvhPOLQ0bEmCZDACBNoam5GeFgo/AICoIxP4O/RMPny8amJAEVgTbIyGZ1dwLbE7bwATeTH6iRAAiNLYKqLC9S7dqG0qBQffnh6ZDu3w97oFE00+omjR1FRVQmDXgsRBcaDBEiABKxNwPflWYiNi8aevXtx/dJNa4sj6/HpFE0wn8h8kZGWBbVKA09mvjCBHKuSAAmMNoHFi1/FguBgxCo3orGpabSHs9n+Oac4RNM2NTZh1aqVCFoQhC1x8UNsxWokQAIkYDkCD+49wBtvrsFzEyYjJSOdy8TMQE+nOARobe3tWBcRAScnR2RmZnEecQjMWIUESMA6BBobGxAWFo7AwPmIjY3j75WJZuDj00GAicCajH370dTUBI1WxwtsEF48TQIkYF0CYs20Sq3GsWMnUHauzLrCyHB0OsVBjPa/iktw6swp7DOkQER58SABEiABqRMI8PfHps2bodaqcf0GA29MsRed4gC0rly6Ap1Bh/j4zXhx5ksD1OQpEiABEpAWgcVLlmD+3PnYrkxAS0uLtISTsDScU+zHOC13WvDG6jfgPWsGdBptP7VYTAIkQALSJSACb1a/sRpuU12xK4X7Mw/FUnSKfVB60NaJmLhIdLS341BOLiO4+mDEIhIgAXkQqK+vR1hoKJYsWoI31kXy92wQs/HxaR+A3juShZobNcaNvscoFH3UYBEJkAAJyIOAu7s7NDod8n6fjz9XVspDaCtKSaf4FPzScyXIzcvDvgP7mPniKTb8SAIkIE8CIvAmKioCarUKn9bWylMJC0lNp/gE6Npbt6DV6hC+di18Z/s+cYZvSYAESEDeBEJCwuDnG4CdSjVEzASPvglwTvE7Lvfa7uG3a1bD08MDeoOB6xH7vl5YSgIkIGMCra2tWLNmNbw8vZCk03F+sQ9b8k4RgFign5iYCIWDA3YkJtIh9nGhsIgESED+BCZNmoT9+w9A7OP8XnaO/BUaBQ3oFAF8UFiAyuoqaDVaTJw4cRQws0sSIAESkAYBEXijVqqQl5dvzPgjDamkI4XdO8XSknNI3ZuKsFWr4DXdSzqWoSQkQAIkMEoE5gcGIiR0JZTxW3Cr7vYojSLPbu16TrHu1i2sCg0xRpk21jciMjoSr61Ywefs8ryWKTUJkIAJBMS0UezGaNz98iu8c+hdjB0/1oTWtlvVbu8UW9tbERcfD28vb/y+sBAajRLZmVnYazAY5xht1+TUjARIgARg/Odfp9HhbkcnkpN24kFHB7EAsEunKP5D2hm/C60tLVBpVMaLIzAoGPve3oeKigokqlQQUVo8SIAESMCWCUyaMgXJGg0qqytw4vhxW1Z1yLrZpVM8cbQQ5VVl0Oq08HD36Ibl6+OLjPR03Lh6BYlJiXSM3WT4hgRIwFYJvOjtDZVShcyMDFRVcccbu3OK5RUVSN2bjuiN6+HvH9DrOvf08sKhw9n4v/91Gzvid/BRai9CLCABErA1AvPnzcdvXltmdI4iSbE9H3YVaNPQUI/QkFDMnu2DZL1hwIAasYnuug0x8HT3wK5duzBmLCeh7fmLQt1JwNYJiGmljVExaGtvw+GDeRgz3u7umYwmthunKDJfvLEuFO33O/D+e0eGFGl1+/ZtREREICDAH1s3qez2IrH1HwPqRwIk8JCA2P4tNCQEvrNnY6tShbGOjnaHxi7+FRD/Ae1O16LuVh0MBsOQHKK4Ejw8PLAvNRXnysrw7gc56OrqsrsLhAqTAAnYDwHnKc7Q7dmL4pJSFP/xD/aj+BOa2oVT/J9nP0TxqbPQqfXwnDbtCfUHfysmoQ0pOuRn5aCoqHjwBqxBAiRAAjImMNPb25hNY++evbhQXS1jTcwT3eadojCq3rAHr7/+OuYFzjWLkt/sAERvikbKPgOu3PjErD7YiARIgATkQiA4eAEWL1oIjVoFEYthT4dNzyk2tzQbM07/4Af/ioMH3x62XdUqDS5evIj33n8X4jEDDxIgARKwVQJiumhteAQcHIED+w4MedpJ7jxs9k5RzCMmxicZ7ZO88+HrcI21afNGPDvBEXt27+VSjeHCZHsSIAFJE3BwcIDOoEP97XrsTdtjN795NusUs94+gOqrF6DX6kfsrk5k0NDoDaiqrMSfPuT8oqS/0RSOBEhg2ASmurhAr9Oj+Mw5/KnEPn7zbNIp/rG0GO8deR/xCQl4yWfmsC+MJzsQgToREeFISU+BWMvIgwRIgARsmYDPbB+sj4+ATqPHlRuXbFlVo242N6f4KPNFYFAQNCr1qBnwd+t+B0cHBXbv2zfgJgCjJgA7JgESIAELEtCoNbhYfRHv5B6Cq8tUC45s2aFsyim2trZhzerVmPDsOBwa5R0ZxML+5cuXY4dGjV8FL7Cs1TgaCZAACViYwIN7D/BWzFsY5zgO+8XNgI3u8mVTj08TE3egtfUO9Hr9qO8+Ixb2R0dG4FB6Fu60tFj48uRwJEACJGBZAiLfok6jRV1dHd5+5x3LDm7B0WzGKWbnZqOqqgr6FIMxabAlGC5e9hoUjsCxE8csMRzHIAESIAGrEnBzc4dGq8Hpo8dRUlxqVVlGa3CbcIoVVZXITM9ERGQEfGf7jharXv2KaNTo6I3Iycm1uwWuvWCwgARIwC4I+Pn6IWJ9FNQaFa7dvGFzOsveKYrdFjRKFQIDAxEeFm5xA/1s7hx4z/BGXv4HFh+bA5IACZCANQi8tnwFAgPnIylejba2NmuIMGpjytopCmPExcXD2cUZSYkjs0DfVNJjFAqjMy4qOo3GhkZTm7M+CZAACciOgPjd275tB5wmOGD79u0QQTi2csjWKYotiNLS09BQXw+tTmfVLYjmzp0Lby8vZGRm2sp1QT1IgARIYEACxsAbXQo+vX4d733wvs3seCNbp3jy5HGcOnEKGq0OXp5eAxrPEidXhoShorIcLXfuWGI4jkECJEACVifg4eEOjU6H/Jxc/Lm83OryjIQAsnSKIvPFnj17ERYWhnnz5owEh2H38R9z58Bl0hScPXVq2H2xAxIgARKQC4EAf3+Eh6+FWqPB9ZqbchG7Xzll5xSbm5uRnJQE33/3RVR0dL+KWfqEeMb++urVOHXmjE09X7c0R45HAiQgPwJhYaEQzjFZpZJ94I2snKKYzE1ISMA///kP7E7ZDakJ7z8nAGJXnb9evii/q5oSkwAJkICZBERGje07dgBdCuxISsQ3D+QbeCM1vzKgSQ6/exhXr16FbvceiDWCljpEUE/puVIcP3p0wMlksR+gv58vyspKIdrwIAESIAF7ISB+k/UGLS6cP48Tp07I9jdQNk6xuLgYubm5UKqUmDljhsnXmXBSmfuzkF9Y2KutmKNUKpX4XUwMVCoVrly50qPOH4qKUVd721iWZkjpce7pD8teW4aSonO4d+/e06f4mQRIgARsmsC0aZ7Qqw1IS0tHRUWFLHWVhVOsqa0x5vRasHAxFi1cZBboyvNVyH0/C7dra3q0LykugTIhAa8tW4aDBw4gOHgBtmyJQ3X14xQpNdeuYOnSxXh1xQp8/fVXPdo//eFln9mYOnUyyv7zo6dP8TMJkAAJ2DyBuYFzEBoaAq1Wg9v1D28m5KS05J2iWKAfv2ULPD2nQaNWmsW2uaUZuZnZmDB5MhQKpx59FBbm49e//jVm+vgYywMC/BEwdy5OPRFFOj84GLvT9hjvJqfPerlH+6c/jHV0xKwZfvhLRYVsHx88rRM/kwAJkIApBCIjozDzpZnYskUpu8AbSTtF8cgzbtsWtLW2QZ+SAjGZa86RkZaOrQkqKBwV6Ozq7NWFWHz6Tefj8i8//9LohB9V9PX1xfqITQgPD0fIylWPivt9nRsUgOoL1f2e5wkSIAESsGUCxsCbxB1ob7+LnTt3o/OJ31ep622el7GQVnm5+bh84aIx88VUFxezRs3LK8CMl2fAzfN5PNP5DICeATDBCxbjLx9/jF16nfE/mvLycrS2tmDZ0qU9xnP3mArPadN6lPX34Wev/Afud7Tj0pXHj2D7q8tyEiABErBFAs5TnJGiS0HV+QocLSyQjYqSdYpV588jLXMvoiKjzM58IeYiG+rrsHz5CuAb4FvFt8BTCzlWrlyOZSuXovhMMVavWYOi4rPIzc3DpClTzDai2P5o1qxZKC+zjR0ezAbBhiRAAnZNwMvbE8r4BGRkZqGislIWLCTpFOsbG6FKSEDQnGCEhoWaBVLMRWaIdFJRkd+1d8AzEHeKvY+w366Ft/d0NDY24Er1FZyvOt+7kokls2f74vLlyya2YnUSIAESsC0CgUFBWLpsMXYmq9HY1CR55STnFMUC/diYGDhPdcGO5B1mAzzy3vtYs2Y1xC38o+NbfAvFow/fvYptieLj4rDvwD68+84hjB83EXHxcTh/YXiOcYbXdDQ3N3Ev1Kd48yMJkIB9ERDzi9ExsfD+15mI3xIn+cCb//bPf/7zn1IykUqpgpjXyy/Mh4e7h1miPejowE/9/LoDcx4tpH8UqCMmfQ8fPgQ/P7FnXxhm+4gt46KMY4m7xbVr12H8eEccPXrUrPFFI7Ed3S9+8SscP1k45LlIswdjQxIgARKQOIGm5maEi+3gAgKwaUs8xNaYUjwkdadYePwoSkpKodGrzXaIArKAfenSJVRXVxv/xPuPPvoIri+4YuniZbh27ZrRIYq6ly9fxQs/eKHbNq6ubtgQE4Xa2loI52ru4eLiguefG4dbNT3XRZrbH9uRAAmQgJwJiGDJlFQDSopLcfb4ScmqIhmnWH3lElL37MXK0JWYN2f+sICJO8K+/r7t/BZweLz0Qgzy8ssv4+O//LXHmsKWOy145ZVXINYcDueY7ReAzz77bDhdsC0JkAAJ2AyBf/N8Ees3bURqRhrOX7ggSb0k4RQbm5uQtD0B/v5+WB81WpkvHPCM4hnjhrVPWiJBmYDaGzVQKZU4efIk9qenQWwpp1QmPFnNrPfTvb1QW1drVls2IgESIAFbIyBuVl5dssS4M5lWI83AG6vPKYrAmnUxv0NT01fIyT0Ec9cjDnbxfHOvCyWlH8LN3R1iMf6Thzj354/L0dTUDDc3N8ydOzI5Gs+VnYNeZ8C5c6VPDsf3JEACJGDXBMRv7rqYtehEFw4eOAixjE0qh9Wd4r60NJwoLERebgHEmhZbOq5/eg1rfvsmPvro/0jK6LbEmLqQAAnIk0BTY5Nxyd38wPlYvyFWMoE3Vn18KhLyHjFmvlDZnEMUl+mUSc5QKLrw+VdfyPOqpdQkQAIkMEoEprpOhUGXijNnzuBPRSWjNIrp3VrNKX5WU2PMSrFo6WIsWLDAdMll0MJR4QiFkxPut96VgbQUkQRIgAQsS+Aln+mIjo5Gyn4DLly4atnB+xnNKk5R7DazOS4O7h4e0KjU/Ygm/2JHJyc4KpzQdrdV/spQAxIgARIYYQIi8GblipUQu97s2rkNIqORtQ+rOMUtW7agta0VOp3W2vqP6vgOY57BGMdn0H6/fVTHYeckQAIkIGcCm9dvwoRnnaGMV+Jem3UTtI/IlgJ1t27hwqVLWLF8+aB2yc7OxsWLF3H48GGIhfK2fEwUd4pjFGi7f9+W1aRuJEACJDAsAiL61GDQI2RVON49ko3oqA0D9if2pxYJH/o6pnl54iezfmx2cOOI3Cme/PA0Tpz4AGJ5xUCH2L4tMzMTUdGRmDlz5kBVbepcR6f5O+PYFAgqQwIkQAL9EBA3SVq9GoWFx3C2qKifWg+L/QP8If6ysrNQWFAAxwlOmDJ5ChRwQk5ONn7+y58bsx09mSd3wA6fODnsJRnCEf7il7/E3btf4513DnZvn/bEGMa3IvPFG6tD0Nz8JRQOjujssh9HoXAAOnumcXwaDz+TAAmQAAl8R2DCsxMgnipO95reLxOxDef/WLEczz47GXl5ed31xF7XsbFxqKgoh1anw4Lg4O5zQ3kz7Menx44dw8KFC3DixAmcOVvcr1Nsv38X27cnQKFwQldXJxwcFMbXoQgp5zpC13FOkyHRvW/ljJaykwAJyJXAd3cK/10xptcdwz8g8t4C45wmDKjdMw4PH3SK31jhCEXQjjjE6+o1vzU6xeoLVZZ1ikKQi1fPY3dyOm7U3EBpSTEioyLh7uraSxkvTy+IPx4kQAIkQAIkMFwCDx2hAg4Ovfeo/uqrL43djx9jerL4Yc0pVl+qhre3D8ZOVCA0NMzorUuLzg5XV7YnARIgARIggQEJ/AMwPm3s7Hgc3f/Ngwe4ebMG6WnpeG7yZCz+zcIB++jr5LAen+ZmZyEhQWfs96c/fgWurs+jqLgYr4eFS2bLnr6UZhkJkAAJkIC8CYjHp2Ia7tat21i+8uHKB5Ert7mhEU5OE6DVaTHN3fStQ82+U7x9ux7OLu5w95hqJCtCan/2s/kQ5R9XVsqbNqUnARIgARKQNIFvu7qMd4oiG9HJoyeNf8eOnEZeQQGmeXhhw4YNOJCRbrIOZt8pFhefQUf7fYjUS11dHXDoUgAKBygUCpw4dRyvzJnDu0WTzcEGJEACJEACQyXg+F3A5qNAmzHjHYyxK4YDWixfshLv5RRgyeLFJiWtN8sptty5g+pLV3AwM6uX47v/9V1cuFSNzxtauu8ih6og65EACZAACZDAUAmIlPGO6P3A03mSC9zcpkIE3NTV1prkFHv3NgRpSs4WYdGvlvRyiKLpzxcGG7c1Kyo+MYSeWIUESIAESIAEzCPwKAL16dbNzc1oaGgy7j3945+88vTpAT+b7BTFZt4fHCvA/KDAPjv2m/UTY3nh0aOou13XZx0WkgAJkAAJkMBwCHR2daH1Tivu372L+voGY1fCSYqd09atizHeJW5WbsTEiRNNGsakHW0aGxuRm5eHz7/4HNOnTcOCRYt63JY2NzXjVPEZ1N54uCedCIldtHw5Xpre/64EJknLyiRAAiRAAnZP4EZNDSrLy1FbWwvhHB/d3Yn34xwd4eE5HfPn/hTTvV80mZVJTtHk3tmABM+Xe+QAAAJeSURBVEiABEiABEaQwKOgmhHsskdXjxxsj0J+IAESMI9AU3Mz0jMy+mwsHvFk5+bg5MnjaGt/vOC4z8osJAES6JPAo+3c+jw5AoV0iiMAkV2QgCAgdtPYEh+HowVHewH5w9kixGyIMU43iM3zV68KRWNDY696LCABErAuAbOWZFhXZI5OAtIkIDbHb6xvgINDz5QotTW1SNao8PbBx1lkxF2jSq3C25mZfUZxS1NDSkUCtk+Ad4q2b2NqaAEC1dXVqKmtg6+fn3Hi/8kh8wvzMfm578F7pk938fx584zJti9fqe4u4xsSIAHrE6BTtL4NKIHMCYicokdy38MOzVYo4AAH9HwAU1V5Hq6urpjo5NStqaubmzGv6NWLn3SX8Q0JkID1CdApWt8GlEDGBEQk3O49u/FmxFsY4zAWnRB53R4/PhWZv7/88gtMnNIzhc2Ecc9C4eiA2/Vcyytj81N0GyRAp2iDRqVKliNwruwcpntPx0szX+oetKvr8dfqi8aHwTSTx/VOmKpwcMD9+4xC7QbHNyQgAQKPv70SEIYikICcCIiMMCXFpVixfEUPsbsgdmR8eDhNeOgMO7t6Or/Ork60d3bAybHno9ZH7fhKAiRgHQL8RlqHO0e1AQKFhYWou30LYWFhwHd7ajTU34bI6RYeHgZfP19ERkThuecmo+VOWw+N77bfRWdHJ6Z5evUo5wcSIAHrEqBTtC5/ji5jAsHBQXhx5o96aHD65Gnc+KQWv351CX7w/R8Yz/n4+KLuVi0edHRgrKOjsaypocn4+vKMGT3a8wMJkIB1Cfx/WSntG/kqKlQAAAAASUVORK5CYII="><em><strong>(A1) (C1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{40}}{{{\text{sin APB}}}} = \frac{{30}}{{{\text{sin 48}}^\circ }}">
<mfrac>
<mrow>
<mn>40</mn>
</mrow>
<mrow>
<mrow>
<mtext>sin APB</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>30</mn>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>sin 48</mtext>
</mrow>
<mo>∘</mo>
</msup>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution into sine rule, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p>(angle APB =) 82.2° (82.2473…°) <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>180 − 48 − 82.2473… <em><strong> (M1)</strong></em></p>
<p>49.8° (49.7526…°) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from parts (a) and (b).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A line, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>, has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \left( {\begin{array}{*{20}{c}} { - 3} \\ 9 \\ {10} \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} 6 \\ 0 \\ 2 \end{array}} \right)">
<mi>r</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {15{\text{,}}\,\,9{\text{,}}\,\,c} \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>15</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>9</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A second line, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span>, is parallel to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> and passes through (1, 2, 3).</p>
<p>Write down a vector equation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>correct equation <em><strong> (A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 6s = 15"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>6</mn> <mi>s</mi> <mo>=</mo> <mn>15</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6s = 18"> <mn>6</mn> <mi>s</mi> <mo>=</mo> <mn>18</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = 3"> <mi>s</mi> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p>substitute their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s"> <mi>s</mi> </math></span> value into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z"> <mi>z</mi> </math></span> component <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10 + 3\left( 2 \right)"> <mn>10</mn> <mo>+</mo> <mn>3</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10 + 6"> <mn>10</mn> <mo>+</mo> <mn>6</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 16"> <mi>c</mi> <mo>=</mo> <mn>16</mn> </math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 6 \\ 0 \\ 2 \end{array}} \right)"> <mi>r</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> (=(<em><strong>i</strong></em> + 2<em><strong>j</strong></em> + 3<em><strong>k</strong></em>) + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span>(6<em><strong>i</strong></em> + 2<em><strong>k</strong></em>)) <em><strong>A2 N2</strong></em></p>
<p><strong>Note:</strong> Accept any scalar multiple of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6 \\ 0 \\ 2 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> for the direction vector.</p>
<p>Award <strong>A1</strong> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 6 \\ 0 \\ 2 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, <em><strong>A1</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2} = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 6 \\ 0 \\ 2 \end{array}} \right)"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, <em><strong>A0</strong> </em>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \left( {\begin{array}{*{20}{c}} 6 \\ 0 \\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right)"> <mi>r</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo></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>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>π</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mo>=</mo><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math></p>
<p>recognising to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></math> (values may be seen in right triangle) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>arctan</mtext><mfenced><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math> (seen anywhere) (accept degrees) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>6</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mn>6</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>12</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>7</mn><mi>π</mi></mrow><mn>12</mn></mfrac></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong></em> if any additional solutions are seen.<br>Award <em><strong>A1A0</strong> </em>for correct answers in degrees.<br>Award <em><strong>A0A0</strong> </em>for correct answers in degrees with additional values.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Determining the composite function was very well done. In part (b) very few candidates showed any recognition that tan (or cot) were required to solve this trigonometric equation. Many saw the 2<em>x</em> and simply employed one of the double angle rules but could not then progress to an answer.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the vectors <em><strong>a</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> and <em><strong>b</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>Find the possible values of <em>p</em> for which <strong><em>a</em></strong> and <strong><em>b</em></strong> are parallel.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1 </strong>(eliminating <em>k</em>)</p>
<p>recognizing parallel vectors are multiples of each other <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em><strong>a</strong></em> = <em>k<strong>b</strong></em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> = <em>k</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{p + 1}}{3} = \frac{8}{{2p}}"> <mfrac> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>3</mn> </mfrac> <mo>=</mo> <mfrac> <mn>8</mn> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mfrac> </math></span>, 3<em>k</em> = <em>p</em> + 1 and 2<em>kp</em> = 8</p>
<p>correct working (must be quadratic) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 2<em>p</em><sup>2</sup> + 2<em>p</em> = 24, <em>p</em><sup>2</sup> + <em>p</em> – 12, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 = \frac{{{p^2} + p}}{4}"> <mn>3</mn> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>p</mi> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p>valid attempt to solve <strong>their</strong> quadratic equation <em><strong>(M1)</strong></em></p>
<p><em>eg </em>factorizing, formula, completing the square</p>
<p>evidence of correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> (<em>p</em> + 4)(<em>p</em> – 3), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{ - 2 \pm \sqrt {4 - 4\left( 2 \right)\left( { - 24} \right)} }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mo>±</mo> <msqrt> <mn>4</mn> <mo>−</mo> <mn>4</mn> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>24</mn> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p><em>p</em> = –4, <em>p</em> = 3 <strong> <em>A1A1 N4</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong> (solving for <em>k</em>)</p>
<p>recognizing parallel vectors are multiples of each other <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em><strong>a</strong></em> = <em>k<strong>b</strong></em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> = <em>k</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, 3<em>k</em> = <em>p</em> + 1 and 2<em>kp</em> = 8</p>
<p>correct working (must be quadratic) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 3<em>k</em><sup>2</sup> – <em>k</em> = 4, 3<em>k</em><sup>2</sup> – <em>k</em> – 4, 4<em>k</em><sup>2</sup> = 3 – <em>k</em></p>
<p>one correct value for <em>k</em> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <em>k</em> = –1, <em>k</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{3}"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span>, <em>k</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{4}"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </math></span></p>
<p>substituting <strong>their</strong> value(s) of <em>k</em> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right) = \frac{3}{4}\left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {\frac{4}{3}} \right) = p + 1"> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {\frac{4}{3}} \right)p = 8"> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mi>p</mi> <mo>=</mo> <mn>8</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 1} \right)\left( {\begin{array}{*{20}{c}} 3 \\ {2p} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {p + 1} \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><em>p</em> = –4, <em>p</em> = 3 <em><strong>A1A1 N4</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong> (working with angles and cosine formula)</p>
<p>recognizing angle between parallel vectors is 0 and/or 180° <em><strong>M1</strong></em></p>
<p><em>eg</em> cos <em>θ</em> = ±1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \bullet b = \left| a \right|\left| b \right|"> <mi>a</mi> <mo>∙</mo> <mi>b</mi> <mo>=</mo> <mrow> <mo>|</mo> <mi>a</mi> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mi>b</mi> <mo>|</mo> </mrow> </math></span></p>
<p>correct substitution of scalar product and magnitudes into equation <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\left( {p + 1} \right) + 2p\left( 8 \right)}}{{\sqrt {{3^2} + {{\left( {2p} \right)}^2}} \sqrt {{{\left( {p + 1} \right)}^2} + {8^2}} }} = \pm 1"> <mfrac> <mrow> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>8</mn> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo>=</mo> <mo>±</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="19p + 3 = \sqrt {4{p^2} + 9} \sqrt {{p^2} + 2p + 65} "> <mn>19</mn> <mi>p</mi> <mo>+</mo> <mn>3</mn> <mo>=</mo> <msqrt> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>9</mn> </msqrt> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>65</mn> </msqrt> </math></span></p>
<p>correct working (must include both ± ) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {p + 1} \right) + 2p\left( 8 \right) = \pm \sqrt {{3^2} + {{\left( {2p} \right)}^2}} \sqrt {{{\left( {p + 1} \right)}^2} + {8^2}} "> <mn>3</mn> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>±</mo> <msqrt> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>8</mn> <mn>2</mn> </msup> </mrow> </msqrt> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="19p + 3 = \pm \sqrt {4{p^2} + 9} \sqrt {{p^2} + 2p + 65} "> <mn>19</mn> <mi>p</mi> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mo>±</mo> <msqrt> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>9</mn> </msqrt> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>65</mn> </msqrt> </math></span></p>
<p>correct quartic equation <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="361\,{p^2} + 114p + 9 = 4{p^4} + 8{p^3} + 269{p^2} + 18p + 585"> <mn>361</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>114</mn> <mi>p</mi> <mo>+</mo> <mn>9</mn> <mo>=</mo> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mrow> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>269</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>18</mn> <mi>p</mi> <mo>+</mo> <mn>585</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{p^4} + 8{p^3} - 92{p^2} - 96p + 576 = 0"> <mn>4</mn> <mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>8</mn> <mrow> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>92</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>96</mn> <mi>p</mi> <mo>+</mo> <mn>576</mn> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p^4} + 2{p^3} - 23{p^2} - 24p + 144 = 0"> <mrow> <msup> <mi>p</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mrow> <msup> <mi>p</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>23</mn> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>24</mn> <mi>p</mi> <mo>+</mo> <mn>144</mn> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {p + 4} \right)^2}{\left( {p - 3} \right)^2} = 0"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>+</mo> <mn>4</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><em>p</em> = –4, <em>p</em> = 3 <em><strong>A2 N4</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the graph of the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 2\,{\text{sin}}\,x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi ">
<mn>2</mn>
<mi>π<!-- π --></mi>
</math></span> . The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> intersects the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1">
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span> exactly twice, at point A and point B. This is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question">
<p>Consider the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 2\,{\text{sin}}\,px"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mi>x</mi> </math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> > 0.</p>
<p>Find the greatest value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> such that the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> does not intersect the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>recognizing period of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is larger than the period of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> <em><strong> (M1)</strong></em></p>
<p><em>eg</em> sketch of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> with larger period (may be seen on diagram), A at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2\pi "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> </math></span>,</p>
<p> image of A when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 2\pi "> <mi>x</mi> <mo>></mo> <mn>2</mn> <mi>π</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{7\pi }}{6} \to 2\pi "> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo stretchy="false">→</mo> <mn>2</mn> <mi>π</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{sin}}\,\left( {2\pi p} \right) = - 1"> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{7\pi }}{6} \times k = 2\pi "> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo>×</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{7\pi }}{6} \cdot \frac{1}{p} = 2\pi "> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo>⋅</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mn>2</mn> <mi>π</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi p = \frac{{7\pi }}{6}"> <mn>2</mn> <mi>π</mi> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{12}}{7}"> <mfrac> <mrow> <mn>12</mn> </mrow> <mn>7</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{7}{12}"> <mi>p</mi> <mo>=</mo> <mfrac> <mn>7</mn> <mn>12</mn> </mfrac> </math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{accept}}\,\,p < \frac{7}{12}\,\,{\text{or}}\,\,p \leqslant \frac{7}{12}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>accept</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo><</mo> <mfrac> <mn>7</mn> <mn>12</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>or</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>⩽</mo> <mfrac> <mn>7</mn> <mn>12</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A buoy is floating in the sea and can be seen from the top of a vertical cliff. A boat is travelling from the base of the cliff directly towards the buoy.</p>
<p>The top of the cliff is 142 m above sea level. Currently the boat is 100 metres from the buoy and the angle of depression from the top of the cliff to the boat is 64°.</p>
<p><img 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"></p>
</div>
<div class="question">
<p>Draw and label the angle of depression on the diagram.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><img 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"> <strong><em>(A1)</em><em> </em></strong><strong><em>(C1)</em></strong></p>
<p><strong>Note:</strong> The horizontal line must be shown and the angle of depression must be labelled. Accept a numerical or descriptive label.</p>
<p><em><strong>[1 mark]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AC</mtext><mo>=</mo><mn>15</mn><mo> </mo><mtext>cm</mtext><mo>,</mo><mo> </mo><mtext>BC</mtext><mo>=</mo><mn>10</mn><mo> </mo><mtext>cm</mtext></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mover><mtext>B</mtext><mo>^</mo></mover><mtext>C</mtext><mo>=</mo><mi>θ</mi></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mtext>C</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>B</mtext><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mover><mtext>B</mtext><mo>^</mo></mover><mtext>C</mtext></math> is acute, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mfenced><mrow><mn>2</mn><mo>×</mo><mtext>C</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>B</mtext></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1 – (sine rule)</strong></p>
<p>evidence of choosing sine rule <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>sin</mi><mo> </mo><mover><mi>A</mi><mo>^</mo></mover></mrow><mi>a</mi></mfrac><mo>=</mo><mfrac><mrow><mi>sin</mi><mo> </mo><mover><mi>B</mi><mo>^</mo></mover></mrow><mi>b</mi></mfrac></math></p>
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mstyle displaystyle="true"><mfrac bevelled="true"><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac></mstyle><mn>10</mn></mfrac><mo>=</mo><mfrac><mrow><mi>sin</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mi>θ</mi></mstyle></mrow><mn>15</mn></mfrac><mo> </mo><mo>,</mo><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>30</mn></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mi>sin</mi><mo> </mo><mi>θ</mi></mstyle><mn>15</mn></mfrac><mo> </mo><mo>,</mo><mo> </mo><mfrac><mstyle displaystyle="true"><msqrt><mn>3</mn></msqrt></mstyle><mstyle displaystyle="true"><mn>30</mn></mstyle></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mi>sin</mi><mo> </mo><mtext>B</mtext></mstyle><mn>15</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2 – (perpendicular from vertex <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">C</mtext></math>)</strong></p>
<p>valid approach to find perpendicular length (may be seen on diagram) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <img src="data:image/png;base64,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">, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>h</mi><mn>15</mn></mfrac><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac></math></p>
<p>correct perpendicular length <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msqrt><mn>3</mn></msqrt></mrow><mn>3</mn></mfrac><mo> </mo><mo>,</mo><mo> </mo><mn>5</mn><msqrt><mn>3</mn></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <em><strong>A</strong></em> mark if candidate goes on to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math>, as this demonstrates a lack of understanding.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute into double-angle formula for cosine <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>2</mn><msup><mfenced><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>,</mo><mo> </mo><mo> </mo><mn>2</mn><msup><mfenced><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo> </mo><msup><mfenced><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>,</mo><mo> </mo><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mn>2</mn><mi>θ</mi></mrow></mfenced><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><msup><mfenced><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>,</mo><mo> </mo><mo> </mo><mn>1</mn><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfenced><mfrac><msqrt><mn>3</mn></msqrt><mn>3</mn></mfrac></mfenced></math></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>2</mn><mo>×</mo><mfrac><mn>3</mn><mn>9</mn></mfrac><mo>,</mo><mo> </mo><mo> </mo><mn>2</mn><mo>×</mo><mfrac><mn>6</mn><mn>9</mn></mfrac><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo> </mo><mfrac><mn>6</mn><mn>9</mn></mfrac><mo>-</mo><mfrac><mn>3</mn><mn>9</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mn>2</mn><mo>×</mo><mtext>C</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>B</mtext></mrow></mfenced><mo>=</mo><mfrac><mn>3</mn><mn>9</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The magnitudes of two vectors, <em><strong>u</strong></em> and <em><strong>v</strong></em>, are 4 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 3 "> <msqrt> <mn>3</mn> </msqrt> </math></span> respectively. The angle between <em><strong>u</strong></em> and <em><strong>v</strong></em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{6}"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span>.</p>
<p>Let <em><strong>w</strong></em> = <em><strong>u</strong></em> − <em><strong>v</strong></em>. Find the magnitude of <em><strong>w</strong></em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1 (cosine rule)</strong></p>
<p>diagram including <em><strong>u</strong></em>, <em><strong>v</strong></em> and included angle of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{6}"> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> <img src="data:image/png;base64,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"></p>
<p>sketch of triangle with <em><strong>w</strong> </em>(does not need to be to scale) <em><strong> (A1)</strong></em></p>
<p><em>eg</em> <img src="data:image/png;base64,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"></p>
<p>choosing cosine rule <em><strong>(M1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} + {b^2} - 2ab\,{\text{cos}}\,C"> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>C</mi> </math></span></p>
<p>correct substitution <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^2} + {\left( {\sqrt 3 } \right)^2} - 2\left( 4 \right)\left( {\sqrt 3 } \right){\text{cos}}\frac{\pi }{6}"> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\frac{\pi }{6} = \frac{{\sqrt 3 }}{2}"> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> 16 + 3 − 12</p>
<p>| <em><strong>w </strong></em>| = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 7 "> <msqrt> <mn>7</mn> </msqrt> </math></span> <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2 (scalar product)</strong></p>
<p>valid approach, in terms of u and v (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em> | <em><strong>w </strong></em>|<sup>2</sup> = (<em><strong>u</strong></em> − <em><strong>v</strong></em>)•(<em><strong>u</strong></em> − <em><strong>v</strong></em>), | <em><strong>w </strong></em>|<sup>2</sup> = <em><strong>u</strong></em>•<em><strong>u </strong></em>− 2<em><strong>u</strong></em>•<strong><em>v </em></strong>+ <strong><em>v</em></strong>•<em><strong>v</strong></em>, | <em><strong>w </strong></em>|<sup>2 </sup>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{u_1} - {v_1}} \right)^2} + {\left( {{u_2}\; - \;{v_2}} \right)^2}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>−</mo> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mspace width="thickmathspace"></mspace> <mo>−</mo> <mspace width="thickmathspace"></mspace> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span>,</p>
<p>| <em><strong>w </strong></em>| = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( {{u_1} - {v_1}} \right)}^2} + {{\left( {{u_2}\; - \;{v_2}} \right)}^2} + {{\left( {{u_3}\; - \;{v_3}} \right)}^2}} "> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>−</mo> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mspace width="thickmathspace"></mspace> <mo>−</mo> <mspace width="thickmathspace"></mspace> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> </mrow> <mspace width="thickmathspace"></mspace> <mo>−</mo> <mspace width="thickmathspace"></mspace> <mrow> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </math></span></p>
<p>correct value for <em><strong>u</strong></em>•<em><strong>u</strong></em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> | <em><strong>u</strong><strong> </strong></em>|<sup>2</sup> = 16, <em><strong>u</strong></em>•<em><strong>u</strong></em> = 16, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}^2 + {u_2}^2 = 16"> <msup> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>16</mn> </math></span></p>
<p>correct value for <strong><em>v</em></strong>•<em><strong>v</strong></em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> | <em><strong>v</strong><strong> </strong></em>|<sup>2</sup> = 16, <strong><em>v</em></strong>•<em><strong>v</strong></em> = 3, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{v_1}^2 + {v_2}^2 + {v_3}^2 = 3"> <msup> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>=</mo> <mn>3</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {\frac{\pi }{6}} \right) = \frac{{\sqrt 3 }}{2}"> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em><strong>u</strong></em>•<strong><em>v</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4 \times \sqrt 3 \times \frac{{\sqrt 3 }}{2}"> <mo>=</mo> <mn>4</mn> <mo>×</mo> <msqrt> <mn>3</mn> </msqrt> <mo>×</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> (= 6) (seen anywhere) <em><strong>A1</strong></em></p>
<p>correct substitution into <em><strong>u</strong></em>•<em><strong>u </strong></em>− 2<em><strong>u</strong></em>•<strong><em>v </em></strong>+ <strong><em>v</em></strong>•<em><strong>v</strong></em> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}^2 + {u_2}^2 + {v_1}^2 + {v_2}^2 - 2\left( {{u_1}{v_1} + {u_2}{v_2}} \right)"> <msup> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> <mn>2</mn> </msup> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> (2 or 3 dimensions) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>16 − 2(6) + 3 (= 7)</p>
<p>| <em><strong>w </strong></em>| = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 7 "> <msqrt> <mn>7</mn> </msqrt> </math></span> <em><strong>A1 N2</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Julio is making a wooden pencil case in the shape of a large pencil. The pencil case consists of a cylinder attached to a cone, as shown.</p>
<p>The cylinder has a radius of <em>r</em> cm and a height of 12 cm.</p>
<p>The cone has a base radius of <em>r</em> cm and a height of 10 cm.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the slant height of the cone <strong>in terms of <em>r</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The total external surface area of the pencil case rounded to 3 significant figures is 570 cm<sup>2</sup>.</p>
<p>Using your graphic display calculator, calculate the value of <em>r</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(slant height<sup>2</sup> =) 10<sup>2</sup> + <em>r </em><sup>2</sup> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> For correct substitution of 10 and <em>r</em> into Pythagoras’ Theorem.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{10}^2} + {r^2}} ">
<msqrt>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi {r^2} + 2\pi r \times 12 + \pi r\sqrt {100 + {r^2}} = 570">
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>r</mi>
<mo>×</mo>
<mn>12</mn>
<mo>+</mo>
<mi>π</mi>
<mi>r</mi>
<msqrt>
<mn>100</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mn>570</mn>
</math></span> <em><strong>(M1)(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution in curved surface area of cylinder and area of the base, <em><strong>(M1)</strong></em> for their correct substitution in curved surface area of cone, <em><strong>(M1)</strong></em> for adding their 3 surface areas and equating to 570. Follow through their part (a).</p>
<p>= 4.58 (4.58358...) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C4)</strong></em></p>
<p><strong>Note:</strong> Last line must be seen to award final <em><strong>(A1)</strong></em>. Follow through from part (a).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4\,{\text{cos}}\left( {\frac{x}{2}} \right) + 1"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> </math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x \leqslant 6\pi "> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>6</mn> <mi>π</mi> </math></span>. Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > 2\sqrt 2 + 1"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <mn>1</mn> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 – FINDING INTERVALS FOR <em>x</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4\,{\text{cos}}\left( {\frac{x}{2}} \right) + 1 > 2\sqrt 2 + 1"> <mn>4</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>></mo> <mn>2</mn> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <mn>1</mn> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4\,{\text{cos}}\left( {\frac{x}{2}} \right) = 2\sqrt 2 "><mn>4</mn><mspace width="thinmathspace"></mspace><mtext>cos</mtext><mfenced><mfrac><mi>x</mi><mn>2</mn></mfrac></mfenced><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {\frac{x}{2}} \right) > \frac{{\sqrt 2 }}{2}"><mtext>cos</mtext><mfenced><mfrac><mi>x</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac></math></span></p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^{ - 1}}\frac{{\sqrt 2 }}{2} = \frac{\pi }{4}"> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p>one additional correct value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{2}"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </math></span> (ignoring domain and equation/inequalities) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4}{\text{, }}315^\circ {\text{, }}\frac{{9\pi }}{4}{\text{, }} - 45^\circ {\text{, }}\frac{{15\pi }}{4}"> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <msup> <mn>315</mn> <mo>∘</mo> </msup> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>15</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p>three correct values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{2}{\text{, }}\frac{{7\pi }}{2}{\text{, }}\frac{{9\pi }}{2}"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span></p>
<p>valid approach to find intervals <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <img src="data:image/png;base64,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"></p>
<p>correct intervals (must be in radians) <em><strong>A1</strong></em><em><strong>A1 N2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{7\pi }}{2} < x < \frac{{9\pi }}{2}"> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> <mo><</mo> <mi>x</mi> <mo><</mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> </p>
<p><strong>Note:</strong> If working shown, award <em><strong>A1A0</strong></em> if inclusion/exclusion of endpoints is incorrect. If no working shown award <em><strong>N1</strong></em>.<br>If working shown, award <em><strong>A1A0</strong></em> if both correct intervals are given, <strong>and</strong> additional intervals are given. If no working shown award <em><strong>N1</strong></em>.<br>Award <em><strong>A0A0</strong></em> if inclusion/exclusion of endpoints are incorrect <strong>and</strong> additional intervals are given.</p>
<p> </p>
<p><strong>METHOD 2 – FINDING INTERVALS FOR <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{2}"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </math></span></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4\,{\text{cos}}\left( {\frac{x}{2}} \right) + 1 > 2\sqrt 2 + 1"><mn>4</mn><mspace width="thinmathspace"></mspace><mtext>cos</mtext><mfenced><mfrac><mi>x</mi><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mo>></mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn></math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4\,{\text{cos}}\left( {\frac{x}{2}} \right) = 2\sqrt 2 "><mn>4</mn><mspace width="thinmathspace"></mspace><mtext>cos</mtext><mfenced><mfrac><mi>x</mi><mn>2</mn></mfrac></mfenced><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {\frac{x}{2}} \right) > \frac{{\sqrt 2 }}{2}"><mtext>cos</mtext><mfenced><mfrac><mi>x</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mfrac><msqrt><mn>2</mn></msqrt><mn>2</mn></mfrac></math></span></p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{s}}^{ - 1}}\frac{{\sqrt 2 }}{2} = \frac{\pi }{4}"> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mfrac> <mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p>one additional correct value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{2}"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </math></span> (ignoring domain and equation/inequalities) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4}{\text{, }}315^\circ {\text{, }}\frac{{9\pi }}{4}{\text{, }} - 45^\circ {\text{, }}\frac{{15\pi }}{4}"> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <msup> <mn>315</mn> <mo>∘</mo> </msup> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <msup> <mn>45</mn> <mo>∘</mo> </msup> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>15</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p>three correct values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{2}"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4}{\text{, }}\frac{{9\pi }}{4}"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p>valid approach to find intervals <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAACWCAYAAADzL0UhAAAbyklEQVR4Ae1dCVCVV5ZOVU9NzWRmamo6NdPT07E76VhTFZPppDvdnfTE9EwymfQY2xUUTUQjm+LK4oJsiisucYm4AIpLIu4bmnYX0SiK4q64I6iogCKLLAKeqe/Cwwf1P+B//P97/72cV/XXe+9f7n/uOee7595zzz33JeIPc4A5oCwHXlK2Zlwx5gBzgBjgrATMAYU5wABXWLhcNeZAI4A/f/6c9B61tbW0IimJYmNj6dGjR7qf1/s+vl+/jJhnavFMT7PVJoBXVFTQYB8fChs/nry8vOjtt9+iq1evMsidaCgZhGqB0Ex5ugzgCxYsoA0bNtCpU6coJDSEcnNyaPSoUQSrbmYFuWwGQ3vWAZcB3AbkzMxMAXAw3XauPQuA684NkJk64DKA2yphD3DbOf5mJWcdMEcHGOA8BuYhksI6wABXWLhsFc2xijLxlQHOAGcLrrAOMMAVFq5MloZpNae3wQBngLMFV1gHGOAKC5etojlWUSa+MsAZ4GzBFdYBBrjCwpXJ0jCt5vQ2GOAMcLbgCusAA1xh4bJVNMcqysRXBjgDnC24wjrAAFdYuDJZGqbVnN4GA5wBzhZcYR1ggCssXLaK5lhFmfjKAGeAswVXWAcY4AoLVyZLw7Sa09tggDPA2YIrrAMMcIWFy1bRHKsoE18Z4AxwtuAK6wADXGHhymRpmFZzehsMcAY4W3CFdYABrrBw2SqaYxVl4qvLAb5u3Tr6+OOPaeXKlbRz507ewogbmIYeREFBAe3YsVNsb5WYmEjLli2jdWvX0onjxwk748gELKvQ6nKATwgLo44dO1K3bt3E0aNHD5oyZQqdzsykqqpKFmI7BPz9vDxavXo19e3bt0EvbPph+x44cCCtWbOG7uTm8oYZOnTE5QA/fPgw+fgMplMnT9LmzZspPDxcCLV79+40JCCAcnJyGOQ6BGgVS+EMHaWlpTRlymTy9PQUOjB69GhK2bGDTmZk0NUrV+jKlSxKP3ZMWPLBgweLe3p7eNDE6Gh6+vQp60kr9MTlANfa2eTmzZv03Xff0RdffCFa8cNpaSy8VgjPGVBZ5Zldu3bRoEGDBGgjIyPpcNphqn72zKHcq6qq6OjRozRp0iTxzFdffUUpKSkO77dKPd1NhyUAbmMCxmADBgwQAly/fj3V1NSwABUDOmS6cGGckDGGZ2mHD+uW8f79+8Xz6L4vW76c9aQZHbEUwAH0/Px8QlcNwps6ZQqVl5frVgBbg8Hf1vIiFxUVUWRkhJDt+PHj6datW07L9nZ2Nk2cOFGUNWnSRHr8+LHTZamsJ5YDOJgtxmaTJwvhwQFXUlLCwmumlZZBQYuLiykqMpLga5k5axbhf1vpLisro5mxsUJPQkNDCQ1IW8tU7XlLAhxMfvbsmfCuw5LDouO/asxvL/UBEAMCAgQQ586da2iXGl3+GTNmiLIDAwOprKyU9cTOGFgW4FD+6poamjp1qhDevHnzCI6W9gIKVeoJmcXG1gEwLm6RKQ019CQ+Pl7oybRp03hYJwvAoeQYg0+qH2shOEYVxW8v9fjmm28E8CZPnmxqA40ens3DPn36dGEc2guPm6unpS24jXB08Xx9fQleVwa5tRxnNhlpfR88eFCMuRHc5IohVmVlJYWFjRcNytatW9kYPH+uB9/0kv3dWgJtzTmtefDWPHcrO5u8vb2F8G7cuMHCs+uGtYZ/rr4n48QJ6tWrFw0ZMtSlwUvZt2/TqFGjRMOSkXGi3euJPWZb+u1WgENBD6WmCsGNGDGCYNVdrbT8vtb1HspKy8jP11fI6tLFiy6XEwKnevbsST4+Pu0+4q0lUNtfNwTgu3ftoq5du9K2bdt0H2uSkykgYAj97re/I38/f93PO/POjRs3ihhoZ56V6ZkNGzaIeqJr2xa68fzAgYOEjIKDg9tUVlvoCAoKEjSMHTO2EQ2bNm0ypJ56aUMcPXRJ73M2/cMireTkZCrIz9fVYNoDuKXfhgA8MiKi0WIT22ICPd8d3+hI//qTn9CHH34ouux6nuV76xb5mMWHP3zwgZBNpzffdLtsOnXqRD/v0IG6dOnidlqM4ndCYqK1AZ6Wlkbde3Sn9PRjuo/t27cTYphTUw/Sn/70J+rcuTNt3LRRdzl63r1v3z6xrFXPMzLeu2fPbmHZjh3TLxdbfWFhIBP00A4dSjVVLrZ3NvcNPfnss88IcevH6vUN9QOdzT1nxjVY4kOHDul+L8Jyofc4EHuPIDA9Q8WWrLb9dUMsuLNOtqaV2rtnr2iVp06ZSrUWdzg1pV3V/wg/xUzH8ePHdSmhmfxA1xg0HTt21DI0mVnfpmXbA7il35YCeHV1NYWFhQmQo5VrWjH+3zqHmFF8gnVBNxShxbW1tZaRR15eHvXu3Zv8/f3bZbx6S6C2v24pgEMxMbc60NtbrEDjlWeuBbR9wwDHT6+ePWnYsGGEuWj7a1b4jWEWGp9FixZZjjaz+WMP4JZ+Ww7gYM6J+vnW6Ohonjpzw1CluKRE9KQw533BDVNirQFIVWUlIYQVlhzBN615RpV7WgK1/XVLAhyC2LJ5s2ihk5KS2pXw3K2E6IrPmjVL8H7Hjh2W5j3iJtDDwGq2K1lZlqbVSLnaA7il35YFONL3eHl5UZ8+fdrlOMtIhdBT1oMHD0S02uCvvnJJKKoe2rTuhfVGV33OnDkMcA20WxbgEOb58+dF9FJUVCTBAaclYD5n3Dj9WVUVjRw5UlhEK3nNm5MxknraHLPtJd2TBo4dnrI0wCHYhQsXiimRixdcHx7ZnGKpeA0NKqxhfHyCVI1pVlYW9fH0pJEjRrSL6VWHaNa4YHmA37h+XSjd0KFDpegyygz8xYsXi8a0sLBQKoCD5wn1a8cPpqZKR7tendHAscNTlgc4Kr98+XIB8rVr1yovPL3CNup+jL379+tHa5OTpeRxQX4BIQ1z//79KScnV8o6tFaWDtGscUEKgD969Eg42/z9/ExNMNBaBqt4H6LDkP1W5ghCLHrCEAM9ERVlZKuTBo4dnpIC4KjY+vXrhPBgzW0V5W9jHGyF9Q0o/B0y8xQzL8jD7+HhQUjXLXNdmqPdIZo1LkgDcERThY4ZI3bMuHfvnrLCa06wZlwTCQ5jY4Xn/OrVq9LzNTU1VRgCJG1UNRJSA8cOT0kDcCg3tkhCFww7ppih7O2xzOvXrlHPHj0pJiZGGZ4iUAfJIW4qmiXIIZo1LkgFcFjx4SNGCCv+8OFDZRTSnQ2LLT2xLPPereEV9siDIZgxfYaSOqKBY4enpAI4hHvq1CkxlYNNDlsjbL7H8Tj9/PlzAgjIjiqzc01LxrYdUrCuQeu6zOccolnjgnQAx7gqbNw4wuZ2MgvJCrRPmzpVLNa4e/eucry8cydXrIYbN24c1VhoqasRctfAscNT0gEcDMKOlBhjXb58WTnFNEIBWlPGuXPnRE8oJmaSks4oGAKsRkRiiDNnziqlJw7RrHFBSoBDgbFvFay4qp7S1oDU2XsQ149VWGgkb9++rZTy2/MkOztb9FBQV1fkcLd/t5m/NXDs8JS0AEc+KywTxKbyZjJTxbLTjx0TY++EBLlizp2RBbashsPthIVSTjlTD/tnHKJZ44K0AL9y5Qp1695dpBPilWaOHWn2ioHf6PGg54PG8VI7GOJgqgw9FWyBpIqeaODY4SlpAY7EBDNn1iUmQFbNporM/7VBv78+1dH8+fMtlWfNLHlBT5YsWSKs+M7vv1dCTxyiWeOCtACHQsCK9+zRQ1hxsxREtXJHDB9Offv0aVepsJ48eUJ9+/YlPz8/JcbiGjh2eEpqgKO7OSY0VHhKb966pUTrbGaDcr1+6S26q2a+x2plw4rbdik9e1Z+j7pDNGtckBrgUKTc3FwxxoJlUmWMZQZA4EUeMmSI8Coj7bAZ77BymYh8RIJGbJhgxSyxeningWOHp6QHOBjz9ddfCyt+ToHWWY+g9dy7d88eMQ5dvnxZuwO3jU/IJwCP+r49e6TmgUM0a1xQAuDoemKhP+bGMd6yCZS/6xxtRUVFNHz4cLGU8uGDB+2WP8XFxWK76qFDhhByDMiqHxo4dnhKCYBDULt37xat88oVK6UVnFkKN3fuXMEbLKU06x2ylJt+/LjgBbKwYmwuC932dDpEs8YFZQCOxf4+gwfToEGDlPCU2gu0Lb+rqqrE6ruQ4GCO+quPAxg9erRIDS1rUggNHDs8pQzAAQLst4wxFuduezEHvnr1asGTvXv3Smmt2tK4OXp2T70/YsWKFVLyxCGaNS4oBXBMm2FDOiwwkDEzqCOFdPb8jRs3BC8Qi80zDC8aPejJsOHDRcOHWRhn+euu5zRw7PCUUgAHw1O2bxeCW79unXSCM1JhML7EclD0aBB7bmTZKpR19lzdWvhZM2dKxxuHaNa4oBzAHz9+LDzqmO8sKmq/HnX0YBC9hai1kuJi6ZTY7EYEcQHo7fXp24cwy2D2+4wsXwPHDk8pB3Aw0jbGkj1LaFuUIjY2VlhvxJ63pRyVn/3hhx8Ej5C2SqZ6OkSzxgUlAY7xZkBAAHkPGEAVFtzb2mxlun//vvCcT5gwgT3nzWy/jDRViJ3ANskyxU9o4NjhKSUBDgBt2rRJtM7wrJsNKKuVv2rVKlH3I0eOtLu665VF2qFDgldpaWnS8MohmjUuKAtwxBvDe4zWOScnRxrh6VXQpvdfu3ZN1HnUqFG8C0wz1tvGN4zFRwcFUUhIqDQ6ooFjh6eUBTgEeODAAdE6r0hKkjZqyaaIrfmG53z27NkiEQbGl615hu95TmfPnhMJMHIsmr4qPz+f8uw2+3CIZo0LlgE4lLPo8WNDlRLRbV5eXtSvX7924UkuLCgQK6bgHUYEm5HgtVJcgdGzApgXh88mLi7OUJ4Zxf/MzEzq/0V/gj6jTD0fywAcXepJEycazuBDaWnCiiMe2yiGW7Uc2yYGZsScBwYGWoJ/cIwtXbrUcFpWrVophjbYI92K8k09mEpdunQhOFD1fBoBHIvicWDdsJ7Dw9OT3v31u7qesS/f3z9AbP2KHGu+vr4UEKDv/fZlaf3u1OkteqPjG+Tn7+c0jVrlWukcdgZ97fXX6J133jG0jgH+AUImn3/+Ofn4+JB/QICh5evhIawsaMC6bmRn0fNsS/di08LXXnuN3n//fUPLbem9eq7Dn7Ro0WI9+KZGAIcVxVFRUaHrSE9Pp9FBo3U9Y/+O8vJyEZQSFRkpuiH214z4vWvXLuratSslJiQ6TaMRdJhZBrbMRR0x9jb0PeXlQiZQRHQRK8rLjS1fj66Vl1PZ06eiKw2dMbSeFRWEBTkwMtgIwuiy21reyZMn6d1336U7ubnOA9zZrgnGCCGhIW3q2mDu+uyZM20qwxH9lZUVFBwUTL09POjSpUumvMPRu11xHtlRe/XuTUFBQaZlKzmenm4JvqGLjlx8ZvAVhgqhvejFWil2/8KFC6KHi2/UW8+nkQV3lmlGANzZd7f2udOZmUJ406ZNU8qjDgdRRESEWFRywaLjx9bKyAr3ITYdILdKgwae3Lp1i65kvWjUGOAa85+Y7/T19aOevXqJPG5WUCYjaIA1w+q5KdhAUNIEBkbwwagybmdnC4CHh4eb0kswgk4GuAbAwViMTwGGqKgoS3XBnBU6Gi34LZDYP+NkhmUV0tn6ueM5NJJhYWEC5JmZpyzJUwa4A4BDeFiAgl09rmRlWVJ4epR63969QhETEhOlr4ueept97/0HD4S/ZlhgoCVj+RngDgAOxbh29aoARXBwsNR7YsPZBKeah4cHJ7doRt7ONgbxS5cKPbHirjkM8BYEHhwUJIS3f/9+aS2fLckkUkY7q8T83IssL015gV1XMd+OuIxii2XqZYC3APBHhYXUp08fMfVgdEhnU0Ux4z/mgBF+i6AP/DbjHbnZ1ynjxIlGR/HTSlPe1Rr6a2tqqKSkhMorXEeDbUVifHy8pRyYDPAWAA6F2rlzp7DiCxYssJTwWqPsKSkpgvajR4+aArhnJffpy949RBy/Vz8v8vLqRf/WoRM9KXMduOz5sG/9cpoXt5JKKmooJTmevl7mmiXACOzBSjNkxbHSlkcM8FYAHAqEHTa7d+tGRyRaC1xaUkIDBw6kuIULTQE3+PLwxjUqqXqRM/zk999SyNRvTHufPZib/q6tKaXOr/+Y7j4uE++vra2mnh++5zJa7t69J/iN0Fj0IJrS547/UgL8wg8HKT5+aaMj8/RpUxmK3S169exFQ4cONXz1lVmC37Jlixhe2FYWmfUe+3Jnj/elI9cKhSwObFhCP375b+lHf/WjRkfqNZPy35Xdop++/Dd07GZBgy58+b/v0Z3SctoUN506dvgZvfqzxseR68bSsm1bXSLPrVu3NtBgzx9X/5YO4GcObKD33v+QfvHaL+mjP35EHf75n+g/P+pMq779znSG2nKpw5q7WlB634cUv/CaY2UcvOh6n3fm/traGurrMUC8q7LgGsVET6fI4KE0eMwMipsdQ3/89HOCx/lekTndd0xt9vn4beo3bLKgoeZZJXV+65d0/855GhY4mvr1+Jy8A0IpIsiHPvivj2lSzCS6UWCsX6KstJQC6jduzDIpTFaPbKQD+PmLdVv/RkTXJb8L/MKDntXUuESB4aSCp9TT05OQkVUPo119L7bbwRz+5cuXXUZn7tG1FB6b0Oh936+eR7svFtLDrHQaGzm10TUzeJJ35Sx5/t8n1K23J8Uv/Jo6/scfGt65+OsYyrj5hHKO76KY6ealQMZ0GXgfHRVFNdXVDe83o74tlSkdwFGhmidXKWLiHME47279XMpAW16u774zv8fQkvAcXcdCAyhYRHi4S4MvRg3qTtfuN067HDPkS7pRXEnX03dQ5JQZVFtTTc+qX4zZHdXBiPPr54bS+GkLGvQj0rcf3SmtosyUZJoWO6vhvBHvsi8DMf+RERGWmBuXEuBLp42h2fGbhIA+/fWnVFpWRBf2ppgmsMbCq6UF8xeIfOquHNva09Dc7/t5eWJnUGSJvXPnjkt4Anpqn96nLr38qcYuxj3vQhq93vFNMfNwM2MXffJnD0qKm0u3ip+ZTteJA3+h37z/CeXUNzg3zxygX/zqM6qqqaHTKWuph99Iyti/kc7eLzWFFizVxLr7L7/8slEKpeZkZ8Y1+QBeVUw/f/llOnSqbsVMYPcP6O9f+SkdrXfsmMGkpmXC4ebR24O2bN7ssvFtUxoc/U9MSBCWY/v27aYorqP3Hti4jObXN7q2e9bMiSTvodGCjmclD+mT3/+K+vqMNJWurPOnKXHOTPINmUQl5S9SUc0M7k++wTHi3eX5V+jf/+UV6tzd21RaGvL8uXFfM/kA/ryWykpKG+ajqysrqbSsLv+UTbFc8Y1UQOgGZ2RYZ+HGkSM/CJqQIVbGoBxXyM2V78ACH4Q5Q0/ctaGjhAB3jUe4JUVAJg+sNkOMtxXAhOw6/n5+Imrt5s2bplqmlnjD11/oKNJw9+/fX3TX3ZGMkgHehikkTEFhwb+7wxPh1Fm0aJGg5dvVqxncbZCpGY1TQv2waeasWS51eqIuej7tJqNLa4WMLtjkyZNFFwze9dY+Z/R9a9asEeBGplTQZHT5XN4Li+wML9C7mlq/e+v2bdtcKh+pAH7u3DkRnIAABa3jkBtAht1BbCuJ3LF/NLqA8NZibh7DBmcU0Nln4hPiNeWgJZum54ze3wspjNcmJzt1OFt/Pc+hvvCoI3TYlbMbUgEc2SbBKEcHruthulH3YiEHHCnYnA7dZaPKbamc4uJiscoNKXKx40ZL96t8HT0oDE+cOVy1OOT06dPCb4MxudENnCPZSgVwR5Vw93mAGl51jMdjYmJEGl2zaUJjFhlZF0yxd8+edg1us3ltZPm2ZaXYzdUVcRQMcIMcMvCkjx0zRoAcY2Iz47/RoCCdFBqUWTNnNQouMVIZuay2jb21+AfZwQhAdvOwTsAuMEjr/raeY4AbBHAI4klRkYhVR3cdoaxmCA9lJiUlCQVBgwIHTluVgJ83HsjN8RSO0JCQECHDJYuXmNpASw3w8se36B//7q8pu8A9Y28tIWJ3xxEjRoixFrpjRo7JkWAfY0y0/oh1RkSdFg1WObc6diR5DR1vCRpTdyTT7v2plqAF8iksKGzo8SUtX26ontjLX1qA19ZUUvjYMLHO2EoAB3Pz8vJo0KBBAohJy43ZjhgNxbz580WZWNFm9dVst08doNdf7WAJgFcXXqdXX/kHCgmfZBmAQ0/yHz4kb29vIdMtm7eYQpu0AF+XOIfOXMq1JMBFC11YKJJDoLseHR3dJq9pQX4+jRs7VigCPPWlpeYskLBv+dvyu7b2KUWHR9HEMUPcD/DqSooKDaYAz48sB3DwuKysjMaNGyfW7iNBh9HDOikBnnMpnZYk1uXaQrYQq1lwGziQtmdyvUNl1KhRhP2s9AgQXfLUgwfF7pjols+ZPZswNWYr36rfCdPC6Oq9x5YA+NlDOynt9FUa4/3flgQ4ZIgGG151yHjhwm8o716eYTKWD+BVT2liRCRVVdfNN1sZ4DYAIvEh4tYhwPDwCVRQ8CKlkO2ept8IoEHLjmewGwnKMNMz3/T9zv4/tW89rfvLD0JB3W3BkZNtVeIKQYuVAQ5eo+G2gRyBU9u2bTNkRx3pAJ40K4p8gic0RFAB4MHjo+j8XWskudMCBqx21uUsQl5yABaplNDVTkxMJOQsv3jxIgHQ2C4JccshoaEC1Ojex8XFUXZ2tmEtuhZ9Rp1D13xGTGwDre4GeOb36ynzfF1GG6sDHDKAdx2rzrD9MvTEe+BAEeKKadd9+/YJHdG7k6l0AL93J4cyT2c2HAD496nH6VHZi7W/Rims0eXAAiNP17Jly4TgRgwfLsAOYdqAHxgYSNjVFN5y7BSpp0tvNL16y6t9kkurv13dcPTq+gl98D9/ps17DjeAXm+Zbbl/QshwAj9x/ObNDvTub39P0xcnu4UWPfXA1OeRI0dowfz5NCEsTCTw6F7fA9ywYYMu+qUDeFNGydBFb0pz0//wkGMcZuSUWtN3uOO/uy24fZ1lsOD29Db9jag3TMFW6wyFlh7gTRnB/10btNEcvw/uSaENO/bqsjjNldeWa7ID3Nm6M8ANjGRzVgj8nHUaJdVkwQBngFvCwqoGLKvUhwHOAGeAK6wDDHCFhWsVK8J0uG8IwgBngLMFV1gHGOAKC5ctp/ssp1V4zwBngLMFV1gHGOAKC9cqVoTpcF9PggHOAGcLrrAOMMAVFi5bTvdZTqvwngHOAGcLrrAOMMAVFq5VrAjT4b6eBAOcAc4WXGEdYIArLFy2nO6znFbhPQOcAc4WXGEdYIArLFyrWBGmw309CQY4A5wtuMI6wABXWLhsOd1nOa3CewY4A5wtuMI6wABXWLhWsSJMh/t6EgxwBjhbcIV1wGmA63mQ72UOMAesz4GXrE8iU8gcYA44ywEGuLOc4+eYAxJwgAEugZCYROaAsxxggDvLOX6OOSABBxjgEgiJSWQOOMuB/we5o1khfb8BKgAAAABJRU5ErkJggg=="></p>
<p>one correct interval for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{2}"> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant \frac{x}{2} < \frac{\pi }{4}{\text{, }}\frac{{7\pi }}{4} < \frac{x}{2} < \frac{{9\pi }}{4}"> <mn>0</mn> <mo>⩽</mo> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> <mo><</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo><</mo> <mfrac> <mi>x</mi> <mn>2</mn> </mfrac> <mo><</mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span></p>
<p>correct intervals (must be in radians) <em><strong>A1</strong></em><em><strong>A1 N2</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{7\pi }}{2} < x < \frac{{9\pi }}{2}"> <mfrac> <mrow> <mn>7</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> <mo><</mo> <mi>x</mi> <mo><</mo> <mfrac> <mrow> <mn>9</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> </p>
<p><strong>Note:</strong> If working shown, award <em><strong>A1A0</strong></em> if inclusion/exclusion of endpoints is incorrect. If no working shown award <em><strong>N1</strong></em>.<br>If working shown, award <em><strong>A1A0</strong></em> if both correct intervals are given, <strong>and</strong> additional intervals are given. If no working shown award <em><strong>N1</strong></em>.<br>Award <em><strong>A0A0</strong></em> if inclusion/exclusion of endpoints are incorrect <strong>and</strong> additional intervals are given.</p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The following diagram shows triangle PQR.</p>
<p><img src="images/Schermafbeelding_2017-08-11_om_09.36.55.png" alt="M17/5/MATME/SP1/ENG/TZ1/03"></p>
<p>Find PR.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1 </strong></p>
<p>evidence of choosing the sine rule <em><strong>(M1)</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{a}{{\sin A}} = \frac{b}{{\sin B}}">
<mfrac>
<mi>a</mi>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>A</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>b</mi>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>B</mi>
</mrow>
</mfrac>
</math></span></p>
<p>correct substitution <em><strong>A1</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{{\sin 30}} = \frac{{13}}{{\sin 45}},{\text{ }}\frac{{13\sin 30}}{{\sin 45}}">
<mfrac>
<mi>x</mi>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>30</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>45</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>13</mn>
<mi>sin</mi>
<mo></mo>
<mn>30</mn>
</mrow>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>45</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 30 = \frac{1}{2},{\text{ }}\sin 45 = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<mn>30</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>45</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>(A1)(A1)</strong></em></p>
<p>correct working <em><strong>A1</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times \frac{{13}}{{\frac{1}{{\sqrt 2 }}}},{\text{ }}\frac{1}{2} \times 13 \times \frac{2}{{\sqrt 2 }},{\text{ }}13 \times \frac{1}{2} \times \sqrt 2 ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mn>13</mn>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>13</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math></span></p>
<p>correct answer <em><strong>A1</strong></em> <em><strong>N3</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PR}} = \frac{{13\sqrt 2 }}{2},{\text{ }}\frac{{13}}{{\sqrt 2 }}{\text{ (cm)}}">
<mrow>
<mtext>PR</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>13</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</math></span></p>
<p><strong>METHOD 2 (using height of Δ</strong><strong>PQR</strong><strong>)</strong></p>
<p>valid approach to find height of ΔPQR <em><strong>(M1)</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 30 = \frac{x}{{13}},{\text{ }}\cos 60 = \frac{x}{{13}}">
<mi>sin</mi>
<mo></mo>
<mn>30</mn>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mn>60</mn>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 30 = \frac{1}{2}{\text{ or }}\cos 60 = \frac{1}{2}">
<mi>sin</mi>
<mo></mo>
<mn>30</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> or </mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mn>60</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{height}} = 6.5">
<mrow>
<mtext>height</mtext>
</mrow>
<mo>=</mo>
<mn>6.5</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>correct working <em><strong>A1</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 45 = \frac{{6.5}}{{{\text{PR}}}},{\text{ }}\sqrt {{{6.5}^2} + {{6.5}^2}} ">
<mi>sin</mi>
<mo></mo>
<mn>45</mn>
<mo>=</mo>
<mfrac>
<mrow>
<mn>6.5</mn>
</mrow>
<mrow>
<mrow>
<mtext>PR</mtext>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mn>6.5</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>6.5</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 45 = \frac{1}{{\sqrt 2 }},{\text{ }}\cos 45 = \frac{1}{{\sqrt 2 }},{\text{ }}\sqrt {\frac{{169 \times 2}}{4}} ">
<mi>sin</mi>
<mo></mo>
<mn>45</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mn>45</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msqrt>
<mfrac>
<mrow>
<mn>169</mn>
<mo>×</mo>
<mn>2</mn>
</mrow>
<mn>4</mn>
</mfrac>
</msqrt>
</math></span></p>
<p>correct answer <em><strong>A1</strong></em> <em><strong>N3</strong></em></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PR}} = \frac{{13\sqrt 2 }}{2},{\text{ }}\frac{{13}}{{\sqrt 2 }}{\text{ (cm)}}">
<mrow>
<mtext>PR</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>13</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>13</mn>
</mrow>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</math></span></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
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