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</div><h2>HL Paper 1</h2><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of <em>x </em>for which the vectors \(\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> {2\cos x} \\ <br> 0 <br>\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> {2\sin x} \\ <br> 1 <br>\end{array}} \right)\) are perpendicular, \(0 \leqslant x \leqslant \frac{\pi }{2}\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">perpendicular when \(\left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> {2\cos x} \\ <br> 0 <br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br> { - 1} \\ <br> {2\sin x} \\ <br> 1 <br>\end{array}} \right) = 0\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow -1 + 4\sin x\cos x = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \sin 2x = \frac{1}{2}\) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 2x = \frac{\pi }{6},\frac{{5\pi }}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x = \frac{\pi }{{12}},\frac{{5\pi }}{{12}}\) <strong> <em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept answers in degrees. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates realised that the scalar product should be used to solve this problem and many obtained the equation \(4\sin x\cos x = 1\). Candidates who failed to see that this could be written as \(\sin 2x = 0.5\) usually made no further progress. The majority of those candidates who used this double angle formula carried on to obtain the solution \(\frac{\pi }{{12}}\) but few candidates realised that \(\frac{{5\pi }}{{12}}\) was also a solution.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\arctan \left( {\frac{1}{5}} \right) + \arctan \left( {\frac{1}{8}} \right) = \arctan \left( {\frac{1}{p}} \right)\), where \(p \in {\mathbb{Z}^ + }\), find <em>p</em>.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the value of \(\arctan \left( {\frac{1}{2}} \right) + \arctan \left( {\frac{1}{5}} \right) + \arctan \left( {\frac{1}{8}} \right)\).</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt at use of \(\tan (A + B) = \frac{{\tan (A) + \tan (B)}}{{1 - \tan (A)\tan (B)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{p} = \frac{{\frac{1}{5} + \frac{1}{8}}}{{1 - \frac{1}{5} \times \frac{1}{8}}}{\text{ }}\left( { = \frac{1}{3}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p = 3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> the value of </span><em style="font-family: 'times new roman', times; font-size: medium;">p</em><span style="font-family: 'times new roman', times; font-size: medium;"> needs to be stated for the final mark.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \left( {\arctan \left( {\frac{1}{2}} \right) + \arctan \left( {\frac{1}{5}} \right) + \arctan \left( {\frac{1}{8}} \right)} \right) = \frac{{\frac{1}{2} + \frac{1}{3}}}{{1 - \frac{1}{2} \times \frac{1}{3}}} = 1\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arctan \left( {\frac{1}{2}} \right) + \arctan \left( {\frac{1}{5}} \right) + \arctan \left( {\frac{1}{8}} \right) = \frac{\pi }{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Those candidates who used the addition formula for the tangent were usually successful.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Some candidates left their answer as the tangent of an angle, rather than the angle itself.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the identity \(\cos 2\theta = 2{\cos ^2}\theta - 1\) to prove that \(\cos \frac{1}{2}x = \sqrt {\frac{{1 + \cos x}}{2}} ,{\text{ }}0 \leqslant x \leqslant \pi \).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find a similar expression for \(\sin \frac{1}{2}x,{\text{ }}0 \leqslant x \leqslant \pi \).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the value of \(\int_0^{\frac{\pi }{2}} {\left( {\sqrt {1 + \cos x} + \sqrt {1 - \cos x} } \right){\text{d}}x} \).</span></p>
<div class="marks">[4]</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos x = 2{\cos ^2}\frac{1}{2}x - 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \frac{1}{2}x = \pm \sqrt {\frac{{1 + \cos x}}{2}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">positive as \(0 \leqslant x \leqslant \pi \) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \frac{1}{2}x = \sqrt {\frac{{1 + \cos x}}{2}} \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos 2\theta = 1 - 2{\sin ^2}\theta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \frac{1}{2}x = \sqrt {\frac{{1 - \cos x}}{2}} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sqrt 2 \int_0^{\frac{\pi }{2}} {\cos \frac{1}{2}x + \sin \frac{1}{2}x{\text{d}}x} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sqrt 2 \left[ {2\sin \frac{1}{2}x - 2\cos \frac{1}{2}x} \right]_0^{\frac{\pi }{2}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sqrt 2 (0) - \sqrt 2 (0 - 2)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\sqrt 2 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></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><span style="font-family: times new roman,times; font-size: medium;">Show that </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{\sin 2\theta }}{{1 + \cos 2\theta }} = \tan \theta \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</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><span style="font-family: times new roman,times; font-size: medium;">Hence find the value of \(\cot \frac{\pi }{8}\) </span><span style="font-family: times new roman,times; font-size: medium;">in the form \(a + b\sqrt 2 \) , where \(a,b \in \mathbb{Z}\).</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><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{\sin 2\theta }}{{1 + \cos 2\theta }} = \frac{{2\sin \theta \cos \theta }}{{1 + 2{{\cos }^2}\theta - 1}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>M1</strong></em><span style="font-family: 'times new roman', times; font-size: medium;"> for use of double angle formulae.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2\sin \theta \cos \theta }}{{2{{\cos }^2}\theta }}\) </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>A1</strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{\sin \theta }}{{\cos \theta }}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \tan \theta \) <em><strong>AG</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\tan \frac{\pi }{8} = \frac{{\sin \frac{\pi }{4}}}{{1 + \cos \frac{\pi }{4}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cot \frac{\pi }{8} = \frac{{1 + \cos \frac{\pi }{4}}}{{\sin \frac{\pi }{4}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{1 + \frac{{\sqrt 2 }}{2}}}{{\frac{{\sqrt 2 }}{2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 1 + \sqrt 2 \) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></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><span style="font-family: times new roman,times; font-size: medium;">The performance in this question was generally good with most candidates answering (a) well; (b) caused more difficulties, in particular the rationalization of the denominator. A number of misconceptions were identified, for example \(\cot \frac{\pi }{8} = \tan \frac{8}{\pi }\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The performance in this question was generally good with most candidates answering (a) well; (b) caused more difficulties, in particular the rationalization of the denominator. A number of misconceptions were identified, for example \(\cot \frac{\pi }{8} = \tan \frac{8}{\pi }\).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\cot \alpha = \tan \left( {\frac{\pi }{2} - \alpha } \right)\) for \(0 < \alpha < \frac{\pi }{2}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find \(\int_{\tan \alpha }^{\cot \alpha } {\frac{1}{{1 + {x^2}}}{\text{d}}x,{\text{ }}0 < \alpha < \frac{\pi }{2}} \).</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 class="p1"><strong>EITHER</strong></p>
<p class="p1">use of a diagram and trig ratios</p>
<p class="p1"><em>eg</em>,</p>
<p class="p1"><img src="images/Schermafbeelding_2017-01-27_om_10.18.06.png" alt="M16/5/MATHL/HP1/ENG/TZ2/03/M"></p>
<p class="p1">\(\tan \alpha = \frac{O}{A} \Rightarrow \cot \alpha = \frac{A}{O}\)</p>
<p class="p1">from diagram, \(\tan \left( {\frac{\pi }{2} - \alpha } \right) = \frac{A}{O}\) <strong><em>R1</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">use of \(\tan \left( {\frac{\pi }{2} - \alpha } \right) = \frac{{\sin \left( {\frac{\pi }{2} - \alpha } \right)}}{{\cos \left( {\frac{\pi }{2} - \alpha } \right)}} = \frac{{\cos \alpha }}{{\sin \alpha }}\) <strong><em>R1</em></strong></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p2">\(\cot \alpha = \tan \left( {\frac{\pi }{2} - \alpha } \right)\) <strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\int_{\tan \alpha }^{\cot \alpha } {\frac{1}{{1 + {x^2}}}{\text{d}}x} = [\arctan x]_{\tan \alpha }^{\cot \alpha }\) <strong><em>(A1)</em></strong></p>
<p class="p1"> </p>
<p class="p1"><strong>Note: </strong>Limits (or absence of such) may be ignored at this stage.</p>
<p class="p1"> </p>
<p class="p1">\( = \arctan (\cot \alpha ) - \arctan (\tan \alpha )\) <strong><em>(M1)</em></strong></p>
<p class="p1">\( = \frac{\pi }{2} - \alpha - \alpha \) <strong><em>(A1)</em></strong></p>
<p class="p1">\( = \frac{\pi }{2} - 2\alpha \) <strong><em>A1</em></strong></p>
<p class="p1"><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;">
<p class="p1">This was generally well done.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was generally well done. Some weaker candidates tried to solve part (b) through use of a substitution, though the standard result \(\arctan x\) was well known. A small number used \(\arctan x + c\) and went on to obtain an incorrect final answer.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the triangle ABC, \({\text{AB}} = 2\sqrt 3 \) , AC = 9 and \({\rm{B\hat AC}} = 150^\circ \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine BC, giving your answer in the form \(k\sqrt 3 \), \(k \in {\mathbb{Z}^ + }\) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point D lies on (BC), and (AD) is perpendicular to (BC). Determine AD.</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{B}}{{\text{C}}^2} = 12 + 81 + 2 \times 2\sqrt 3 \times 9 \times \frac{{\sqrt 3 }}{2} = 147\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{BC}} = 7\sqrt 3 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of triangle \({\text{ABC}} = \frac{1}{2} \times 9 \times 2\sqrt 3 \times \frac{1}{2}{\text{ }}\left( { = \frac{{9\sqrt 3 }}{2}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore \(\frac{1}{2} \times {\text{AD}} \times 7\sqrt 3 = \frac{{9\sqrt 3 }}{2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{AD}} = \frac{9}{7}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></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 \(z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve \(2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus and argument of \(z\) in terms of \(\theta \). Express each answer in its simplest form.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cube roots of \(z\) in modulus-argument form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(2\sin (x + 60^\circ ) = \cos (x + 30^\circ )\)</p>
<p>\(2(\sin x\cos 60^\circ + \cos x\sin 60^\circ ) = \cos x\cos 30^\circ - \sin x\sin 30^\circ \) <strong><em>(M1)(A1)</em></strong></p>
<p>\(2\sin x \times \frac{1}{2} + 2\cos x \times \frac{{\sqrt 3 }}{2} = \cos x \times \frac{{\sqrt 3 }}{2} - \sin x \times \frac{1}{2}\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow \frac{3}{2}\sin x = - \frac{{\sqrt 3 }}{2}\cos x\)</p>
<p>\( \Rightarrow \tan x = - \frac{1}{{\sqrt 3 }}\) <strong><em>M1</em></strong></p>
<p>\( \Rightarrow x = 150^\circ \) <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>choosing two appropriate angles, for example 60° and 45° <strong><em>M1</em></strong></p>
<p>\(\sin 105^\circ = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \) and</p>
<p>\(\cos 105^\circ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ \) <strong><em>(A1)</em></strong></p>
<p>\(\sin 105^\circ + \cos 105^\circ = \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}\) <strong><em>A1</em></strong></p>
<p>\( = \frac{1}{{\sqrt 2 }}\) <strong><em>AG</em></strong></p>
<p><strong>OR</strong></p>
<p>attempt to square the expression <strong><em>M1</em></strong></p>
<p>\({(\sin 105^\circ + \cos 105^\circ )^2} = {\sin ^2}105^\circ + 2\sin 105^\circ \cos 105^\circ + {\cos ^2}105^\circ \)</p>
<p>\({(\sin 105^\circ + \cos 105^\circ )^2} = 1 + \sin 210^\circ \) <strong><em>A1</em></strong></p>
<p>\( = \frac{1}{2}\) <strong><em>A1</em></strong></p>
<p>\(\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>\(z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta \)</p>
<p>\(\left| z \right| = \sqrt {{{(1 - \cos 2\theta )}^2} + {{(\sin 2\theta )}^2}} \) <strong><em>M1</em></strong></p>
<p>\(\left| z \right| = \sqrt {1 - 2\cos 2\theta + {{\cos }^2}2\theta + {{\sin }^2}2\theta } \) <strong><em>A1</em></strong></p>
<p>\( = \sqrt 2 \sqrt {(1 - \cos 2\theta )} \) <strong><em>A1</em></strong></p>
<p>\( = \sqrt {2(2{{\sin }^2}\theta )} \)</p>
<p>\( = 2\sin \theta \) <strong><em>A1</em></strong></p>
<p>let \(\arg (z) = \alpha \)</p>
<p>\(\tan \alpha = - \frac{{\sin 2\theta }}{{1 - \cos 2\theta }}\) <strong><em>M1</em></strong></p>
<p>\( = \frac{{ - 2\sin \theta \cos \theta }}{{2{{\sin }^2}\theta }}\) <strong><em>(A1)</em></strong></p>
<p>\( = - \cot \theta \) <strong><em>A1</em></strong></p>
<p>\(\arg (z) = \alpha = - \arctan \left( {\tan \left( {\frac{\pi }{2} - \theta } \right)} \right)\) <strong><em>A1</em></strong></p>
<p>\( = \theta - \frac{\pi }{2}\) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>\(z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta \)</p>
<p>\( = 2{\sin ^2}\theta - 2{\text{i}}\sin \theta \cos \theta \) <strong><em>M1A1</em></strong></p>
<p>\( = 2\sin \theta (\sin \theta - {\text{i}}\cos \theta )\) <strong><em>(A1)</em></strong></p>
<p>\( = - 2{\text{i}}\sin \theta (\cos \theta + {\text{i}}\sin \theta )\) <strong><em>M1A1</em></strong></p>
<p>\( = 2\sin \theta \left( {\cos \left( {\theta - \frac{\pi }{2}} \right) + {\text{i}}\sin \left( {\theta - \frac{\pi }{2}} \right)} \right)\) <strong><em>M1A1</em></strong></p>
<p>\(\left| z \right| = 2\sin \theta \) <strong><em>A1</em></strong></p>
<p>\(\arg (z) = \theta - \frac{\pi }{2}\) <strong><em>A1</em></strong></p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to apply De Moivre’s theorem <strong><em>M1</em></strong></p>
<p>\({(1 - \cos 2\theta - {\text{i}}\sin 2\theta )^{\frac{1}{3}}} = {2^{\frac{1}{3}}}{(\sin \theta )^{\frac{1}{3}}}\left[ {\cos \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right)} \right]\) <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>for modulus, <strong><em>A1 </em></strong>for dividing argument of \(z\) by 3 and <strong><em>A1 </em></strong>for \(2n\pi \).</p>
<p> </p>
<p>Hence cube roots are the above expression when \(n = - 1,{\text{ }}0,{\text{ }}1\). Equivalent forms are acceptable. <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi \) <span class="s1">where \(k \in \mathbb{Z}\).</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 class="p1">Use the principle of mathematical induction to prove that</p>
<p class="p1">\(\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi \) where \(k \in \mathbb{Z}\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence or otherwise solve the equation \(\sin x + \sin 3x = \cos x\) <span class="s1">in the interval \(0 < x < \pi \).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4} = \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2} - \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2} = \frac{{\sqrt 2 }}{2}\) </span><strong><em>(M1)A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>M1 </em></strong>for 5 equal terms with \) + \) or \( - \) signs.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \frac{{1 - (1 - 2{{\sin }^2}x)}}{{2\sin x}}\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \frac{{2{{\sin }^2}x}}{{2\sin x}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \sin x\) </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">let \({\text{P}}(n):\sin x + \sin 3x + \ldots + \sin (2n - 1)x \equiv \frac{{1 - \cos 2nx}}{{2\sin x}}\)</p>
<p class="p1">if \(n = 1\)</p>
<p class="p1">\({\text{P}}(1):\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x\) which is true (as proved in part (b)) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><span class="s1">assume \({\text{P}}(k)\)</span> true, \(\sin x + \sin 3x + \ldots + \sin (2k - 1)x \equiv \frac{{1 - \cos 2kx}}{{2\sin x}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Notes: </strong>Only award <strong><em>M1 </em></strong>if the words “assume” and “true” appear. Do not award <strong><em>M1 </em></strong>for “let \(n = k\)<em>” </em>only. Subsequent marks are independent of this <strong><em>M1</em></strong><em>.</em></p>
<p class="p3"> </p>
<p class="p4">consider \({\text{P}}(k + 1)\)<span class="s2">:</span></p>
<p class="p1">\({\text{P}}(k + 1):\sin x + \sin 3x + \ldots + \sin (2k - 1)x + \sin (2k + 1)x \equiv \frac{{1 - \cos 2(k + 1)x}}{{2\sin x}}\)</p>
<p class="p1"><span class="Apple-converted-space">\(LHS = \sin x + \sin 3x + \ldots + \sin (2k - 1)x + \sin (2k + 1)x\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \frac{{1 - \cos 2kx}}{{2\sin x}} + \sin (2k + 1)x\) </span><strong><em>A1</em></strong></p>
<p class="p1">\( \equiv \frac{{1 - \cos 2kx + 2\sin x\sin (2k + 1)x}}{{2\sin x}}\)</p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \frac{{1 - \cos 2kx + 2\sin x\cos x\sin 2kx + 2{{\sin }^2}x\cos 2kx}}{{2\sin x}}\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \frac{{1 - \left( {(1 - 2{{\sin }^2}x)\cos 2kx - \sin 2x\sin 2kx} \right)}}{{2\sin x}}\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \frac{{1 - (\cos 2x\cos 2kx - \sin 2x\sin 2kx)}}{{2\sin x}}\) </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \equiv \frac{{1 - \cos (2kx + 2x)}}{{2\sin x}}\) </span><strong><em>A1</em></strong></p>
<p class="p1">\( \equiv \frac{{1 - \cos 2(k + 1)x}}{{2\sin x}}\)</p>
<p class="p1">so if true for \(n = k\) , then also true for \(n = k + 1\)</p>
<p class="p1">as true for \(n = 1\) then true for all \(n \in {\mathbb{Z}^ + }\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Accept answers using transformation formula for product of sines if steps are shown clearly.</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>R1 </em></strong>only if candidate is awarded at least 5 marks in the previous steps.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\sin x + \sin 3x = \cos x \Rightarrow \frac{{1 - \cos 4x}}{{2\sin x}} = \cos x\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow 1 - \cos 4x = 2\sin x\cos x,{\text{ }}(\sin x \ne 0)\) </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow 1 - (1 - 2{\sin ^2}2x) = \sin 2x\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow \sin 2x(2\sin 2x - 1) = 0\) </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="s1">\( \Rightarrow \sin 2x = 0\) or \(\sin 2x = \frac{1}{2}\)</span> <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(2x = \pi ,{\text{ }}2x = \frac{\pi }{6}\) and \(2x = \frac{{5\pi }}{6}\)</p>
<p class="p1"><strong>OR</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\sin x + \sin 3x = \cos x \Rightarrow 2\sin 2x\cos x = \cos x\) </span><strong><em>M1A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow (2\sin 2x - 1)\cos x = 0,{\text{ }}(\sin x \ne 0)\) </span><strong><em>M1A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( \Rightarrow \sin 2x = \frac{1}{2}\) of \(\cos x = 0\) </span><strong><em>A1</em></strong></p>
<p class="p1">\(2x = \frac{\pi }{6},{\text{ }}2x = \frac{{5\pi }}{6}\) and \(x = \frac{\pi }{2}\)</p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1"><span class="s1">\(\therefore x = \frac{\pi }{2},{\text{ }}x = \frac{\pi }{{12}}\) and \(x = \frac{{5\pi }}{{12}}\)</span> <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Do not award the final <strong><em>A1 </em></strong>if extra solutions are seen.</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="font: 28px Helvetica; margin: 0px; text-align: justify;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows a tangent, (TP) , to the circle with centre O and radius <em>r</em> . The size of \({\rm{P\hat OA}}\) is \(\theta \) radians.</span></p>
<p style="font: normal normal normal 28px/normal Helvetica; text-align: center; margin: 0px;"> </p>
<p style="font: normal normal normal 28px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of triangle AOP in terms of <em>r</em> and \(\theta \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of triangle POT in terms of <em>r</em> and \(\theta \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using your results from part (a) and part (b), show that \(\sin \theta < \theta < \tan \theta \) .</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of \({\text{AOP}} = \frac{1}{2}{r^2}\sin \theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]<br></em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{TP}} = r\tan \theta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of POT \( = \frac{1}{2}r(r\tan \theta )\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}{r^2}\tan \theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]<br></em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of sector OAP \( = \frac{1}{2}{r^2}\theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of triangle OAP < area of sector OAP < area of triangle POT <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{2}{r^2}\sin \theta < \frac{1}{2}{r^2}\theta < \frac{1}{2}{r^2}\tan \theta \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \theta < \theta < \tan \theta \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to find the area of Triangle <em>AOP</em> correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to find the area of Triangle <em>AOP</em> correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates were able to find the area of Triangle <em>AOP</em> correctly. Most were then able to get an expression for the other triangle. In the final section, few saw the connection between the area of the sector and the relationship. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The first three terms of a geometric sequence are \(\sin x,{\text{ }}\sin 2x\) and \(4\sin x{\cos ^2}x,{\text{ }} - \frac{\pi }{2} < x < \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the common ratio <em>r</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the set of values of <em>x </em>for which the geometric series \(\sin x + \sin 2x + 4\sin x{\cos ^2}x + \ldots \) converges.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider \(x = \arccos \left( {\frac{1}{4}} \right),{\text{ }}x > 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that the sum to infinity of this series is \(\frac{{\sqrt {15} }}{2}\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\sin x,{\text{ }}\sin 2x{\text{ and }}4\sin x{\cos ^2}x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \frac{{2\sin x\cos x}}{{\sin x}} = 2\cos x\)<em> </em><strong>A1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica; min-height: 26.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \(\frac{{\sin 2x}}{{\sin x}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| r \right| < 1 \Rightarrow \left| {2\cos x} \right| < 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 1 < r < 1 \Rightarrow - 1 < 2\cos x < 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 < \cos x < \frac{1}{2}{\text{ for }} - \frac{\pi }{2} < x < \frac{\pi }{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - \frac{\pi }{2} < x < - \frac{\pi }{3}{\text{ or }}\frac{\pi }{3} < x < \frac{\pi }{2}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \({S_\infty } = \frac{{\sin x}}{{1 - 2\cos x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty } = \frac{{\sin \left( {\arccos \left( {\frac{1}{4}} \right)} \right)}}{{1 - 2\cos \left( {\arccos \left( {\frac{1}{4}} \right)} \right)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\frac{{\sqrt {15} }}{4}}}{{\frac{1}{2}}}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct numerator and <strong><em>A1 </em></strong>for correct denominator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sqrt {15} }}{2}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p class="p1">Solve the equation \(\sin 2x - \cos 2x = 1 + \sin x - \cos x\) for \(x \in [ - \pi ,{\text{ }}\pi ]\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\((\sin 2x - \sin x) - (\cos 2x - \cos x) = 1\)</p>
<p>attempt to use both double-angle formulae, in whatever form <strong><em>M1</em></strong></p>
<p>\((2\sin x\cos x - \sin x) - (2{\cos ^2}x - 1 - \cos x) = 1\)</p>
<p>or \((2\sin x\cos x - \sin x) - (2{\cos ^2}x - \cos x) = 0\) for example <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow any rearrangement of the above equations.</p>
<p> </p>
<p>\(\sin x(2\cos x - 1) - \cos x(2\cos x - 1) = 0\)</p>
<p>\((\sin x - \cos x)(2\cos x - 1) = 0\) <strong><em>(M1)</em></strong></p>
<p>\(\tan x = 1{\text{ and }}\cos x = \frac{1}{2}\) <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>These <strong><em>A </em></strong>marks are dependent on the <strong><em>M </em></strong>mark awarded for factorisation.</p>
<p> </p>
<p>\(x = - \frac{{3\pi }}{4},{\text{ }} - \frac{\pi }{3},{\text{ }}\frac{\pi }{3},{\text{ }}\frac{\pi }{4}\) <strong><em>A2</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for two correct answers, which could be for both tan or both cos solutions, for example.</p>
<p> </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 class="p1">In triangle \({\text{ABC, BC}} = \sqrt 3 {\text{ cm}}\), \({\rm{A\hat BC}} = \theta \) and \({\rm{B\hat CA}} = \frac{\pi }{3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that length \({\text{AB}} = \frac{3}{{\sqrt 3 \cos \theta + \sin \theta }}\).</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 class="p1">Given that \(AB\) has a minimum value, determine the value of \(\theta \) <span class="s1">for which this occurs.</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>any attempt to use sine rule <strong><em>M1</em></strong></p>
<p>\(\frac{{{\text{AB}}}}{{\sin \frac{\pi }{3}}} = \frac{{\sqrt 3 }}{{\sin \left( {\frac{{2\pi }}{3} - \theta } \right)}}\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{\sqrt 3 }}{{\sin \frac{{2\pi }}{3}\cos \theta - \cos \frac{{2\pi }}{3}\sin \theta }}\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone use of degrees.</p>
<p> </p>
<p>\( = \frac{{\sqrt 3 }}{{\frac{{\sqrt 3 }}{2}\cos \theta + \frac{1}{2}\sin \theta }}\) <strong><em>A1</em></strong></p>
<p>\(\frac{{{\text{AB}}}}{{\frac{{\sqrt 3 }}{2}}} = \frac{{\sqrt 3 }}{{\frac{{\sqrt 3 }}{2}\cos \theta + \frac{1}{2}\sin \theta }}\)</p>
<p>\(\therefore {\text{AB}} = \frac{3}{{\sqrt 3 \cos \theta + \sin \theta }}\) <strong><em>AG</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><strong>METHOD 1</strong></p>
<p>\(({\text{AB}})' = \frac{{ - 3\left( { - \sqrt 3 \sin \theta + \cos \theta } \right)}}{{{{\left( {\sqrt 3 \cos \theta + \sin \theta } \right)}^2}}}\) <strong><em>M1A1</em></strong></p>
<p>setting \(({\text{AB}})' = 0\) <strong><em>M1</em></strong></p>
<p>\(\tan \theta = \frac{1}{{\sqrt 3 }}\)</p>
<p>\(\theta = \frac{\pi }{6}\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\({\text{AB}} = \frac{{\sqrt 3 \sin \frac{\pi }{3}}}{{\sin \left( {\frac{{2\pi }}{3} - \theta } \right)}}\)</p>
<p>\(AB\) minimum when \(\sin \left( {\frac{{2\pi }}{3} - \theta } \right)\) is maximum <strong><em>M1</em></strong></p>
<p>\(\sin \left( {\frac{{2\pi }}{3} - \theta } \right) = 1\) <strong>(<em>A1)</em></strong></p>
<p>\(\frac{{2\pi }}{3} - \theta = \frac{\pi }{2}\) <strong><em>M1</em></strong></p>
<p>\(\theta = \frac{\pi }{6}\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>shortest distance from \(B\) to \(AC\) is perpendicular to \(AC\) <strong><em>R1</em></strong></p>
<p>\(\theta = \frac{\pi }{2} - \frac{\pi }{3} = \frac{\pi }{6}\) <strong><em>M1A2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [8 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 class="p1">The following diagram shows the triangle ABC where \({\text{AB}} = 2,{\text{ AC}} = \sqrt 2 \) and \({\rm{B\hat AC}} = 15^\circ \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_08.33.57.png" alt="M16/5/MATHL/HP1/ENG/TZ1/05.c"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Expand and simplify \({\left( {1 - \sqrt 3 } \right)^2}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By writing \(15^\circ \) as \(60^\circ - 45^\circ \) find the value of \(\cos (15^\circ )\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find BC </span>in the form \(a + \sqrt b \) where \(a,{\text{ }}b \in \mathbb{Z}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\left( {1 - \sqrt 3 } \right)^2} = 4 - 2\sqrt 3 \) </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A0 </em></strong>for \(1 - 2\sqrt 3 + 3\).</p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\cos (60^\circ - 45^\circ ) = \cos (60^\circ )\cos (45^\circ ) + \sin (60^\circ )\sin (45^\circ )\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{1}{2} \times \frac{{\sqrt 2 }}{2} + \frac{{\sqrt 3 }}{2} \times \frac{{\sqrt 2 }}{2}{\text{ }}\left( {{\text{or }}\frac{1}{2} \times \frac{1}{{\sqrt 2 }} + \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}} \right)\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{\sqrt 2 + \sqrt 6 }}{4}{\text{ }}\left( {{\text{or }}\frac{{1 + \sqrt 3 }}{{2\sqrt 2 }}} \right)\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(B{C^2} = 2 + 4 - 2 \times \sqrt 2 \times 2\cos (15^\circ )\) </span><strong><em>M1</em></strong></p>
<p class="p2">\( = 6 - \sqrt 2 \left( {\sqrt 2 + \sqrt 6 } \right)\)</p>
<p class="p2"><span class="Apple-converted-space">\( = 4 - \sqrt {12} {\text{ }}\left( { = 4 - 2\sqrt 3 } \right)\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(BC = \pm \left( {1 - \sqrt 3 } \right)\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\(BC = - 1 + \sqrt 3 \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept \(BC = \sqrt 3 - 1\).</p>
<p class="p1"><strong>Note</strong>: <span class="Apple-converted-space"> </span>Award <strong><em>M1A0 </em></strong>for \(1 - \sqrt 3 \).</p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Valid geometrical methods may be seen.</p>
<p class="p1"><strong><em>[4 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;">
<p class="p1">The main error here was to fail to note the word ‘simplify’ in the question and some candidates wrote \(1 + 3\) in their final answer rather than 4.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This was well done by the majority of candidates, though a few wrote \(\cos (60 - 45) = \cos 60 - \cos 45\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates were able to use the cosine rule correctly but then failed to notice the result obtained was the same as that obtained in part (a).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The triangle ABC is equilateral of side 3 cm. The point D lies on [BC] such that BD = 1 cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\cos {\rm{D\hat AC}}\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{A}}{{\text{D}}^2} = {2^2} + {3^2} - 2 \times 2 \times 3 \times \cos 60^\circ \) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(or \({\text{A}}{{\text{D}}^2} = {1^2} + {3^2} - 2 \times 1 \times 3 \times \cos 60^\circ \))</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman'; min-height: 26.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>M1 </em></strong>for use of cosine rule with 60° angle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{A}}{{\text{D}}^2} = 7\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos {\rm{D\hat AC}} = \frac{{9 + 7 - 4}}{{2 \times 3 \times \sqrt 7 }}\) <em><strong>M1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong> </strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> <strong>M1</strong> f</span><span style="font-family: 'times new roman', times; font-size: medium;">or use of cosine rule involving \({\rm{D\hat AC}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman'; min-height: 26.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{2}{{\sqrt 7 }}\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">let point E be the foot of the perpendicular from D to AC</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">EC = 1 (by similar triangles, or triangle properties) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(or AE = 2)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{DE}} = \sqrt 3 \) a</span><span style="font-family: 'times new roman', times; font-size: medium;">nd \({\text{AD}} = \sqrt 7 \) (by Pythagoras) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos {\rm{D\hat AC}} = \frac{2}{{\sqrt 7 }}\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> If first <strong><em>M1 </em></strong>not awarded but remainder of the question is correct award <strong><em>M0A0M1A1A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
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[N/A]
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{\sin 3x}}{{\sin x}} - \frac{{\cos 3x}}{{\cos x}}\)<em>.</em></span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For what values of <em>x </em>does \(f(x)\) not exist?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Simplify the expression \(\frac{{\sin 3x}}{{\sin x}} - \frac{{\cos 3x}}{{\cos x}}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos x = 0{\text{, }}\sin x = 0\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{{n\pi }}{2},n \in \mathbb{Z}\) <strong><em>A1</em></strong></span></p>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sin 3x\cos x - \cos 3x\sin x}}{{\sin x\cos x}}\)<strong> </strong><strong><em>M1 A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin (3x - x)}}{{\frac{1}{2}\sin 2x}}\) <strong><em>A1 A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><strong>OR</strong><em><strong><br></strong></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sin 2x\cos x + \cos 2x\sin x}}{{\sin x}} - \frac{{\cos 2x\cos x - \sin 2x\sin x}}{{\cos x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2\sin x{{\cos }^2}x + 2{{\cos }^2}x\sin x - \sin x}}{{\sin x}} - \frac{{2{{\cos }^3}x - \cos x - {{\sin }^2}x\cos x}}{{\cos x}}\) <strong><em>A1 A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4{\cos ^2}x - 1 - 2{\cos ^2}x + 1 + 2{\sin ^2}x\) <strong><em>A1</em></strong></span><br><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2{\cos ^2}x + 2{\sin ^2}x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]<br></em></strong></span></p>
<div class="question_part_label">b.</div>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was well answered, although many candidates lost a mark through not giving sufficient solutions. It was rare for a student to receive no marks for part (b), but few solved the question by the easiest route, and as a consequence, there were frequently errors in the algebraic manipulation of the expression. </span></p>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was well answered, although many candidates lost a mark through not giving sufficient solutions. It was rare for a student to receive no marks for part (b), but few solved the question by the easiest route, and as a consequence, there were frequently errors in the algebraic manipulation of the expression. </span></p>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The angle of the largest sector is twice the angle of the smallest sector.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the size of the angle of the smallest sector.</span></p>
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<h2 style="margin-top: 1em">Markscheme</h2>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If the areas are in arithmetic sequence, then so are the angles. <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow {S_n} = \frac{n}{2}(a + l) \Rightarrow \frac{{12}}{2}(\theta + 2\theta ) = 18\theta \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px 'Hiragino Kaku Gothic ProN';"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 18\theta = 2\pi \) <em><strong>(A1)</strong></em><span style="font: 32.0px Helvetica;"><br></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = \frac{\pi }{9}\) (accept \(20^\circ \)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{a}}_{12}} = 2{a_1}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{12}}{2}({a_1} + 2{a_1}) = \pi {r^2}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(3{a_1} = \frac{{\pi {r^2}}}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{3}{2}{r^2}\theta = \frac{{\pi {r^2}}}{6}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = \frac{{2\pi }}{{18}} = \frac{\pi }{9}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"> (accept \(20^\circ \))</span></span> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let smallest angle = <em>a</em> , common difference = <em>d</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a + 11d = 2a\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 11d\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_n} = \frac{{12}}{2}(2a + 11d) = 2\pi \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(6(2a + a) = 2\pi \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(18a = 2\pi \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = \frac{\pi }{9}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"> (accept \(20^\circ \))</span> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Stronger candidates had little problem with this question, but a significant minority of weaker candidates were unable to access the question or worked with area and very quickly became confused. Candidates who realised that the area of each sector was proportional to the angle usually gained the correct answer.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows a circular lake with centre O, diameter AB and radius 2 km.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 29px/normal Helvetica; text-align: center; margin: 0px;"><img 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alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Jorg needs to get from A to B as quickly as possible. He considers rowing to point P and then walking to point B. He can row at \(3{\text{ km}}\,{{\text{h}}^{ - 1}}\) and walk at \(6{\text{ km}}\,{{\text{h}}^{ - 1}}\). Let \({\rm{P\hat AB}} = \theta \) radians, and <em>t</em> be the time in hours taken by Jorg to travel from A to B.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(t = \frac{2}{3}(2\cos \theta + \theta )\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\theta \) for which \(\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = 0\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">What route should Jorg take to travel from A to B in the least amount of time?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Give reasons for your answer.</span></p>
<div class="marks">[3]</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">angle APB is a right angle</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \cos \theta = \frac{{{\text{AP}}}}{4} \Rightarrow {\text{AP}} = 4\cos \theta \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Allow correct use of cosine rule.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{arc PB}} = 2 \times 2\theta = 4\theta \) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{{{\text{AP}}}}{3} + \frac{{{\text{PB}}}}{6}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Allow use of their AP and their PB for the </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow t = \frac{{4\cos \theta }}{3} + \frac{{4\theta }}{6} = \frac{{4\cos \theta }}{3} + \frac{{2\theta }}{3} = \frac{2}{3}(2\cos \theta + \theta )\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>AG</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = \frac{2}{3}( - 2\sin \theta + 1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{2}{3}( - 2\sin \theta + 1) = 0 \Rightarrow \sin \theta = \frac{1}{2} \Rightarrow \theta = \frac{\pi }{6}\) (or 30 degrees) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{{\text{d}}^2}t}}{{{\text{d}}{\theta ^2}}} = - \frac{4}{3}\cos \theta < 0\,\,\,\,\left( {{\text{at }}\theta = \frac{\pi }{6}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow t\) is maximized at \(\theta = \frac{\pi }{6}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">time needed to walk along arc AB is \(\frac{{2\pi }}{6}{\text{ (}} \approx {\text{1 hour)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">time needed to row from A to B is \(\frac{4}{3}{\text{ (}} \approx {\text{1.33 hour)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence, time is minimized in walking from A to B <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The fairly easy trigonometry challenged a large number of candidates.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: small;">Part (b) was very well done.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Satisfactory answers were very rarely seen for (c). Very few candidates realised that a minimum can occur at the beginning or end of an interval.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the expansion of \({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3}\) in the form \(a + {\text{i}}b\) , where \(a\) and \(b\) </span><span style="font-family: times new roman,times; font-size: medium;">are in terms of \({\sin \theta }\) and \({\cos \theta }\) .</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><span style="font-family: times new roman,times; font-size: medium;">Hence show that \(\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta \) .</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><span style="font-family: times new roman,times; font-size: medium;">Similarly show that \(\cos 5\theta = 16{\cos ^5}\theta - 20{\cos ^3}\theta + 5\cos \theta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong> solve the equation \(\cos 5\theta + \cos 3\theta + \cos \theta = 0\) , where \(\theta \in \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">By considering the solutions of the equation \(\cos 5\theta = 0\) , show that </span><span style="font-family: times new roman,times; font-size: medium;">\(\cos \frac{\pi }{{10}} = \sqrt {\frac{{5 + \sqrt 5 }}{8}} \)</span><span style="font-family: times new roman,times; font-size: medium;"> and state the value of \(\cos \frac{{7\pi }}{{10}}\)</span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3} = {\cos ^3}\theta + 3{\cos ^2}\theta \left( {{\text{i}}\sin \theta } \right) + 3\cos \theta {\left( {{\text{i}}\sin \theta } \right)^2} + {\left( {{\text{i}}\sin \theta } \right)^3}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {\cos ^3}\theta - 3\cos \theta {\sin ^2}\theta + {\text{i}}\left( {3{{\cos }^2}\theta \sin \theta - {{\sin }^3}\theta } \right)\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">from De Moivre’s theorem</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3} = \cos 3\theta + {\text{i}}\sin 3\theta \) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos 3\theta + {\text{i}}\sin 3\theta = \left( {{{\cos }^3}\theta - 3\cos \theta {{\sin }^2}\theta } \right) + {\text{i}}\left( {3{{\cos }^2}\theta \sin \theta - {{\sin }^3}\theta } \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equating real parts <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos 3\theta = {\cos ^3}\theta - 3\cos \theta {\sin ^2}\theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {\cos ^3}\theta - 3\cos \theta \left( {1 - {{\cos }^2}\theta } \right)\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {\cos ^3}\theta - 3\cos \theta + 3{\cos ^3}\theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 4{\cos ^3}\theta - 3\cos \theta \) <em><strong>AG</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Do not award marks if part (a) is not used.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^5} = \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\cos ^5}\theta + 5{\cos ^4}\theta \left( {{\text{i}}\sin \theta } \right) + 10{\cos ^3}\theta {\left( {{\text{i}}\sin \theta } \right)^2} + 10{\cos ^2}\theta {\left( {{\text{i}}\sin \theta } \right)^3} + 5\cos \theta {\left( {{\text{i}}\sin \theta } \right)^4} + {\left( {{\text{i}}\sin \theta } \right)^5}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">from De Moivre’s theorem</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos 5\theta = {\cos ^5}\theta - 10{\cos ^3}\theta {\sin ^2}\theta + 5\cos \theta {\sin ^4}\theta \) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {\cos ^5}\theta - 10{\cos ^3}\theta \left( {1 - {{\cos }^2}\theta } \right) + 5\cos \theta {\left( {1 - {{\cos }^2}\theta } \right)^2}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {\cos ^5}\theta - 10{\cos ^3}\theta + 10{\cos ^5}\theta + 5\cos \theta - 10{\cos ^3}\theta + 5{\cos ^5}\theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\therefore \cos 5\theta = 16{\cos ^5}\theta - 20{\cos ^3}\theta + 5\cos \theta \) <em><strong>AG</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> If compound angles used in (b) and (c), then marks can be allocated in (c) only.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos 5\theta + \cos 3\theta + \cos \theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \left( {16{{\cos }^5}\theta - 20{{\cos }^3}\theta + 5\cos \theta } \right) + \left( {4{{\cos }^3}\theta - 3\cos \theta } \right) + \cos \theta = 0\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(16{\cos ^5}\theta - 16{\cos ^3}\theta + 3\cos \theta = 0\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \theta \left( {16{{\cos }^4}\theta - 16{{\cos }^2}\theta + 3} \right) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \theta \left( {4{{\cos }^2}\theta - 3} \right)\left( {4{{\cos }^2}\theta - 1} \right) = 0\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\therefore \cos \theta = 0\)</span><span style="font-family: times new roman,times; font-size: medium;">; \( \pm \frac{{\sqrt 3 }}{2}\); \( \pm \frac{1}{2}\) <em><strong> A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\therefore \theta = \pm \frac{\pi }{6}\); \(\pm \frac{\pi }{3}\); \( \pm \frac{\pi }{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos 5\theta = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(5\theta = ...\frac{\pi }{2}\); \(\left( {\frac{{3\pi }}{2};\frac{{5\pi }}{2}} \right)\); \(\frac{{7\pi }}{2}\); \(...\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = ...\frac{\pi }{{10}}\); \(\left( {\frac{{3\pi }}{{10}};\frac{{5\pi }}{{10}}} \right)\); </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{7\pi }}{10}\); \(...\)</span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> These marks can be awarded for verifications later in the question.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">now consider \(16{\cos ^5}\theta - 20{\cos ^3}\theta + 5\cos \theta = 0\) </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>M1</strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \theta \left( {16{{\cos }^4}\theta - 20{{\cos }^2}\theta + 5} \right) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\cos ^2}\theta = \frac{{20 \pm \sqrt {400 - 4\left( {16} \right)\left( 5 \right)} }}{{32}}\)</span><span style="font-family: times new roman,times; font-size: medium;">; \(\cos \theta = 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \theta = \pm \sqrt {\frac{{20 \pm \sqrt {400 - 4\left( {16} \right)\left( 5 \right)} }}{{32}}} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \frac{\pi }{{10}} = \sqrt {\frac{{20 + \sqrt {400 - 4\left( {16} \right)\left( 5 \right)} }}{{32}}} \) since max value of cosine \( \Rightarrow \) angle closest to zero</span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong><strong><span style="font-family: times new roman,times; font-size: medium;">R1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \frac{\pi }{{10}} = \sqrt {\frac{{4.5 + 4\sqrt {25 - 4\left( 5 \right)} }}{{4.8}}} = \sqrt {\frac{{5 + \sqrt 5 }}{8}} \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos \frac{{7\pi }}{{10}} = - \sqrt {\frac{{5 - \sqrt 5 }}{8}} \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved to be very difficult for most candidates. Many had difficulties in following the instructions and attempted to use addition formulae rather than binomial expansions. A small number of candidates used the results given and made a good attempt to part (d) but very few answered part (e).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved to be very difficult for most candidates. Many had difficulties in following the instructions and attempted to use addition formulae rather than binomial expansions. A small number of candidates used the results given and made a good attempt to part (d) but very few answered part (e).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved to be very difficult for most candidates. Many had difficulties in following the instructions and attempted to use addition formulae rather than binomial expansions. A small number of candidates used the results given and made a good attempt to part (d) but very few answered part (e).</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved to be very difficult for most candidates. Many had difficulties in following the instructions and attempted to use addition formulae rather than binomial expansions. A small number of candidates used the results given and made a good attempt to part (d) but very few answered part (e).</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved to be very difficult for most candidates. Many had difficulties in following the instructions and attempted to use addition formulae rather than binomial expansions. A small number of candidates used the results given and made a good attempt to part (d) but very few answered part (e).</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">From a vertex of an equilateral triangle of side \(2x\), a circular arc is drawn to divide the triangle into two regions, as shown in the diagram below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that the areas of the two regions are equal, find the radius of the arc in terms of <em>x</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of triangle \( = \frac{1}{2}{(2x)^2}\sin \frac{\pi }{3}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {x^2}\sqrt 3 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> A \(0.5 \times {\text{base}} \times {\text{height}}\) calculation is acceptable.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of sector \({\text{ = }}\frac{\theta }{2}{r^2} = \frac{\pi }{6}{r^2}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">area of triangle is twice the area of the sector</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 2\left( {\frac{\pi }{6}{r^2}} \right) = {x^2}\sqrt 3 \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow r = x\sqrt {\frac{{3\sqrt 3 }}{\pi }} \,\,\,\,\,\)or equivalent <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of candidates obtained the correct answer. A small minority of candidates used degree measure rather than radian measure, or failed to notice that the triangle was equilateral.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The angle \(\theta \) lies in the first quadrant and \(\cos \theta = \frac{1}{3}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(\sin \theta \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\tan 2\theta \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\cos \left( {\frac{\theta }{2}} \right)\) , giving your answer in the form \(\frac{{\sqrt a }}{b}\) where <em>a</em> , \(b \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[3]</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \theta = \frac{{\sqrt 8 }}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">\(\tan 2\theta = \frac{{2 \times \sqrt 8 }}{{1 - 8}} = - \frac{{2\sqrt 8 }}{7}\,\,\,\,\,\left( { - \frac{{4\sqrt 2 }}{7}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\cos ^2}\left( {\frac{\theta }{2}} \right) = \frac{{1 + \frac{1}{3}}}{2} = \frac{2}{3}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \left( {\frac{\theta }{2}} \right) = \frac{{\sqrt 6 }}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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 class="p1"><span class="s1">The following diagram shows the curve \(y = a\sin \left( {b(x + c)} \right) + d\), where \(a\)</span>, <span class="s1">\(b\)</span>, <span class="s1">\(c\) and \(d\) </span>are all positive constants. The curve has a maximum point at \((1,{\text{ }}3.5)\) and a minimum point at \((2,{\text{ }}0.5)\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_07.33.57.png" alt="M16/5/MATHL/HP1/ENG/TZ1/03"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \(a\) and the value of \(d\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the smallest possible value of \(c\), given \(c > 0\).</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 class="p1"><span class="Apple-converted-space">\(a = 1.5\,\,\,d = 2\) </span><strong><em>A1A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(b = \frac{{2\pi }}{2} = \pi \) </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to solve an appropriate equation or apply a horizontal translation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(c = 1.5\) </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award a follow through mark for the final <strong><em>A1</em></strong>.</p>
<p class="p1">Award <strong><em>(M1)A0 </em></strong>for \(c = - 0.5\).</p>
<p class="p1"><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;">
<p class="p1">Parts (a) and (b) were largely done successfully, but there was still a large minority who did not score well, and this is something that teachers need to be aware of for the future.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were largely done successfully, but there was still a large minority who did not score well, and this is something that teachers need to be aware of for the future.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This part was less successfully done. Some attempted the question by putting in a point and solving the equation. Others did it through realizing it represented a horizontal translation. Of these many failed to heed the instruction (given in the stem of the question, as well as repeated in part (c)) that \(c\) had to be greater than zero.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The logo, for a company that makes chocolate, is a sector of a circle of radius \(2\) cm, shown as shaded in the diagram. The area of the logo is \(3\pi {\text{ c}}{{\text{m}}^2}\)<span class="s1">.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2015-12-22_om_11.15.57.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find, in radians, the value of the angle \(\theta \), as indicated on the diagram.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the total length of the perimeter of the logo.</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><strong>METHOD 1</strong></p>
<p>\({\text{area}} = \pi {2^2} - \frac{1}{2}{2^2}\theta \;\;\;( = 3\pi )\) <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for using area formula.</p>
<p> </p>
<p>\( \Rightarrow 2\theta = \pi \Rightarrow \theta = \frac{\pi }{2}\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Degrees loses final A1</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p> </p>
<p>let \(x = 2\pi - \theta \)</p>
<p>\({\text{area}} = \frac{1}{2}{2^2}x\;\;\;( = 3\pi )\) <strong><em>M1</em></strong></p>
<p>\( \Rightarrow x = \frac{3}{2}\pi \) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow \theta = \frac{\pi }{2}\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>Area of circle is \(4\pi \) <strong><em>A1</em></strong></p>
<p>Shaded area is \(\frac{3}{4}\) of the circle <strong><em>(R1)</em></strong></p>
<p>\( \Rightarrow \theta = \frac{\pi }{2}\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{arc length}} = 2\frac{{3\pi }}{2}\) <strong><em>A1</em></strong></p>
<p>\({\text{perimeter}} = 2\frac{{3\pi }}{2} + 2 \times 2\)</p>
<p>\( = 3\pi + 4\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p><strong><em>Total [5 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;">
<p class="p1">Good methods. Some candidates found the larger angle.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally good, some forgot the radii.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the diagram below, AD is perpendicular to BC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">CD = 4, BD = 2 and AD = 3. \({\rm{C}}\hat {\rm{A}}{\rm{D}} = \alpha \) and \({\rm{B}}\hat {\rm{A}}{\rm{D}} = \beta \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 36px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the exact value of \(\cos (\alpha - \beta )\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{AC}} = 5\) and \(\text{AB} = \sqrt {13}\) (may be seen on diagram) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \alpha = \frac{3}{5}\) <strong>and</strong> \(\sin \alpha = \frac{4}{5}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \beta = \frac{3}{{\sqrt {13} }}\) <strong>and</strong> \(\sin \beta = \frac{2}{{\sqrt {13} }}\) <strong><em>(A1)</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> If only the two cosines are correctly given award <strong><em>(A1)(A1)(A0)</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of \(\cos (\alpha - \beta ) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{3}{5} \times \frac{3}{{\sqrt {13} }} + \frac{4}{5} \times \frac{2}{{\sqrt {13} }}\) (substituting) <strong><em>M1</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{17}}{{5\sqrt {13} }}\) \(\left( { = \frac{{17\sqrt {13} }}{{65}}} \right)\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">METHOD 2</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{AC}} = 5\) amd \({\text{AB}} = \sqrt {13} \) (may be seen on diagram) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of \(\cos (\alpha + \beta ) = \frac{{{\text{A}}{{\text{C}}^2} + {\text{A}}{{\text{B}}^2} - {\text{B}}{{\text{C}}^2}}}{{{\text{2(AC)(AB)}}}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{25 + 13 - 36}}{{2 \times 5 \times \sqrt {13} }}\,\,\,\,\,\left( { = \frac{1}{{5\sqrt {13} }}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of \(\cos (\alpha + \beta ) + \cos (\alpha - \beta ) = 2\cos \alpha \cos \beta \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \alpha = \frac{3}{5}\) <strong>and</strong> \(\cos \beta = \frac{3}{{\sqrt {13} }}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos (\alpha - \beta ) = \frac{{17}}{{5\sqrt {13} }}\,\,\,\,\,\left( { = 2 \times \frac{3}{5} \times \frac{3}{{\sqrt {13} }} - \frac{1}{{5\sqrt {13} }}} \right){\text{ }}\left( { = \frac{{17\sqrt {13} }}{{65}}} \right)\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates used a lot of space answering this question, but were generally successful. A few candidates incorrectly used the formula for the cosine of the difference of angles. An interesting alternative solution was noted, in which the side AB is reflected in AD and the required result follows from the use of the cosine rule.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the curve defined by the equation \({x^2} + \sin y - xy = 0\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the gradient of the tangent to the curve at the point \((\pi ,{\text{ }}\pi )\) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, show that \(\tan \theta = \frac{1}{{1 + 2\pi }}\), where \(\theta \) is the acute angle between the tangent to the curve at \((\pi ,{\text{ }}\pi )\) and the line <em>y </em>= <em>x </em>.</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to differentiate implicitly <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2x + \cos y\frac{{{\text{d}}y}}{{{\text{d}}x}} - y - x\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\) <strong><em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>A1 </em></strong>for differentiating \({x^2}\) and sin <em>y </em>; <strong><em>A1 </em></strong>for differentiating <em>xy</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">substitute <em>x </em>and <em>y </em>by \(\pi \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2\pi - \frac{{{\text{d}}y}}{{{\text{d}}x}} - \pi - \pi \frac{{{\text{d}}y}}{{{\text{d}}x}} = 0 \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{\pi }{{1 + \pi }}\) <strong><em>M1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: <em>M1 </em></strong>for attempt to make d<em>y</em>/d<em>x </em>the subject. This could be seen earlier.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = \frac{\pi }{4} - \arctan \frac{\pi }{{1 + \pi }}\) (or seen the other way) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \theta = \tan \left( {\frac{\pi }{4} - \arctan \frac{\pi }{{1 + \pi }}} \right) = \frac{{1 - \frac{\pi }{{1 + \pi }}}}{{1 + \frac{\pi }{{1 + \pi }}}}\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \theta = \frac{1}{{1 + 2\pi }}\) <strong><em>AG</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part a) proved an easy 6 marks for most candidates, while the majority failed to make any headway with part b), with some attempting to find the equation of their line in the form <em>y</em> = <em>mx</em> + <em>c</em> . Only the best candidates were able to see their way through to the given answer.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Part a) proved an easy 6 marks for most candidates, while the majority failed to make any headway with part b), with some attempting to find the equation of their line in the form <em>y</em> = <em>mx</em> + <em>c</em> . Only the best candidates were able to see their way through to the given answer.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{\cos A + \sin A}}{{\cos A - \sin A}} = \sec 2A + \tan 2A\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\cos A + \sin A}}{{\cos A - \sin A}} = \sec 2A + \tan 2A\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">consider right hand side</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sec 2A + \tan 2A = \frac{1}{{\cos 2A}} + \frac{{\sin 2A}}{{\cos 2A}}\) <strong> <em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{\cos }^2}A + 2\sin A\cos A + {{\sin }^2}A}}{{{{\cos }^2}A - {{\sin }^2}A}}\) <strong> <em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for recognizing the need for single angles and <strong><em>A1 </em></strong>for recognizing \({\cos ^2}A + {\sin ^2}A = 1\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{(\cos A + \sin A)}^2}}}{{(\cos A + \sin A)(\cos A - \sin A)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\cos A + \sin A}}{{\cos A - \sin A}}\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\cos A + \sin A}}{{\cos A - \sin A}} = \frac{{{{(\cos A + \sin A)}^2}}}{{(\cos A + \sin A)(\cos A - \sin A)}}\) <strong> <em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{\cos }^2}A + 2\sin A\cos A + {{\sin }^2}A}}{{{{\cos }^2}A - {{\sin }^2}A}}\) <strong> <em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correct numerator and <strong><em>A1 </em></strong>for correct denominator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{1 + \sin 2A}}{{\cos 2A}}\) <strong> <em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sec 2A + \tan 2A\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Solutions to this question were good in general with many candidates realising that multiplying the numerator and denominator by \((\cos A + \sin A)\) might be helpful.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"> </p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Prove the trigonometric identity \(\sin (x + y)\sin (x - y) = {\sin ^2}x - {\sin ^2}y\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given \(f(x) = \sin({x + \frac{\pi }{6}})\sin({x - \frac{\pi }{6}}),{\text{ }}x \in \left[ {0,{\text{ }}\pi } \right]\), find the range of \(f\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given \(g(x) = \csc( {x + \frac{\pi }{6}})\csc( {x - \frac{\pi }{6}}),{\text{ }}x \in \left[ {0,{\text{ }}\pi } \right],{\text{ }}x \ne \frac{\pi }{6},{\text{ }}x \ne \frac{{5\pi }}{6}\), find the range of \(g\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\sin (x + y)\sin (x - y)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = (\sin x\cos y + \cos x\sin y)(\sin x\cos y - \cos x\sin y)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\sin ^2}x{\cos ^2}y + \sin x\sin y\cos x\cos y - \sin x\sin y\cos x\cos y - {\cos ^2}x{\sin ^2}y\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\sin ^2}x{\cos ^2}y - {\cos ^2}x{\sin ^2}y\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\sin ^2}x(1 - {\sin ^2}y) - {\sin ^2}y(1 - {\sin ^2}x)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\sin ^2}x - {\sin ^2}x{\sin ^2}y - {\sin ^2}y + {\sin ^2}x{\sin ^2}y\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\sin ^2}x - {\sin ^2}y\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(f(x) = {\sin ^2}x - \frac{1}{4}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">range is \(f \in \left[ { - \frac{1}{4},{\text{ }}\frac{3}{4}} \right]\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for each end point. Condone incorrect brackets.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(g(x) = \frac{1}{{{{\sin }^2}x - \frac{1}{4}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">range is \(g \in \left] { - \infty ,{\text{ }} - 4} \right] \cup \left[ {\frac{4}{3},{\text{ }}\infty } \right[\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for each part of range. Condone incorrect brackets.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part a) often proved to be an easy 4 marks for candidates. A number were surprisingly content to gain the first 3 marks but were unable to make the final step by substituting \(1 - {\sin ^2}y\) for \({\cos ^2}y\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts b) and c) were more often than not, problematic. Some puzzling ‘working’ was often seen, with candidates making little headway. Otherwise good candidates were able to answer part b), though correct solutions for c) were a rarity. The range \(g \in \left[ { - 4,{\text{ }}\frac{4}{3}} \right]\) was sometimes seen, but gained no marks.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the equation \(\frac{{\sqrt 3 - 1}}{{\sin x}} + \frac{{\sqrt 3 + 1}}{{\cos x}} = 4\sqrt 2 ,{\text{ }}0 < x < \frac{\pi }{2}\). Given that \(\sin \left( {\frac{\pi }{{12}}} \right) = \frac{{\sqrt 6 - \sqrt 2 }}{4}\) and \(\cos \left( {\frac{\pi }{{12}}} \right) = \frac{{\sqrt 6 + \sqrt 2 }}{4}\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">verify that \(x = \frac{\pi }{{12}}\) is a solution to the equation;</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">hence find the other solution to the equation for \(0 < x < \frac{\pi }{2}\).</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 class="p1"><strong>EITHER</strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{LHS}} = \frac{{\sqrt 3 - 1}}{{\frac{{\sqrt 6 - \sqrt 2 }}{4}}} + \frac{{\sqrt 3 + 1}}{{\frac{{\sqrt 6 + \sqrt 2 }}{4}}}\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = \frac{{\sqrt 3 - 1}}{{\frac{{\sqrt 3 - 1}}{{2\sqrt 2 }}}} + \frac{{\sqrt 3 + 1}}{{\frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 2\sqrt 2 + 2\sqrt 2 \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s2">\({\text{LHS}} = 4\sqrt 2 \Rightarrow x = \frac{\pi }{{12}}\) </span>is a solution <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{LHS}} = \frac{{\sqrt 3 - 1}}{{\frac{{\sqrt 6 - \sqrt 2 }}{4}}} + \frac{{\sqrt 3 + 1}}{{\frac{{\sqrt 6 + \sqrt 2 }}{4}}}\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = \frac{{\left( {\sqrt 3 - 1} \right)\left( {\frac{{\sqrt 6 + \sqrt 2 }}{4}} \right) + \left( {\sqrt 3 + 1} \right)\left( {\frac{{\sqrt 6 - \sqrt 2 }}{4}} \right)}}{{\left( {\frac{{\sqrt 6 - \sqrt 2 }}{4}} \right)\left( {\frac{{\sqrt 6 + \sqrt 2 }}{4}} \right)}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s2">\( = 2\sqrt {18} - 2\sqrt 2 \) </span>(or equivalent) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="s2">\({\text{LHS}} = 4\sqrt 2 \Rightarrow x = \frac{\pi }{{12}}\) </span>is a solution <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\frac{{\sqrt 2 }}{4}\left( {\frac{{\sqrt 3 - 1}}{{\sin x}} + \frac{{\sqrt 3 + 1}}{{\cos x}}} \right) = 2 \Rightarrow \frac{{\sin \frac{\pi }{{12}}}}{{\sin x}} + \frac{{\cos \frac{\pi }{{12}}}}{{\cos x}} = 2\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{\sin \frac{\pi }{{12}}\cos x + \cos \frac{\pi }{{12}}\sin x}}{{\sin x\cos x}} = 2\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1">\(\sin \frac{\pi }{{12}}\cos x + \cos \frac{\pi }{{12}}\sin x = 2\sin x\cos x\)</p>
<p class="p1"><span class="Apple-converted-space">\(\sin \left( {\frac{\pi }{{12}} + x} \right) = \sin 2x\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\(\frac{\pi }{{12}} + x = \pi - 2x\) or \(\pi - \left( {\frac{\pi }{{12}} + x} \right) = 2x\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\(x = \frac{{11\pi }}{{36}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[5 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;">
<p class="p1">This question proved to be the most problematic question in the paper.</p>
<p class="p1">Part (a) was generally well done, with competent fraction and surd manipulation seen successfully in leading to the given answer.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question proved to be the most problematic question in the paper.</p>
<p class="p1">The number of scripts seen where part (b) was tackled with complete success numbered in the single figures; solutions were rarely if ever seen. Some candidates scored one mark by finding, or using, the common denominator \(\sin x\cos x\).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the functions \(f(x) = \tan x,{\text{ }}0 \le \ x < \frac{\pi }{2}\) and \(g(x) = \frac{{x + 1}}{{x - 1}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \(g \circ f(x)\), stating its domain.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence show that \(g \circ f(x) = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(y = g \circ f(x)\)<span class="s1">, find an exact value for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) </span>at the point on the graph of \(y = g \circ f(x)\) where \(x = \frac{\pi }{6}\), expressing your answer in the form \(a + b\sqrt 3 ,{\text{ }}a,{\text{ }}b \in \mathbb{Z}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area bounded by the graph of \(y = g \circ f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = \frac{\pi }{6}\) is \(\ln \left( {1 + \sqrt 3 } \right)\).</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 class="p1">\(g \circ f(x) = \frac{{\tan x + 1}}{{\tan x - 1}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\(x \ne \frac{\pi }{4},{\text{ }}0 \le x < \frac{\pi }{2}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\frac{{\tan x + 1}}{{\tan x - 1}} = \frac{{\frac{{\sin x}}{{\cos x}} + 1}}{{\frac{{\sin x}}{{\cos x}} - 1}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p1">\( = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{(\sin x - \cos x)(\cos x - \sin x) - (\sin x + \cos x)(\cos x + \sin x)}}{{{{(\sin x - \cos x)}^2}}}\) <strong><em>M1(A1)</em></strong></p>
<p>\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{(2\sin x\cos x - {{\cos }^2}x - {{\sin }^2}x) - (2\sin x\cos x + {{\cos }^2}x + {{\sin }^2}x)}}{{{{\cos }^2}x + {{\sin }^2}x - 2\sin x\cos x}}\)</p>
<p>\( = \frac{{ - 2}}{{1 - \sin 2x}}\)</p>
<p>Substitute \(\frac{\pi }{6}\) into any formula for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) <strong><em>M1</em></strong></p>
<p>\(\frac{{ - 2}}{{1 - \sin \frac{\pi }{3}}}\)</p>
<p>\( = \frac{{ - 2}}{{1 - \frac{{\sqrt 3 }}{2}}}\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{ - 4}}{{2 - \sqrt 3 }}\)</p>
<p>\( = \frac{{ - 4}}{{2 - \sqrt 3 }}\left( {\frac{{2 + \sqrt 3 }}{{2 + \sqrt 3 }}} \right)\) <strong><em>M1</em></strong></p>
<p>\( = \frac{{ - 8 - 4\sqrt 3 }}{1} = - 8 - 4\sqrt 3 \) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{(\tan x - 1){{\sec }^2}x - (\tan x + 1){{\sec }^2}x}}{{{{(\tan x - 1)}^2}}}\) <strong><em>M1A1</em></strong></p>
<p>\( = \frac{{ - 2{{\sec }^2}x}}{{{{(\tan x - 1)}^2}}}\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{ - 2{{\sec }^2}\frac{\pi }{6}}}{{{{\left( {\tan \frac{\pi }{6} - 1} \right)}^2}}} = \frac{{ - 2\left( {\frac{4}{3}} \right)}}{{{{\left( {\frac{1}{{\sqrt 3 }} - 1} \right)}^2}}} = \frac{{ - 8}}{{{{\left( {1 - \sqrt 3 } \right)}^2}}}\) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for substitution \(\frac{\pi }{6}\).</p>
<p> </p>
<p>\(\frac{{ - 8}}{{{{\left( {1 - \sqrt 3 } \right)}^2}}} = \frac{{ - 8}}{{\left( {4 - 2\sqrt 3 } \right)}}\frac{{\left( {4 + 2\sqrt 3 } \right)}}{{\left( {4 + 2\sqrt 3 } \right)}} = - 8 - 4\sqrt 3 \) <strong><em>M1A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Area \(\left| {\int_0^{\frac{\pi }{6}} {\frac{{\sin x + \cos x}}{{\sin x - \cos x}}{\text{d}}x} } \right|\) <strong><em>M1</em></strong></p>
<p>\( = \left| {\left[ {\ln \left| {\sin x - \cos x} \right|} \right]_0^{\frac{\pi }{6}}} \right|\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone absence of limits and absence of modulus signs at this stage.</p>
<p> </p>
<p>\( = \left| {\ln \left| {\sin \frac{\pi }{6} - \cos \frac{\pi }{6}} \right| - \ln \left| {\sin 0 - \cos 0} \right|} \right|\) <strong><em>M1</em></strong></p>
<p>\( = \left| {\ln \left| {\frac{1}{2} - \frac{{\sqrt 3 }}{2}} \right| - 0} \right|\)</p>
<p>\( = \left| {\ln \left( {\frac{{\sqrt 3 - 1}}{2}} \right)} \right|\) <strong><em>A1</em></strong></p>
<p>\( = - \ln \left( {\frac{{\sqrt 3 - 1}}{2}} \right) = \ln \left( {\frac{2}{{\sqrt 3 - 1}}} \right)\) <strong><em>A1</em></strong></p>
<p>\( = \ln \left( {\frac{2}{{\sqrt 3 - 1}} \times \frac{{\sqrt 3 + 1}}{{\sqrt 3 + 1}}} \right)\) <strong><em>M1</em></strong></p>
<p>\( = \ln \left( {\sqrt 3 + 1} \right)\) <strong><em>AG</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<p><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows the boundary of the cross-section of a water channel.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The equation that represents this boundary is \(y = 16\sec \left( {\frac{{\pi x}}{{36}}} \right) - 32\) where <em>x</em> and <em>y</em> are both measured in cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The top of the channel is level with the ground and has a width of 24 cm. The maximum depth of the channel is 16 cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the width of the water surface in the channel when the water depth is 10 cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Give your answer in the form \(a\arccos b\) where \(a,{\text{ }}b \in \mathbb{R}\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">10 cm water depth corresponds to \(16\sec \left( {\frac{{\pi x}}{{36}}} \right) - 32 = - 6\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Rearranging to obtain an equation of the form \(\sec \left( {\frac{{\pi x}}{{36}}} \right) = k\) or equivalent</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>i.e.</em> making a trignometrical function the subject of the equation. <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \left( {\frac{{\pi x}}{{36}}} \right) = \frac{8}{{13}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\pi x}}{{36}} = \pm \arccos \frac{8}{{13}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \pm \frac{{36}}{\pi }\arccos \frac{8}{{13}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not penalise the omission of ±.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Width of water surface is \(\frac{{72}}{\pi }\arccos \frac{8}{{13}}{\text{ (cm)}}\) <strong><em>R1</em></strong> <strong><em>N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Candidate who starts with 10 instead of −6 has the potential to gain the two <strong><em>M1</em></strong> marks and the <strong><em>R1</em></strong> mark.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a question in context which proved difficult for many candidates. Many appeared not to have fully comprehended the implications of the details of the diagram. A few candidates attempted integration, for no apparent reason.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">In the triangle ABC, </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{A\hat BC}} = 90^\circ\)</span> , \({\text{AC}} = \sqrt {\text{2}}\) and AB = BC + 1.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that cos \(\hat A - \sin \hat A = \frac{1}{{\sqrt 2 }}\).</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><span style="font-family: 'times new roman', times; font-size: medium;">By squaring both sides of the equation in part (a), solve the equation to find the angles in the triangle.</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><span style="font-family: 'times new roman', times; font-size: medium;">Apply Pythagoras’ theorem in the triangle ABC to find BC, and hence show that \(\sin \hat A = \frac{{\sqrt 6 - \sqrt 2 }}{4}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, calculate the length of the perpendicular from B to [AC].</span></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><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \hat A = \frac{{{\text{BA}}}}{{\sqrt 2 }}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \hat A = \frac{{{\text{BC}}}}{{\sqrt 2 }}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \hat A - \sin \hat A = \frac{{{\text{BA}} - {\text{BC}}}}{{\sqrt 2 }}\) <strong><em>R1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{{\sqrt 2 }}\) <strong><em>AG</em></strong></span></p>
<p><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\cos ^2}\hat A - 2\cos \hat A\sin \hat A + {\sin ^2}\hat A = \frac{1}{2}\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - 2\sin \hat A\cos \hat A = \frac{1}{2}\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin 2\hat A = \frac{1}{2}\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(2\hat A = 30^\circ \) <strong><em>A1</em></strong></span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">angles in the triangle are 15° and 75° <strong><em>A1A1</em></strong></span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong>Accept answers in radians.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{B}}{{\text{C}}^2} + {({\text{BC}} + 1)^2} = 2\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(2{\text{B}}{{\text{C}}^2} + 2{\text{BC}} - 1 = 0\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{BC}} = \frac{{ -2 + \sqrt {12} }}{4}\left( { = \frac{{\sqrt 3 - 1}}{2}} \right)\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \hat A = \frac{{{\text{BC}}}}{{\sqrt 2 }} = \frac{{\sqrt 3 - 1}}{{2\sqrt 2 }}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sqrt 6 - \sqrt 2 }}{4}\) <strong><em>AG</em></strong></span></p>
<p><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(h = {\text{ABsin}}\hat A\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = ({\text{BC}} + 1)\sin \hat A\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sqrt 3 + 1}}{2} \times \frac{{\sqrt 6 - \sqrt 2 }}{4} = \frac{{\sqrt 2 }}{4}\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\tfrac{1}{2}AB.BC = \tfrac{1}{2}AC.h\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sqrt 3 - 1}}{2} \cdot \frac{{\sqrt {3 + 1} }}{2} = \sqrt {2h} \) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{2}{4} = \sqrt 2 h\) <strong><em>M1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(h = \frac{1}{{2\sqrt 2 }}\) <strong><em>A1</em></strong></span></p>
<p><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span></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;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many good solutions to this question, although some students incorrectly stated the value of \(\arcsin \left( {\frac{1}{2}} \right)\). A surprising number of students had greater difficulties with part (d).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Many good solutions to this question, although some students incorrectly stated the value of \(\arcsin \left( {\frac{1}{2}} \right)\). A surprising number of students had greater difficulties with part (d).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Many good solutions to this question, although some students incorrectly stated the value of \(\arcsin \left( {\frac{1}{2}} \right)\). A surprising number of students had greater difficulties with part (d).</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Many good solutions to this question, although some students incorrectly stated the value of \(\arcsin \left( {\frac{1}{2}} \right)\). A surprising number of students had greater difficulties with part (d).</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows a curve with equation \(y = 1 + k\sin x\) , defined for \(0 \leqslant x \leqslant 3\pi \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 29px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point \({\text{A}}\left( {\frac{\pi }{6}, - 2} \right)\) lies on the curve and \({\text{B}}(a,{\text{ }}b)\) is the maximum point.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>k</em> = – 6 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence, find the values of <em>a</em> and <em>b</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \( - 2 = 1 + k\sin \left( {\frac{\pi }{6}} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 3 = \frac{1}{2}k\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = - 6\) <strong><em>AG</em></strong> <strong><em>N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">maximum \( \Rightarrow \sin x = - 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = \frac{{3\pi }}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(b = 1 - 6( - 1)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 7\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y' = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k\cos x = 0 \Rightarrow x = \frac{\pi }{2},{\text{ }}\frac{{3\pi }}{2},{\text{ }} \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = \frac{{3\pi }}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(b = 1 - 6( - 1)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 7\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1A1</em></strong> for \(\left( {\frac{{3\pi }}{2},{\text{ }}7} \right)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was the most successfully answered question in the paper. Part (a) was done well by most candidates. In part (b), a small number of candidates used knowledge about transformations of functions to identify the coordinates of B. Most candidates used differentiation.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Show that \({(1 + {\text{i}}\tan \theta )^n} + {(1 - {\text{i}}\tan \theta )^n} = \frac{{2\cos n\theta }}{{{{\cos }^n}\theta }},\;\;\;\cos \theta \ne 0\).</p>
<p>(ii) Hence verify that \({\text{i}}\tan \frac{{3\pi }}{8}\) is a root of the equation \({(1 + z)^4} + {(1 - z)^4} = 0,\;\;\;z \in \mathbb{C}\).</p>
<p>(iii) State another root of the equation \({(1 + z)^4} + {(1 - z)^4} = 0,\;\;\;z \in \mathbb{C}\).</p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Use the double angle identity \(\tan 2\theta = \frac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}\) to show that \(\tan \frac{\pi }{8} = \sqrt 2 - 1\).</p>
<p>(ii) Show that \(\cos 4x = 8{\cos ^4}x - 8{\cos ^2}x + 1\).</p>
<p>(iii) Hence find the value of \(\int_0^{\frac{\pi }{8}} {\frac{{2\cos 4x}}{{{{\cos }^2}x}}{\text{d}}x} \).</p>
<div class="marks">[13]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) <strong> METHOD 1</strong></p>
<p>\({(1 + {\text{i}}\tan \theta )^n} + {(1 - {\text{i}}\tan \theta )^n} = {\left( {1 + {\text{i}}\frac{{\sin \theta }}{{\cos \theta }}} \right)^n} + {\left( {1 - {\text{i}}\frac{{\sin \theta }}{{\cos \theta }}} \right)^n}\) <strong><em>M1</em></strong></p>
<p>\( = {\left( {\frac{{\cos \theta + i\sin \theta }}{{\cos \theta }}} \right)^n} + {\left( {\frac{{\cos \theta - i\sin \theta }}{{\cos \theta }}} \right)^n}\) <strong><em>A1</em></strong></p>
<p>by de Moivre’s theorem (<strong><em>M1)</em></strong></p>
<p>\({\left( {\frac{{\cos \theta + i\sin \theta }}{{\cos \theta }}} \right)^n} = \frac{{\cos n\theta + i\sin n\theta }}{{{{\cos }^n}\theta }}\) <strong><em>A1</em></strong></p>
<p>recognition that \(\cos \theta - i\sin \theta \) is the complex conjugate of \(\cos \theta + i\sin \theta \) (<strong><em>R1)</em></strong></p>
<p>use of the fact that the operation of complex conjugation commutes with the operation of raising to an integer power:</p>
<p>\({\left( {\frac{{\cos \theta - i\sin \theta }}{{\cos \theta }}} \right)^n} = \frac{{\cos n\theta - i\sin n\theta }}{{{{\cos }^n}\theta }}\) <strong><em>A1</em></strong></p>
<p>\({(1 + {\text{i}}\tan \theta )^n} + {(1 - {\text{i}}\tan \theta )^n} = \frac{{2\cos n\theta }}{{{{\cos }^n}\theta }}\) <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\({(1 + {\text{i}}\tan \theta )^n} + {(1 - {\text{i}}\tan \theta )^n} = {(1 + {\text{i}}\tan \theta )^n} + {\left( {1 + {\text{i}}\tan ( - \theta )} \right)^n}\) <strong><em>(M1)</em></strong></p>
<p>\( = \frac{{{{(\cos \theta + i\sin \theta )}^n}}}{{{{\cos }^n}\theta }} + \frac{{{{\left( {\cos ( - \theta ) + i\sin ( - \theta )} \right)}^n}}}{{{{\cos }^n}\theta }}\) <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for converting to cosine and sine terms.</p>
<p> </p>
<p>use of de Moivre’s theorem <strong><em>(M1)</em></strong></p>
<p>\( = \frac{1}{{{{\cos }^n}\theta }}\left( {\cos n\theta + {\text{i}}\sin n\theta + \cos ( - n\theta ) + {\text{i}}\sin ( - n\theta )} \right)\) <strong><em>A1</em></strong></p>
<p>\( = \frac{{2\cos n\theta }}{{{{\cos }^2}\theta }}\;\;\;{\text{as}}\;\;\;\cos ( - n\theta ) = \cos n\theta \;\;\;{\text{and}}\;\;\;\sin ( - n\theta ) = - \sin n\theta \) <strong><em>R1AG</em></strong></p>
<p>(ii) \({\left( {1 + {\text{i}}\tan \frac{{3\pi }}{8}} \right)^4} + {\left( {1 - {\text{i}}\tan \frac{{3\pi }}{8}} \right)^4} = \frac{{2\cos \left( {4 \times \frac{{3\pi }}{8}} \right)}}{{{{\cos }^4}\frac{{3\pi }}{8}}}\) <strong><em>(A1)</em></strong></p>
<p>\( = \frac{{2\cos \frac{{3\pi }}{2}}}{{{{\cos }^4}\frac{{3\pi }}{8}}}\) <strong><em>A1</em></strong></p>
<p>\( = 0\;\;\;{\text{as}}\;\;\;\cos \frac{{3\pi }}{2} = 0\) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>The above working could involve theta and the solution of \(\cos (4\theta ) = 0\).</p>
<p> </p>
<p>so \({\text{i}}\tan \frac{{3\pi }}{8}\) is a root of the equation <strong><em>AG</em></strong></p>
<p>(iii) either \( - {\text{i}}\tan \frac{{3\pi }}{8}\;\;\;{\text{or}}\;\;\; - {\text{i}}\tan \frac{\pi }{8}\;\;\;{\text{or}}\;\;\;{\text{i}}\tan \frac{\pi }{8}\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept \({\text{i}}\tan \frac{{5\pi }}{8}\;\;\;{\text{or}}\;\;\;{\text{i}}\tan \frac{{7\pi }}{8}\).</p>
<p>Accept \( - \left( {1 + \sqrt 2 } \right){\text{i}}\;\;\;{\text{or}}\;\;\;\left( {1 - \sqrt 2 } \right){\text{i}}\;\;\;{\text{or}}\;\;\;\left( { - 1 + \sqrt 2 } \right){\text{i}}\).</p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(\tan \frac{\pi }{4} = \frac{{2\tan \frac{\pi }{8}}}{{1 - {{\tan }^2}\frac{\pi }{8}}}\) <strong><em>(M1)</em></strong></p>
<p>\({\tan ^2}\frac{\pi }{8} + 2\tan \frac{\pi }{8} - 1 = 0\) <strong><em>A1</em></strong></p>
<p>let \(t = \tan \frac{\pi }{8}\)</p>
<p>attempting to solve \({t^2} + 2t - 1 = 0\;\;\;{\text{for}}\;\;\;t\) <strong><em>M1</em></strong></p>
<p>\(t = - 1 \pm \sqrt 2 \) <strong><em>A1</em></strong></p>
<p>\(\frac{\pi }{8}\) is a first quadrant angle and tan is positive in this quadrant, so</p>
<p>\(\tan \frac{\pi }{8} > 0\) <strong><em>R1</em></strong></p>
<p>\(\tan \frac{\pi }{8} = \sqrt 2 - 1\) <strong><em>AG</em></strong></p>
<p>(ii) \(\cos 4x = 2{\cos ^2}2x - 1\) <strong><em>A1</em></strong></p>
<p>\( = 2{\left( {2{{\cos }^2}x - 1} \right)^2} - 1\) <strong><em>M1</em></strong></p>
<p>\( = 2\left( {4{{\cos }^4}x - 4{{\cos }^2}x + 1} \right) - 1\) <strong><em>A1</em></strong></p>
<p>\( = 8{\cos ^4}x - 8{\cos ^2}x + 1\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept equivalent complex number derivation.</p>
<p> </p>
<p>(iii) \(\int_0^{\frac{\pi }{8}} {\frac{{2\cos 4x}}{{{{\cos }^2}x}}{\text{d}}x = 2} \int_0^{\frac{\pi }{8}} {\frac{{8{{\cos }^4}x - 8{{\cos }^2}x + 1}}{{{{\cos }^2}x}}{\text{d}}x} \)</p>
<p>\( = 2\int_0^{\frac{\pi }{8}} {8{{\cos }^2}x - 8 + {{\sec }^2}x{\text{d}}x} \) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>The <strong><em>M1 </em></strong>is for an integrand involving no fractions.</p>
<p> </p>
<p>use of \({\cos ^2}x = \frac{1}{2}(\cos 2x + 1)\) <strong><em>M1</em></strong></p>
<p>\( = 2\int_0^{\frac{\pi }{8}} {4\cos 2x - 4 + {{\sec }^2}x{\text{d}}x} \) <strong><em>A1</em></strong></p>
<p>\( = [4\sin 2x - 8x + 2\tan x]_0^{\frac{\pi }{8}}\) <strong><em>A1</em></strong></p>
<p>\( = 4\sqrt 2 - \pi - 2\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p><strong><em>[13 marks]</em></strong></p>
<p><strong><em>Total [23 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;">
<p class="p1">Fairly successful.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Most candidates attempted to use the hint. Those who doubled the angle were usually successful – but many lost the final mark by not giving a convincing reason to reject the negative solution to the intermediate quadratic equation. Those who halved the angle got nowhere.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>The majority of candidates obtained full marks.</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>This was poorly answered, few candidates realising that part of the integrand could be re-expressed using \(\frac{1}{{{{\cos }^2}x}} = {\sec ^2}x\), which can be immediately integrated.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the triangle PQR, PQ = 6 , PR = <em>k </em>and <span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{P\hat QR}} = 30^\circ \) .</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For the case <em>k </em>= 4 , find the two possible values of QR.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the values of <em>k </em>for which the conditions above define a unique triangle.</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to apply cosine rule <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({4^2} = {6^2} + {\text{Q}}{{\text{R}}^2} - 2 \cdot {\text{QR}} \cdot 6\cos 30^\circ\) ( <strong>or </strong></span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Q}}{{\text{R}}^2} - 6\sqrt 3 {\text{ QR}} + 20 = 0\)</span> ) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{QR}} = 3\sqrt 3 + \sqrt 7 {\text{ or QR}} = 3\sqrt 3 - \sqrt 7 \) <em><strong>A1A1</strong></em><br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k \geqslant 6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 6\sin 30^\circ = 3\) <em><strong>M1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">The </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">in (b) is for recognizing the right-angled triangle case.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k \geqslant 6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">use of discriminant: \(108 - 4(36 - {k^2}) = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>k </em>= 3 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><em style="font-family: 'times new roman', times; font-size: medium;">k </em><span style="font-family: 'times new roman', times; font-size: medium;">= ±3 is </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A0</em>.</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates using the sine rule here made little or no progress. With the cosine rule, the two values are obtained quite quickly, which was the case for a majority of candidates. A small number were able to write down the correct quadratic equation to be solved, but then made arithmetical errors en route to their final solution(s). Part b) was often left blank. The better candidates were able to deduce <em>k</em> = 3 , though the solution \(k \geqslant 6\) was rarely, if at all, seen by examiners.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates using the sine rule here made little or no progress. With the cosine rule, the two values are obtained quite quickly, which was the case for a majority of candidates. A small number were able to write down the correct quadratic equation to be solved, but then made arithmetical errors en route to their final solution(s). Part b) was often left blank. The better candidates were able to deduce <em>k</em> = 3 , though the solution \(k \geqslant 6\) was rarely, if at all, seen by examiners.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The following diagram shows a sector of a circle where \({\rm{A\hat OB}} = x\) radians and the length of the \({\text{arc AB}} = \frac{2}{x}{\text{ cm}}\).</p>
<p class="p1">Given that the area of the sector is \(16{\text{ c}}{{\text{m}}^2}\), find the length of the arc \(AB\)<span class="s1">.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2016-01-28_om_15.11.09.png" alt></span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">\({\text{arc length}} = \frac{2}{x} = rx\;\;\;\left( { \Rightarrow r = \frac{2}{{{x^2}}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(16 = \frac{1}{2}{\left( {\frac{2}{{{x^2}}}} \right)^2}x\;\;\;\left( { \Rightarrow \frac{2}{{{x^3}}} = 16} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <em><strong>M1</strong></em>s for attempts at the use of arc-length and sector-area formulae.</p>
<p class="p4"> </p>
<p class="p1">\(x = \frac{1}{2}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\({\text{arc length}} = {\text{4 (cm)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p5"><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\frac{\pi }{2} < \alpha < \pi \) and \(\cos \alpha = - \frac{3}{4}\), find the value of sin 2<span style="font: 12.5px Times;">α </span>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \alpha = \sqrt {1 - {{\left( { - \frac{3}{4}} \right)}^2}} = \frac{{\sqrt 7 }}{4}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to use double angle formula <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin 2\alpha = 2\frac{{\sqrt 7 }}{4}\left( { - \frac{3}{4}} \right) = - \frac{{3\sqrt 7 }}{8}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sqrt 7 }}{4}\) seen would normally be awarded </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates scored full marks on this question, though their explanations for part a) often lacked clarity. Most preferred to use some kind of right-angled triangle rather than (perhaps in this case) the more sensible identity \({\sin ^2}\alpha + {\cos ^2}\alpha = 1\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The vectors <strong><em>a</em></strong> , <strong><em>b</em></strong> , <strong><em>c</em></strong> satisfy the equation <strong><em>a</em></strong> + <strong><em>b</em></strong> + <strong><em>c</em></strong> = <strong>0</strong> . Show that <strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong> = <strong><em>c</em></strong> \( \times \) <strong><em>a</em></strong> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">taking cross products with <strong><em>a</em></strong>, <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>a</em></strong> \( \times \) (<strong><em>a</em></strong> + <strong><em>b</em></strong> + <strong><em>c</em></strong>) = <strong><em>a</em></strong> \( \times \) <strong><em>0</em></strong> = <strong><em>0</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">using the algebraic properties of vectors and the fact that <strong><em>a</em></strong> \( \times \) <strong><em>a</em></strong> = <strong><em>0</em></strong> , <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> + <strong><em>a</em></strong> \( \times \) <strong><em>c</em></strong> = <strong><em>0</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = <strong><em>c</em></strong> \( \times \) <strong><em>a</em></strong> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">taking cross products with <strong><em>b</em></strong>, <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>b</em></strong> \( \times \) (<strong><em>a</em></strong> + <strong><em>b</em></strong> + <strong><em>c</em></strong>) = <strong><em>0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>b</em></strong> \( \times \) <strong><em>a</em></strong> + <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong> = <strong><em>0</em></strong> <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>a</em></strong> \( \times \) <strong><em>b</em></strong> = <strong><em>b</em></strong> \( \times \) <strong><em>c</em></strong> <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">this completes the proof</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following functions:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(h(x) = \arctan (x),{\text{ }}x \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(g(x) = \frac{1}{x}\), \(x\in \mathbb{R}\)</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">, \({\text{ }}x \ne 0\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = h(x)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for the composite function \(h \circ g(x)\) and state its domain.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f(x) = h(x) + h \circ g(x)\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) find \(f'(x)\) in simplified form;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) show that \(f(x) = \frac{\pi }{2}\) for \(x > 0\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Nigel states that \(f\) is an odd function and Tom argues that \(f\) is an even function.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State who is correct and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence find the value of \(f(x)\) for \(x < 0\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/maths_14a_markscheme.png" alt> <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> <strong><em>A1</em></strong> for correct shape, <strong><em>A1 </em></strong>for asymptotic behaviour at \(y = \pm \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(h \circ g(x) = \arctan \left( {\frac{1}{x}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">domain of \(h \circ g\) is equal to the domain of \(g:x \in \circ ,{\text{ }}x \ne 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(f(x) = \arctan (x) + \arctan \left( {\frac{1}{x}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = \frac{1}{{1 + {x^2}}} + \frac{1}{{1 + \frac{1}{{{x^2}}}}} \times - \frac{1}{{{x^2}}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = \frac{1}{{1 + {x^2}}} + \frac{{ - \frac{1}{{{x^2}}}}}{{\frac{{{x^2} + 1}}{{{x^2}}}}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{{1 + {x^2}}} - \frac{1}{{1 + {x^2}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>f </em>is a constant <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">when \(x > 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(1) = \frac{\pi }{4} + \frac{\pi }{4}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{\pi }{2}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/maths_14c_markscheme.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">from diagram</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = \arctan \frac{1}{x}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha = \arctan x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta + \alpha = \frac{\pi }{2}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">hence \(f(x) = \frac{\pi }{2}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \left( {f(x)} \right) = \tan \left( {\arctan (x) + \arctan \left( {\frac{1}{x}} \right)} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{x + \frac{1}{x}}}{{1 - x\left( {\frac{1}{x}} \right)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">denominator = 0, so \(f(x) = \frac{\pi }{2}{\text{ (for }}x > 0)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Nigel is correct. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arctan (x)\) is an odd function and \(\frac{1}{x}\) is an odd function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">composition of two odd functions is an odd function and sum of two odd functions is an odd function <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f( - x) = \arctan ( - x) + \arctan \left( { - \frac{1}{x}} \right) = - \arctan (x) - \arctan \left( {\frac{1}{x}} \right) = - f(x)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore <em>f </em>is an odd function. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(f(x) = - \frac{\pi }{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\sin \left( {\theta + \frac{\pi }{2}} \right) = \cos \theta \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Consider \(f(x) = \sin (ax)\) where \(a\) is a constant. Prove by mathematical induction that \({f^{(n)}}(x) = {a^n}\sin \left( {ax + \frac{{n\pi }}{2}} \right)\) where \(n \in {\mathbb{Z}^ + }\) and \({f^{(n)}}(x)\) represents the \({{\text{n}}^{{\text{th}}}}\) derivative of \(f(x)\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\sin \left( {\theta + \frac{\pi }{2}} \right) = \sin \theta \cos \frac{\pi }{2} + \cos \theta \sin \frac{\pi }{2}\) <strong><em>M1</em></strong></p>
<p>\( = \cos \theta \) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept a transformation/graphical based approach.</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 class="p1">consider \(n = 1,{\text{ }}f'(x) = a\cos (ax)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="s1">since </span>\(\sin \left( {ax + \frac{\pi }{2}} \right) = \cos ax\) then the proposition is true for \(n = 1\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">assume that the proposition is true for \(n = k\) so \({f^{(k)}}(x) = {a^k}\sin \left( {ax + \frac{{k\pi }}{2}} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\({f^{(k + 1)}}(x) = \frac{{{\text{d}}\left( {{f^{(k)}}(x)} \right)}}{{{\text{d}}x}}\;\;\;\left( { = a\left( {{a^k}\cos \left( {ax + \frac{{k\pi }}{2}} \right)} \right)} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = {a^{k + 1}}\sin \left( {ax + \frac{{k\pi }}{2} + \frac{\pi }{2}} \right)\) (using part (a)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\( = {a^{k + 1}}\sin \left( {ax + \frac{{(k + 1)\pi }}{2}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">given that the proposition is true for \(n = k\) then we have shown that the proposition is true for \(n = k + 1\). Since we have shown that the proposition is true for \(n = 1\) then the proposition is true for all \(n \in {\mathbb{Z}^ + }\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award final <strong><em>R1 </em></strong>only if all prior <em><strong>M</strong></em> and <em><strong>R</strong></em> marks have been awarded.</p>
<p class="p3"><em><strong>[7 marks]</strong></em></p>
<p class="p3"><em><strong>Total [8 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>Solve the equation \({\sec ^2}x + 2\tan x = 0,{\text{ }}0 \leqslant x \leqslant 2\pi \).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>use of \({\sec ^2}x = {\tan ^2}x + 1\) <em><strong>M1</strong></em></p>
<p>\({\tan ^2}x + 2\tan x + 1 = 0\)</p>
<p>\({(\tan x + 1)^2} = 0\) <em><strong>(M1)</strong></em></p>
<p>\(\tan x = - 1\) <em><strong>A1</strong></em></p>
<p>\(x = \frac{{3\pi }}{4},{\text{ }}\frac{{7\pi }}{4}\) <em><strong>A1A1</strong></em></p>
<p><strong>METHOD 2</strong></p>
<p>\(\frac{1}{{{{\cos }^2}x}} + \frac{{2\sin x}}{{\cos x}} = 0\) <em><strong>M1</strong></em></p>
<p>\(1 + 2\sin x\cos x = 0\)</p>
<p>\(\sin 2x = - 1\) <em><strong>M1A1</strong></em></p>
<p>\(2x = \frac{{3\pi }}{2},{\text{ }}\frac{{7\pi }}{2}\)</p>
<p>\(x = \frac{{3\pi }}{4},{\text{ }}\frac{{7\pi }}{4}\) <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1A0 </strong></em>if extra solutions given or if solutions given in degrees (or both).</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(\sin 2nx = \sin \left( {(2n + 1)x} \right)\cos x - \cos \left( {(2n + 1)x} \right)\sin x\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong> prove, by induction, that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\cos x + \cos 3x + \cos 5x + \ldots + \cos \left( {(2n - 1)x} \right) = \frac{{\sin 2nx}}{{2\sin x}},\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for all \(n \in {\mathbb{Z}^ + }{\text{, }}\sin x \ne 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Solve the equation \(\cos x + \cos 3x = \frac{1}{2},{\text{ }}0 < x < \pi \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\sin (2n + 1)x\cos x - \cos (2n + 1)x\sin x = \sin (2n + 1)x - x\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \sin 2nx\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) if <em>n</em> = 1 <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{LHS}} = \cos x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{RHS}} = \frac{{\sin 2x}}{{2\sin x}} = \frac{{2\sin x\cos x}}{{2\sin x}} = \cos x\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so LHS = RHS and the statement is true for <em>n</em> = 1 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">assume true for <em>n</em> = <em>k</em> <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only award <strong><em>M1</em></strong> if the word <strong>true</strong> appears.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> Do <strong>not</strong> award <strong><em>M1</em></strong> for ‘let <em>n</em> = <em>k</em>’ only.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> Subsequent marks are independent of this <strong><em>M1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(\cos x + \cos 3x + \cos 5x + \ldots + \cos (2k - 1)x = \frac{{\sin 2kx}}{{2\sin x}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">if <em>n</em> = <em>k</em> + 1 then</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos x + \cos 3x + \cos 5x + \ldots + \cos (2k - 1)x + \cos (2k + 1)x\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin 2kx}}{{2\sin x}} + \cos (2k + 1)x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin 2kx + 2\cos (2k + 1)x\sin x}}{{2\sin x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin (2k + 1)x\cos x - \cos (2k + 1)x\sin x + 2\cos (2k + 1)x\sin x}}{{2\sin x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin (2k + 1)x\cos x + \cos (2k + 1)x\sin x}}{{2\sin x}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin (2k + 2)x}}{{2\sin x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin 2(k + 1)x}}{{2\sin x}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so if true for <em>n</em> = <em>k</em>, then also true for <em>n</em> = <em>k</em> + <em>1</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as true for <em>n</em> = 1 then true for all \(n \in {\mathbb{Z}^ + }\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Final <strong><em>R1</em></strong> is independent of previous work.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[12 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\frac{{\sin 4x}}{{2\sin x}} = \frac{1}{2}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin 4x = \sin x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4x = x \Rightarrow x = 0\) but this is impossible</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4x = \pi - x \Rightarrow x = \frac{\pi }{5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4x = 2\pi + x \Rightarrow x = \frac{{2\pi }}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4x = 3\pi - x \Rightarrow x = \frac{{3\pi }}{5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for not including any answers outside the domain <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award the first <strong><em>M1A1</em></strong> for correctly obtaining \(8{\cos ^3}x - 4\cos x - 1 = 0\) or equivalent and subsequent marks as appropriate including the answers \(\left( { - \frac{1}{2},\frac{{1 \pm \sqrt 5 }}{4}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [20 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question showed the weaknesses of many candidates in dealing with formal proofs and showing their reasoning in a logical manner. In part (a) just a few candidates clearly showed the result and part (b) showed that most candidates struggle with the formality of a proof by induction. The logic of many solutions was poor, though sometimes contained correct trigonometric work. Very few candidates were successful in answering part (c) using the unit circle. Most candidates attempted to manipulate the equation to obtain a cubic equation but made little progress. A few candidates guessed \(\frac{{2\pi }}{3}\) as a solution but were not able to determine the other solutions.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A triangle has sides of length \(({n^2} + n + 1)\), \((2n + 1)\) and \(({n^2} - 1)\) where \(n > 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Explain why the side \(({n^2} + n + 1)\) must be the longest side of the triangle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that the largest angle, \(\theta \), of the triangle is \(120^\circ \).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) a reasonable attempt to show either that \({n^2} + n + 1 > 2n + 1\) or</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({n^2} + n + 1 > {n^2} - 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">complete solution to each inequality </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\cos \theta = \frac{{{{(2n + 1)}^2} + {{({n^2} - 1)}^2} - {{({n^2} + n + 1)}^2}}}{{2(2n + 1)({n^2} - 1)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{ - 2{n^3} - {n^2} + 2n + 1}}{{2(2n + 1)({n^2} - 1)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \frac{{(n - 1)(n + 1)(2n + 1)}}{{2(2n + 1)({n^2} - 1)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \frac{1}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\theta = 120^\circ \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There were very few complete and accurate answers to part a). The most common incorrect response was to state the triangle inequality and feel that this was sufficient.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many substituted a particular value for n and illustrated the result. Most students recognised the need for the Cosine rule and applied it correctly. Many then expanded and simplified to the correct answer. There was significant fudging in the middle on some papers. There were many good responses to this question.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\sin x + \cos x = \frac{2}{3}\), find \(\cos 4x\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\sin ^2}x + {\cos ^2}x + 2\sin x\cos x = \frac{4}{9}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">using \({\sin ^2}x + {\cos ^2}x = 1\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(2\sin x\cos x = - \frac{5}{9}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">using \(2\sin x\cos x = \sin 2x\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin 2x = - \frac{5}{9}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos 4x = 1 - 2{\sin ^2}2x\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award this <strong><em>M1 </em></strong>for decomposition of cos 4<em>x </em>using double angle formula anywhere in the solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - 2 \times \frac{{25}}{{81}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{31}}{{81}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Find all solutions to the equation \(\tan x + \tan 2x = 0\) where \(0^\circ \le x < 360^\circ\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\(\tan x + \tan 2x = 0\)</p>
<p>\(\tan x + \frac{{2\tan x}}{{1 - {{\tan }^2}x}} = 0\) <strong><em>M1</em></strong></p>
<p>\(\tan x - {\tan ^3}x + 2\tan x = 0\) <strong><em>A1</em></strong></p>
<p>\(\tan x(3 - {\tan ^2}x) = 0\) <strong>(<em>M1)</em></strong></p>
<p>\(\tan x = 0 \Rightarrow x = 0,{\text{ }}x = 180^\circ \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>If \(x = 360^\circ \) seen anywhere award <strong><em>A0</em></strong></p>
<p> </p>
<p>\(\tan x = \sqrt 3 \Rightarrow x = 60^\circ ,{\text{ }}240^\circ \) <strong><em>A1</em></strong></p>
<p>\(\tan x = - \sqrt 3 \Rightarrow x = 120^\circ ,{\text{ }}300^\circ \) <strong><em>A1</em></strong></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="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows two straight lines intersecting at O and two circles, each with centre O. The outer circle has radius <em>R</em> and the inner circle has radius <em>r</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"><img 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n3qavmPzQXFioIXWEGRilxhPtnpoZIkxWButLABBdHmnzneferIEiectxuYGvOeHkrp4GEJ5reKn1iBR+hMPN7+RT7aLP55NTkSCYUjoXCkd+qKUtwsd3gZhmGAQ+Fs96kj6+SJyemzZzt8Y54A+RlhH3W+pIdSKtrTsuGanZ0TLRhUDyS/25t/Vt4oLIxEhpOZiuyUt0lhNTkSGf59YfNRuxfQ7vB5jxeEhNp9Ux4jcnTPl/SA8IvfKTZ8dFiKhEWbybK/F5fGIr1z2Y1qudl+cGV4T2T4anG7DXPI7e7T7heEhAT3dIaiUQHDjj6mBxxg8RPzEj2d7M8/K5OV0VC4L1nYqvvJVmGhLzQQV76zI0Lcy39sLqgSEvx7xPS8x/RArZ3IGzdnTm4VPKVihYlCejm9e5T6R7IyGQkNXi78UP8zkKfIyAppJUJedp92v/bt7RctgVhDMpHAgYpe0gP1I9y3ZvMlfkC6Ia2SZT8UkoO7aRDdKlzuDUd6FwpbzUSIlf+Ib8nCOnli8pWXX3Hm83UH+Eh5X0WAkEOD4IQG3p9VJzRt6ehKg1j3affD571c4oelUYM8pmcxlToydIT71my+Tp6YlNc83n3+WWtfrFGoiFKu3addLnicCFJF1RAsU/SYnmQiIX6B4q9f+LXUYULWsVWTLGsVk24gT2x4yAvjLwgexWu4ILQncmoMyxQ9pufihYtuatBXn1+/NHLqD//Z3euIX1/bkl3mn+2Smy9/lz0/0Hde2aj2w5iWzc7OSeR/1WuQyOkF1CCPcVmD1q+dG3jiH84tf9G1Bgme0LVD4/ln2w+zyZoaxbtXzgz0nUl/U12jKE73aZdL8G8TNchjetxs1FCunvvHSGjPrxb+vZvXgU4xwedj2KfR/LMtUviPK2cGKr0aw8nVL6t6NVhbrCDdp/7WIKja9c1+E58e1xo17vxpYfzJULh7DRI/gtAubc0/E7P7tJslfomQ4CrpM9zSoNtrb4wc/s3li79+MvT0ufe+RA2qAVo6hqLRJs9ba/ept+U/X14993TFKGNr4NTF1OrnTrz+yROTgmcYUIO8xB0NWr92buDn5977PLMwFgn98uKHXUVPfalB1NJf2rBLm3f36Vf/mTr9ZGjvZOrz3J27ufWPF04+Gwn9dGRh7bYTGvTapUvef+D2QQ3yEjc0SLl67tivLl7/vPI4HVv49C/dvCCMcJWuSNoOux0WlM/nh6JRrt2noEE7z4/bH84PhsKRY29knNCgixcu8vrM7YAa5CWOaxALA5mrd+bq7a5eU94iaZtAsuzExMTFCxcuXrhw4V//VYDu08/fObk3cnj+es4qSU8MXvzYETsINQhhOKxBt9feGDl85p1P7+Tu3M3lPr54+AnLPkYN2pX19XVr8OXChVf4qc/d3J27udvL53rDlaKK3M2rCzO/+skTT5984/r6V92/OGoQYsVRDaqEgRTrPu7eeg+CBkFNClvcG1ArntfO+sedrxU1CHEUpzToTua9N84d22ux1b/6/L1/eToUfvKf3/gT2kFN0XX9zG9/a9Wgd1J/4KpBuesXf2kG8u5k3ktMDjwRGTDNW9QgxFGc0CCwdyr3D2Ti/33h2B7LTdVVWNrfGsTKfz5ZXV1MpX73+u9eevFF3sOAGgaDwk+e7LbnBjUIqce9GkXHlhQDQDuAtW7UjMFn88/m5i7waQr7/A+TPwlbO2y+eG/mHxxKiuVkGCGEGuQlEmiQL+uDWPfpboMWQZ6eHx3zXoZufzg/yCqD7tw1e26cSYrlsEYRqQY1iAMwoabRRKEqoKXD6yN6cmsL//zTyE9Ov/P5V7k7d29/eu2Vk89GQuEnj136v04kxXJ37j43MooahDBc7VlFDaqFdZ9W96zuCkSLDh0c8KZXvuJz1Xdp/G45011iwboEv8OxZ9Vj3J4f5MBaz+YE37U2YeHntt4L89qEPbHHZxqEszs8RgINEn/X2gG6TydisQ42N2vpENxitbPgeB+RDzJFDfIYOWa5Duw/IO+xc6z7dDGV6niOMkuiiXmAqv0lvmeNGuQxcsy0/81LvxU8irkb0H06FI068uQHY+r42Li8k8ygA1nkOxxn2nuMHGf7SHHYfA2szKem/KdLIFl2cP8BGQfa52SoOMWzfTxGDg0S/0CYGgghEEh2YxQ/HBb09MAhGaPUU1PTx58fc/wzcRDUII+R46xnuUqlYRO3LP/pBhZjku58sedGRgUfYIZnPXtMD0TguDdqN18w1l7kZArAzj6tOjnDHSRNlomf4kwmEpIGHyWlByqyxLfq64cNigYr//FSKyFZJstxz5AUE7z8bygaFXyn+YweSqn47Rq5O3dfPP2SsBYyS5xPxGLez5wF10+KY1fffCsl/lhe8S01n9FDKRW/XSN35+78/GtihqVbdp96ABw/PyB8smx2du7ZZw5z+Yhsgo0a3tNDKX11fl78MkUIS4t2RjDrPuW+a+GwIMHd6iNDR955+22+H1RzsEDRe3oopenl9L69/dw3aPMFXWPinDrfbvepB7CIuJietfhdGtR0bAX5QgNCDzU/d/GDmi+MvyBISAgqdAQMHDBl5Db/bPclxRlNUInO+yqCRQ817U/B0/M5YUJCsE2HolFhLXaO88+arKmpacGDQRQT8zzooWYcTvxqN+6ZXUe6T72Bz/yzpkv8Y+YpJuZ50AP/w3uIut116MBBXg30kHtyqvvUA9j8MxEsXEgpcI/cN0fXdQH9a99T0aBXXn5F/NRY7s7d2dk5790xl7pPPUCcZBmXL65dMCnGhYoGQYyDu8S0XN4/Tq3dp4LfQg0RpKVj395+wbPy1EzOyPgtS01Fg6ToXIXlpTvmQfepB3CffyaFI0YpXUylJmIx3lcROCoaJEvXWM606t2u4NB1HZLcHnSfegObf+a9DEmREaOUTsRi8o7rlJce9v+4m+s2F/TQuxo47Gz4vPhwmX8GpYniJ5swIM2LHQ16dX7+uZFR7hJjZx0fG3fJZubbfeoB3s8/g/lz4n+YINAYkPaeHQ2SpVo6Z1bcOh6jYeU/HLtPPYA12XpTEXZk6Mi56Wneb7o1WCHNix0N0jQtEgp7c5Zel2s9mzt04KCz9azidJ96gGfJMnhaSPGRzsTjWCHNhR0NMgxDlkrFnGnhO2I5C9h96g0ezD97bmRU8OnRAGx+DAZxocf6H+emp2UJCTkV6WTdp+J05HsJm3/mxmFB0FsjxY0NTgAGg7hQpUGwI2U5u6r7JL343aceALFYN+afnTwxOXj4sBSRNTAJeV9FQKnSIKhVl6JKKGeaQp0VdLBRO+J3n3qAGy0dEhlBlNKJWAyDQbzoqfnvZ585PDU1zV1fbC4whdq1YqD7VPx5Wl7i+Pyz50ZGZTGCoEAXNwMvajXonbffFn+mIlvr2dzTA4fsP8Hk7T71AOv8sy6/F0iHyWIE4eHOfKnVIIgOSJGhhwUJMju1QqwuRtLuU2/ofv7ZejZ3ZOjI+PHjvN+KXWbicWwT40itBsmVoYcdP3x0eCIWay4r/ug+9QawCzqef2b/qSAC0KIRzKyoINRqEKX01fn5I0NHuIuL/QVt2btZ/tbuU7S3bdLx/DNIFAh+mrMVMPzRMedIAw2Cx6AUczzYOnXqdMOd5NfuUw/oLFl28sSkXLEVnNfBnQYaBNapRO4YPH4PHRywnrphGAaU//i1+9QDWEuHzc0Ax6hKJPcQeUBHjC8NNIhS+uabb8rljtXcABINnxcclklsOf/sxs1b+/v3SdGeygCTHx9RfGmsQdJlx2CBR8bCz1K0SkoBJMuazz97bmRUuvsZM2Ii0FiDwEaVYqSZdcF4swB2n3pA8/lnkAuTq8wPM2KC0FiDKKXvX7smyzghWNc/XD24/wBokPhT+2SEzT+rMZChLUOiXBgA4UJ8VnFnVw2CTmJZesfgCFboPgXHQa5nsiywOk+2MW7cvLVvb3/02Weli7tNxGKCHB0ecHbVIErp+PHj4o/yuHHz1vGx8ZrwMwSk5YpNyIJ1/tl6NidjGIiaj1iMGIpAMw2C4K7IhUJ/XHr30MGBeqtH13U4ExUtbTdgU7d/fuTnkt7Ji6nUUDQqne3mS5ppEATtxIxMr2dzc3MXmnSfQjP0TDyO+8wloPpcxtAbbGwZr9yXNNMgKmpk+pPMZ0d/cbRl9ymkclCG3AA+W0ln7mA0WihaaBBMNXNqpowjC9LANrtPoQgNn3jOIrW4G4YxFI1Kqp6+pIUGUUrPTU8LUjO9ns1BFWJb3acQuUAZcgqpBYiajyVZ2vqDQGsNgj3HPUl/7f0PBvYf6KwdCWXIKcAuljqaOxGLYW20ULTWIErp8efHOJpC69kclP9MxGIdD59HGeoeeBoNRaPSZeIZ8BZkTOT5GFsaxNEU+iTzWX35T2cwGZL3Gc4R5oLJK0AUjSAhsaVBhmEMHj7sfb3im2+lnh445GD3KciQvLEMXsgeAwLgXWABvWjY0iBq1iu2O1WvG/9revq8Gw9eCEliU6t9fCPcE7GY1JEsv2JXg8AU8iYqxLpPXepp9kFcwxvY8CAfOLDwpUs0Xy042NUgappCbkeFWPmPq2efskGlaJnvBpsD54/pFmgECUsbGmQYhqsJMtZ9mkwkPNgr7FQ/HzzkHQeOgfTNHDhMh4lMGxpE3UyQ7dZ96jYQ7MCZ0wzWjyp7CswK13TYo83iX9aXzo+vPCzT8vbDzNnecN95ZaPM6XLEoz0NopS+MD6+b2+/gx1kLbtP3QbP3mAQQmA8kJ9sQ4ghcDKCyluFhb5QOBL6pyvFbepnDXq0+SCzev+HBj8pb35zLXN/a9c33LYGOdtBxrpP+W56dgaZnx7+bcGOIfHZMZD8u8PKD1dH9kR6Fwq734Tys0WUhWOhyVXyY+1Pth9mkyORkRWy+7tvW4Mopa9duhQJhW/cvOVU+FmQTZ/P55lB5BsrwA7M/PHfMSTgV7qa32hIeVO9emYgEhr7t5V3RkPhvmRhi9LyZvG28tHl4ZfYvVr5td7Zjz96uS8U7l8qPqaU0vI2KawmRyKhcCQ0cll5uF15VXDrpvqT9zYeZs72Dl4u/ACv8s27U32hgfhH6Uu94cjPzFfZLN6+tXR2bIWUQSPg9x9tPkif7d1zbOnBdoPLZv/k0Wbx08vDe6pNNsuF9U5dfbBZpuVtolwe3gMzlOFtmsAfDUdC4UjIvFS6RQor8Pt9Z9LfbD6inWkQjOaZmpruxv/qoPvUA5hBNBGLCaKMrqLrOmTfxXkSOAhs1MVUyuO/W95Q4hWH64dCcrByBz4uXvlZOBKyaMSGEu/dc2zh3ubfC5d7d25U+OeR3vOrD7foFvwoHBlZ+fbBUj/cz/fur47siYSeOqt8T8vfZc8PRIZ/X9jcKCQHd4TA/HN9yb98q8z1hcKRULh/Sf2+8PuKLpxRNmuue+ef3NsoXjXlYza7+ZhSSumjzcLvj4UGzq78zzb9oZAcNI27H8nKpEVlrB9EtQ1Y3igsjFjfF1xqJxpEzQksnZUsdtN96g2apoFdkEwkvH+EeoNhGBAoaTmGSV6SiQSHWZogCr1z2Y1HFO5V5qRsKmeZRlh/DW7+qlva/DXQoIov83hTmY2ERi4tKdrm11dGXs5ubG8oc32hgbjyXZluF5f+yaoFj4tL/aHBSx99dPVPD4ork5HQ4KWP0q+/cW/j4cporc0CVNQEfq2ijKYbVVHGyn9+nz3zVCXIBUKzI1WWT4KsjO6I3aMNZa4vtGd05WHZ/ChAjjvUIFay2G5wuvvuU2+AWxRcM9GMtS7x8VuzAgXxnj/nKkFoU2gero7sMe9b+BFoRHm7ePVYY6GxSokZ0q6YTj8UkoOR4dmrhe8r7tH2gyvDzNCo1jtTUEYX/rxRBskYHE3eeLj9/3Y1WyqvAP8E/rQpGdRi0LELBt2puvj6j8J8Bfi1yqWa7+uMstmxHUTNPL394LSD3aeewRLVcNmC62ZLatRH9rfTBAhF88jHW+/VRrdxlV40/DXInfINklcAAAmZSURBVA1eLvxQkZiQGbuB23j4anEbbvdqvavRgoofNJfdeGTe/3PZjUfNYuTWX6tcIaTzGitI33mlTqrqPwrQxOpLrfwIzLcuNIiawWk7Q+9Z96mMdcm6rjMlSiYSMsZN4C0EQX0ACHJxeJuVkMpsdvNxxdsyDZNKkKhyE4JhYv01i2GyTf7r3X97/dVK6PeKUtzcMaOsdzu4Zk+dVb43PZ0dD6tMVkYr4lX5hxUhAyeoPhjUzGyp9qoqFzx2pfj3WvvIStUrgFEGl2q6daaYdqVBuq4/+0yLfnr3uk89xmpETMRiWUWRwotRVRWi7P4w5ewA5/bwaTHZCY7olQRQ78J/PSzeJ38zPaC/kvv/vVlmGmT+Wmh2vfjgPtmi2w+uDD91LPlR9kuQHkaNq0UtGvSdma566uytr4v3yXZVnBicu0qBUkVoPrr3zX1SnReDf1J5/cfFpf5QuC/5l4fFr8l22WJkbT1cOd8XGnm9sFGmlqjzxsP71dcLstWXvLdBvi5u/rgjcNv/s3oGguiP4De70iBqet27eWTXP1xlw+e7/EOCYBiGqqrQ5AF3taqqArqWpVKJGT5D0ehaJiOFYnYP88I4fSlgjwycXbpLtr5dHdlj5rArEdljSYVsl6n1PzfuXe5lqW5K4Rat5KTCkeGFLNmi1DQrzJwaYBoUC1nyV/Buzr6rbparE1JVzpdpE1Uuw0L1r1XCVWb63MzohyOhPceSKwWyVfWvqgoIzJ+A6cSuv1JAEI6ERi6vFKx/vVsNort7ZKKV/zhLqVRay2TgJmdixP0+1zRtLZOBpB67Kr6X5DHcvDBHKG8WlauXF24UCn/O3lo6a2blmxT4+QAHNKjeI7tx89bzo2OedZ/yRdM0ZnGAy7mWyWia5tkbL5VK+Xwe7j0mPfl83veffD08vTAHACPFkuQGK6Pa9vEfDmgQrfbIrr3/AZfuU+4QQtYyGeamgR6ll9P5fJ4Q4qAilEolVVWzisICPRCiSi+nxXQMvYG3F9Y94AGZ/hSl5c17rw8/1bCg2U84o0HU9MhefPE32IZuGIamaaARzD4CtzSZSCymUllFySqKpmnExPpxGYZBLOTz+ayirGUyyUSCOVlWjRPBBxQBUGRZvbAKO60M1UkxP+OYBhmGse+ppyKh8NV335X2QeQKuq6DJKWX0/U6Yn8lE4lkIpFVFFVVJb/TnAdqvoWtvEea4JgGUbOlHk+wtInV2AHfii3rj9DGaQluPKlxUoMoPo4Qz4EwEM5plReHNYj6xC1HpAH3m+w4r0HsuYROBOI2ML8B7W6pcV6DqDm3xQcnUiEiA13T3o8HQpzFFQ2iuD8Ql4HTmfA51x3WefvccEuDqGknS1u0ioiLruvg7we5DK1r2Lz9RnOgPcRFDaJm807QWpYQVzEMA4rRMQ7dNT+SlUnuvSDuahBuF8RxIBEW9Acba0Pvnbpy61a2+DfzB9ah8ZmHlfb0Std73/lrH79qmUxWfrg6sqdqqOs2KVR6ZWFutBe4q0HUkiZDsxnpHhgmF/hEmGXeWHmjsHDe1BQYGj919cHm4817rw+PXClum3NCRl4vfP+/4HxVTSZj48fghIyx1wsbj4pL/Y3nnLmC6xpEzfAhyhDSJSBA6eU07wvhjjkaFSydrW+Lf/3R1BoYb/gjWZmEOWGWQz7YZDIwfH4kK5M7erTTHwvDzCqDVj3ACw2iFhnCLAbSGTCbAQWoAhs1vXCv0tTaeLb834tLY40HV8NgkIqxA782uUr+RpSFY9aXdR+PNIias10wmYp0AJR6zMTjvC9EIMqb914f3rPLYHyTqqHOVdNgYeB01fToEIxJ/ChbIF5OC/FOg6hlJ6EMIfbBbVND+a/q/Y1H1Jzl2r9UfFw1396CZYK9eXSidch81cFnvAY2eqpBFPcT0ia4YeoobxX+zygMhN4qXO6tPomscljFFil8ebu4WWYatE0KK8ljoXDkzI1i8evKrOuRFfJ4s/jnYuU4w8qpPuVtcr9uor6LeK1BFHcVYhvcKo14tEkIKX56ebh6mPzO0PiBs0v/URk7XzmHZ+Sy8u1GJYyd/mbzkRlOYjMbLWfDDydXfeyLMXBvIS3BTRIQ+GgQNXdYwKe+IrsBg6hQgIIANw2ilKqqinVDSD1QBxSEQ1kQyleDKJYvInVgIWLQ4KxB1JQh7ABCWHchClCg4K9BlFJd12HzBb4PKLiUSiXYA0E7lg4RQoMopYZhQD90ejmNUYCgwVxytIUDiCgaBEAsALMhgQIypBgTDCxiaRDFHRkw2FMHT0AILMJpEKWUEAJRagwN+BgWBETvO+CIqEHUskEXUyncoP5D0zR8zCCAoBoEwFR8rKX2Gfi1IlaE1iBqeWBi2t4HoHmL1CO6BlFKdV2HtH0ykcDIpbzk83n0v5B6JNAgAJoYh6JR3MHSwZ4iM/E4+l9IDdJoELWU0qJBJBHM/EFvGmmITBoEoEEkC2j+IHaQT4OoxSDCzS0s8KhA8wdpiZQaBDAjfy2TwSSLOBBCJmIxdJkRm0isQZRSXdfhSHt0zUSAOV/YfYrYR24NAggh4JpNxGJ4sD0XDMOAykNwvtAsRezjBw0CYDIsuACoRJ5hGEZWUeCTX0yl0PlC2sU/GkSr7wdUIrfBTxtxBF9pEAD3BvgFeG+4gVV90P9FusSHGgToug6zaVCJHKRGfTDwjHSPbzUIsN4zM/E43jMdA5qOnhfiOD7XIACUCIpWJmKxrKJg6NQ+mqZBAQREnVF9EGcJhAYxVFWFLD6M79M0jfcViYuu6/l8HoR7KBpNL6dRuBE3CJYGAaVSibkVaBbVYzV8IOiD9T6IewRRgwDDMKxm0Uw8ns/ngyxGhBAmzWD4YC8e4gHB1SCGrutrmQw4HRBwDZQYaZqWXk6zt7+YSmHkHvES1KAdSqWSVYxm4vGsovjSFgAbkFk9THrQ50K8BzWoASBGzE0bikbhFpXdOCKENHxfKD0IR1CDmqHruqqqi6kUsxcmYrHFVCqfz0uRotZ1XdO0tUwG2tmZfbeWyUhx/UgQQA2yS6lUyufzVj2C4FF6OQ2SJII1USqVNE3LKkoykWBOJdMdTdNEuEgEsYIa1AlgH9XYF+DdJBOJtUwmqyiEEFdtDV3XCSGqqmYVZTGVYh6W1V6Dy3DvGhCke1CDHIBZH4upVI0qMUVIJhJMnmCpqkpawX6ZvXiT11/LZMSxyBDEJqhBrmAYBiEEhAk8o93kw/5iQpZeTmcVBeTGl2k7JFCgBnGjVCq1tIPQokF8z/8HyB7WU7ypUqoAAAAASUVORK5CYII=" alt></p>
<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the shaded regions with areas <em>A</em> and <em>B</em> . Given that \(A:B = 2:1\), find the <strong>exact</strong> value of the ratio \(R:r\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A = \frac{\theta }{2}({R^2} - {r^2})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(B = \frac{\theta }{2}{r^2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">from \(A:B = 2:1\) , we have \({R^2} - {r^2} = 2{r^2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(R = \sqrt 3 r\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence exact value of the ratio \(R:r{\text{ is }}\sqrt 3 :1\) <strong><em>A1 N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was successfully answered by most candidates using a variety of correct approaches. A few candidates, however, did not use a parameter for the angle, but instead substituted an angle directly, e.g., \(\frac{\pi }{2}\) or \(\frac{\pi }{4}\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following system of equations:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x + y + z = 1\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[2x + 3y + z = 3\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x + 3y - z = \lambda \]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where \(\lambda \in \mathbb{R}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that this system does not have a unique solution for any value of \(\lambda \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the value of \(\lambda \) for which the system is consistent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) For this value of \(\lambda \) , find the general solution of the system.</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">using row operations, <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">to obtain 2 equations in the same 2 variables <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for example \(y - z = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2y - 2z = \lambda - 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the fact that one of the left hand sides is a multiple of the other left hand side indicates that the equations do not have a unique solution, or equivalent <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\lambda = 3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) put \(z = \mu \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then \(y = 1 + \mu \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and \(x = - 2\mu \) or equivalent <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|\) for \(0 \leqslant x \leqslant 8\pi \).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Solve \(\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}\) for \(0 \leqslant x \leqslant 8\pi \).</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/maths_5a_markscheme.png" alt> <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct shape and <strong><em>A1 </em></strong>for correct domain and range.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{{4\pi }}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to find any other solutions <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>if at least one of the other solutions is correct (in radians or degrees) or clear use of symmetry is seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 8\pi - \frac{{4\pi }}{3} = \frac{{20 \pi }}{3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 4\pi - \frac{{4\pi }}{3} = \frac{{8\pi }}{3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 4\pi + \frac{{4\pi }}{3} = \frac{{16\pi }}{3}\) <strong><em>A</em>1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for all other three solutions correct and no extra solutions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> If working in degrees, then max <strong><em>A0M1A0</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In triangle ABC, AB = 9 cm , AC = 12 cm , and \(\hat B\) is twice the size of \({\hat C}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the cosine of \({\hat C}\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{9}{{\sin C}} = \frac{{12}}{{\sin B}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{9}{{\sin C}} = \frac{{12}}{{\sin 2C}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using double angle formula \(\frac{9}{{\sin C}} = \frac{{12}}{{2\sin C\cos C}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 9(2\sin C\cos C) = 12\sin C\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 6\sin C(3\cos C - 2) = 0\,\,\,\,\,{\text{or equivalent}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((\sin C \ne 0)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \cos C = \frac{2}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There were many totally correct solutions to this question, but again a significant minority did not make much progress. The most common reasons for this were that candidates immediately assumed that because the question asked for the cosine of \({\hat C}\) that they should use the cosine rule, or they did not draw a diagram and then confused which angles were opposite which sides.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider \(w = 2\left( {{\text{cos}}\frac{\pi }{3} + {\text{i}}\,{\text{sin}}\frac{\pi }{3}} \right)\)</p>
</div>
<div class="specification">
<p>These four points form the vertices of a quadrilateral, <em>Q</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup> in modulus-argument form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch on an Argand diagram the points represented by <em>w</em><sup>0</sup> , <em>w</em><sup>1</sup> , <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area of the quadrilateral <em>Q</em> is \(\frac{{21\sqrt 3 }}{2}\).</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 \(z = 2\left( {{\text{cos}}\frac{\pi }{n} + {\text{i}}\,{\text{sin}}\frac{\pi }{n}} \right),\,\,n \in {\mathbb{Z}^ + }\). The points represented on an Argand diagram by \({z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}\) form the vertices of a polygon \({P_n}\).</p>
<p>Show that the area of the polygon \({P_n}\) can be expressed in the form \(a\left( {{b^n} - 1} \right){\text{sin}}\frac{\pi }{n}\), where \(a,\,\,b\, \in \mathbb{R}\).</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>\({w^2} = 4\text{cis}\left( {\frac{{2\pi }}{3}} \right){\text{;}}\,\,{w^3} = 8{\text{cis}}\left( \pi \right)\) <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> Accept Euler form.</p>
<p><strong>Note:</strong> <em><strong>M1</strong></em> can be awarded for either both correct moduli or both correct arguments.</p>
<p><strong>Note:</strong> Allow multiplication of correct Cartesian form for <em><strong>M1</strong></em>, final answers must be in modulus-argument form.</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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hjcY4890oMPPpinwYeCswyY8e0gSik/NbrmgZq/VaUcKj/SZ0lZZd0aID/96U+z38KShVtuuWXe70UcvE2ePDnzZG4PedjKQbMlAHaoeUa8AhwhiXIecxLgJAUMG264YV4x7PTTT89lVCu3StEk0UxwHT0T++yzT1p++eXThz70ofSNb3wjWaAYASk8hCWgOzSaEK3yEM/hgfU077zzppVXXjlNmjQpN4923HHH9NRTT/UNuTcwLiwvQOE5fDm3Cm4FOOItlPOYlADlth4pX8fVV1+dHnroob6uSDVxWB9DLbz4FB8QUDyKyPlqmcCDDjoofeUrX8nT+63PYX1RihnxXOvR0KxpB0mP1YBYHniyFeUmm2ySj2eeeSZZWOiiiy5KSy+9dDuy7EtjZBcv6MumXBQJjJ4EKCpQoNSUFHhYpWvxxRfPq49b3JjJDgTU2M2Y69KirM6e52PQ7bvFFlvkAlJeeW699daJb8GYCfHwxNowBD54G65EpMmakR+rgk+Dn2OXXXbJm1ZpNv34xz/OixsttNBCw81uiueLxTGFOMrNWJCAmpgjNMg9BTMc/frrr0/2nqX4KHpbIu5gZyCApIeAQVUpNV2kOddcc2WAAUwUHA+ugUazeeaM6vxEOZyfeOKJtMACC+S8+D2sts7XopmCh3ZSsTjaKc2SVldIIGr4sArinh/ABk5GlJpJu8gii2QAqILMYAWQZlgMzhQyfCmhnMCJBWIZQ705an5kX5X55ptvsCya+j+aKs6AcYUVVsjPs6jstctJivDaTDkHY6LsqzKYhMr/PScBCqzmV8trhoSyM+lfe+21PL4DiPB7IPGaIenrymRtsDxCIeWj98IkMjvN4cEK6LZwEE/Txfqk3/zmN5vJrmFcvAAFywhMM800ubzCWDd8MQAE4U14s1TvmQIczUqyxO9ZCVAw4PGTn/wkHXLIIVnJ3QtnlTgCCCg9M78/UUBAAWxCqYRxerI8bIqtF4ezVBzKK760gQgfiDjNglV/PtxLQ7pxjXdhyhT8yy+6hgPgaqVVLyzK2P//4uPoL5FyP2YlEIpjQ+g555wz7bvvvrnLUoEpoFoaOdcCDf9RJApKIQEORXUPPE477bTshDXcXbj/peOQvmeFt4uCl7Am5CGvAA18sbgCuNqVr3QKcLRTmiWtrpYABUP8Adttt11eZvBHP/pRVmhKzmlJ+ZzrKThlpYgUVBz3ekqM3TCxTrenfHSD3nzzzbn2l2Yc4reL5BNlcgZmwEJeCI+RX73ytMpLAY5WJVee6zkJhEI9//zzackll8zH8ccfn7dsjMJUFTHCqmcAo0ZHmgTIAsnGckhLDwaLxjyZcIrmSP//aacCB4gpl2uACCzCspKl0azRXKryMdzrgY244aZYni8S6FIJUFrAYElBZG1S4y3OPffcvPsaPwWHJ3CoKl+1OJTUf2p26VFKu7YZ0i5c+g6KbOyIOFHrS6d6XU23lWtpOeThQLfffnu69NJLcznMSdHLgteqJdJKXv2fKc7R/hIp92NWApSLAkUzgy/j1ltvTV/+8pfzzvImfoU1IU49oqzAAYhU0wQ4wvkVNBsAibhh6ey5554ZaNrlHA0eAhTMUdl0003zeA6WBkAzAGzWWWft83vUK1O9cPzXotJUqSWVEjYmJUAJAhBcM+0/8pGPZN+E2aWGiCNNkKjBawnCf56POADCtXM86xyA4b+IWyu9VsOCh0j/6aefTg8//HC2mAyzN1/F0cjZ22reBThalVx5bsxI4BOf+EQ6+eST0w033NCn4COh6O0WGOAIq8cZKFrRy3gOVgirRzhnr7jtpAIc7ZRmSasnJWCzal2olhmkbGrosBa6uUDAQfMomkYcvpoltocwb2WxxRbLQ9CVgf+mnVSAo53SLGn1pATM79BUueWWW/KaFuED6fbCADdAF76OBx54IIPesccem8eU6O2xKhiA0SxrJ9X3ALUzl5JWkUAXSwBQcCoaEn7SSSflkZ/tNu1Hovh4BAzAgzN01113zfNTbIWJAIZww85dt7NMxeIYiTda0uwpCaixOTOtjGXOh8lo0bvSzQXRpEKaKqbT6yGaMGFCnkcjXLl0L6N2+2wKcGSxlp/xLgE1svEdFuDZdtttM4CoofkQKJ1zO2vsZuWNh6rV4J6lFF3AmiWf+tSn0qqrrpqbJfF/9PLo8WknFeBopzRLWj0rAbWzGtw09JdeeikZio4oIOokaMg//Bhx7R4YAA/T+O+44470ne98p+2+jFz4Gj8FOGoIpQSNLwkAhRh3Mcccc2R/x89//vPcXNEMUNNHLwuF7QQBCDwAsgAzZxYHZ6geFd2u7keDCnCMhpRLHl0tAWAAHCiea05GSwIadWlIeYBGJ5srQAJvzkDEwQ+z/fbb52HmO+ywQ/ZtBK8jLfACHCMt4ZJ+T0ggLAmWh3VC9bIcffTRefFfFomDUnaqyRLAEcJ0r4ly3nnn5XVUraeKd4AyGlSAYzSkXPLoagkABAoXjkZKaTSphXn4DXTTqt3Fi1q/f4E8g5xZL/Xi9X+umfvIwzNWILP+x5e+9KW+hZLxV5oqzUi0xC0SaIMEOBvDorBq+KGHHppsMWDKPF8HxY3/ZefaEeHhCwFC1bTawFoGBBZFAMNVV12VAU0X8nTTTdfHS5W/duRbL41icdSTTAkf9xKwudJnPvOZdPnll2dZsCICLGopaIAI5dZDY4GfdhFrAuHBZDaWkObJ7LPPnsMj71p8tYuHajoFOKrSKNdFAv+XACuC8ttgybUFiFkU/cl/cRg8Zo4Ia4OjNQZf9X+m1XsOW6NEOW/vv//+NHHixAwk+MQDwBot4BgdT0qrkirPFQl0SAIUkPLzbWyzzTZ5UNiMM86Y1l133Sk4orCsgDPPPDNddtll2Q9iYSBh7ST5mPV6ySWX5HxYHNbbABbRNGp3no34LxZHI+mU/8atBMK6oLD2hH3f+96XTjzxxL7FjfsLZu21187rjfJDRK0fafSPO5x7G0t//OMfzyNcAUWABj5ZOs6jQQU4RkPKJY+ek0AoP6eoqeqHH354uuuuu3KNrzAUlOKKx/8AWCxWHM+FUletAIpdva8llFB88VyHM1QeFj82QvTAAw/MI0TDASsekJJ35F8r7XaGFeBopzRLWmNGAhSV7wCxIjRRrCtqx/uwJMSxwRPFdYQiV4UQyuwMhDzTiMSTlniOeF4+/C0sG7vBmSZfBSF5I/FHgxqXYjQ4KHkUCXSpBCgjH4dan6NT1yfFNKFMmMVxKHAjMDj44IPT5ptvnj7/+c9n34T4ATz1iu1/4CEPPDjbROrZZ59Nm222Wd9j4nSKinO0U5Iv+Xa1BNTm/AehxADE2heGeOuitVapBYA4UBsBgeZLbMOoG5Wyi18PbOQLLPzvGlDZo+WUU05JRxxxRDKXBl+Gwsu7U1SAo1OSL/l2tQRCscPqoKyU3oZLpt9fffXVeWm+UOBqs6FasI033jg/pxvVKuSshxhMVo0X1/KQJ3KW7m233ZbvV1tttQwabkarSZIzrvFTmio1hFKCigQosIOlQYFdU1Zrk9qPRdPhueeey/6PeqBBip5DgIevRHoRVkvKACvGZbBMjBC1fcNWW22Vu2PDEmFtNMq3VtrtDBs2cEBQAiUQZ4dx9IWKBHpZAr5jROHju6bwFNdubfPOO28eiFW1Dnz3AQrOkYazeAEIEV5LPp4DCtLStLEXiyaS5hEdk7+jmn6tdGqFRTlYP/QWP414qZVGhA0bOKCoobXOPL/2piDsQkUCY1UCFNdKWzE71f0vf/nL9Ktf/So9+uij6Te/+U3WAcrZLEkLAY/77rsvPfbYY3mXuQCwSC9AIO6HcgY8wEJTSXoOINIKDXsnN8JR2Jtuuildd911eeKNxVKXW265VvgpzxQJ9IQEXnzxxTwr1czZK664Ijs8VaAUkzOUH8TkM82JqNW/+tWvDrqTm+aJ+CyKjTbaKC2yyCJ5Q+sAirA0Ik4zwuKP0TvE2WrpALzS3UYVffDeP59hmwYsDQx94AMfSCuvvHI655xzptjEt3+G5b5IoJclQHEpm53S7FCvCWGIuTEW5qrYRjKsBrV5I6WsJQeKqlljSLklDA866KApemHk3yoBjZ/97Gd5nguA2mmnnXJLoZU0h91UARr6pqGr5dasEL3wwgu3WrbyXJFAV0uAYqupKZuh6NbDMH9EMwCYBGiwxFWqzZJ0NU84YFdZZZX8ePhRmk2rf3xNKN26NsO2a90TTzzRt+1l/7iD3Q8bOAgHenHmABHLtFuVqFCRwFiWAAABFno7WBnHH398djjShbA0Wun1oEOf/vSnMwCxZsKRCVCi2VOv+TCYvC2FCCw++MEP5iHywMOWka3QAOAgjDgwC10xqv3m7IgwZwRdOVzWXHPNdNRRR/WhbisMlWeKBLpdAjHwii6YsQo87AR30UUX9fV20J1GRGfEATQBMO4vvvjirNwHHHBAmmuuufocmdKSHxIv0tcZIZzOOkvLiNaIG+HiW5TInBoWjGYLkGq2KRVlGgAcEnWwJMz/v/vuu/sWMpGRgmJW+0s8E2+sQWDW3rXXXps9wfamKFQkMBYlEApLMUNRDczSy6LrlMXNSoj/6skgLPVQXPEp/BlnnJF22223PCeF0iM6V4vwwmIAFtEsEleYs/9V7gFMupGBB/19/vnnc5dy/Fcr/UZhA4ADEjoUhPeV32KTTTbJo9f4MoAFpxCkBST2oeD9XWuttdJKK62UPvaxj6VZZpmlUZ7lvyKBnpUAvUCUki5Qbj0UtiiYeeaZ84I/FJfuBCjUKqznIy3/U3CbXtM9K5azCBDFlk8tkgbA8OzkyZPTvffem/NU4XvGOBB6yu+iq3idddbJ+nrqqafm4fLLLLNMy62DAcCBEYcCzDnnnBk0CET3E3rqqady9xPBiHfIIYfkBUVcEwS0DVOuVmFLWJFAL0uAIlPYqKl99+7pyhe/+MXsdFQ+YBJxapWXrgRwUPLbb789HXfccdnSADisgtDDWs8LAwj00JD0fffdN/eSCAcWwOPkk0/Oj5oct9dee6VJkyYlQyVU7rvvvnvuNqbnrdAA4IhEFNphLYJFF100mzYPP/xwRlYFspktc4gHeOedd87ISwCEESZWpFXORQJjRQKh8JRbJUlHAAeyMvr73//+7O8wGHKoZCwI3yD9Yb1LM3aZl089fZKvnhwT7nbcccc88AyQaIZoNrFAWCNWLtNqiPiLLbZYbs5Ey2KofFbjDQCOqmBkpDBmAWobQUT7U1okleXB/GHuWNQV+sUI0moG5bpIYKxJgCLTE988xY5aW03/2c9+NvewcHI2ogAble+NN96YrXhdpSpq6UaarlXUtSh8KawT7gFgBTiOOeaYPKZEK4H1z6UgHU0VY00CjORNv1uhAcCBYYeMJOraGA2rH80zzzy5K0cPCqYNr+UYIgTxoaRrjBUqEhiLEgAY8X3TD997KLba3RgJTXtNeFY5vWBBeA45i+egW9IAMvwaK6ywQk5PmrHOh2tHLaKH0jFSle7JR4XOt2GUKmcrUumLI504e9ZRL+1a+VXDBgAHoQQiYcQ1xIJU0JTTRr81VDNatFCRQJFA6vPtcYweeeSRWW+Mm1DBAg9En1yzAgADpRWHsu+yyy598YYqTyBE+QEQQAAUtnKweFA0R8ynufLKK3OrYKjpDiXeAOBQOEegKHOGpaFg2lNI02XrrbfO1sdQMilxigTGugSqFoXxTPvss0/uYQnFVn6gIZ7BkiwFvg1drypkTRQVdLPEqlG5AyJk1i7HrGbLCy+8kNcNsfrY9NNP32zSDeMPAA6mkwNBRNeGvuqnBiiIE2jDDTfM1+WnSKBI4H+gEPpBHlYiR5deemmuhOkSEOEfoej0Sk+KilkvB9AI5W9Gnvwq0mRtsDKsSSptrQK+SM5WgNVqk6QeLwOAQ0Ec4efwoOYJEwwDGDSijSNUnEJFAkUCrzdDWBSaJ0DCfiy6PR955JFc6dIXCsyapxyn878AAAqWSURBVD/GU+hGnXvuuXPzpQo8Q5UpwJEmi+Kss87KPSiaQiwaE+S4E1yHRTTUdAeLNwA4oJUDBUopqEIpOCcMhAsgGSyD8n+RwHiQQDTtWRMqWMCxwQYbZF06++yz81kvB72hPyeddFIO4xSlWxTbc82Q5+QrL5U7AKoSFwMdDsCq/jfc6wHdH2FFyDAKhDH3kDRAhVUirNnCDpfh8vzYlYDvCVEiR/W+Vqnj/1r/jXZYlRc6QXf0RgIIU+75BGebbbY8sMuyg2a/mjofz4WuxX0z/IeFE00gacQRlb1zPV0Vlz4DIS0KIOQ+dL0WL3UX8qmaNjKVSIQppDCMtlLQWoyUsCKBAIz4aN371tzXonqKUCtup8LojW0VNBemnnrqvGCx+V3WrmGJaLLoEUF0SXmboVBuMvIsnSQ3+QWQBGiE/vZPnxPVchg6Pwz2BB533nlnjobPWjTA4gjGMSRDDMULMuDLEml8HBw68YJrJVzCigSalYBvzzeHXEdlFc2AZtPrlvh6WTRJWCDKYji44enKx5rXdBHu7H6oBGg8g1x71krqF154YZ4Fa1wIK0fPiyYSIKlFLJYLLrggz3cxfP3WW29Nq6++enau0vlaNMDiCEYUBBNhdhnnHoNKLItmzLujHjO1MithRQKNJODj971Va17fl/taVC+8VtxOhanlOTC32267XIvTK5NE7Qyn4lUGcQCmc1gQQ+XXMw5pGWtlqPkvfvGLbHGwFg499NBsSfBN6oGpRXQemOk53XTTTfMwC3GB94orrljrkTQAOCKWpd/tGmV+isIY1ipzqMYMckRXUDxTzkUCw5FAVFKUgILpnqRQasRephjVGRPXZphhhjyfhC9Bs0A5/acLNSruoZQ3QIeCAw+6abkL4zj8ZyyHNMmTLh922GE1k/UsOdtsyjIaRrKGdeS/WjTAdglTUeHMQ1liiSUyE9plASISMqzW5jTNmFa1GChhRQIhgag53etSNO9C09i0hloUTeha/3VLGIuJH4MiWtBbJay3Q9OBrlFwYa6jx7IZ3kNfPWt1r5jYBiwAiAlwZNlolDeLSBy9Mhy3QJteByDV4mcAcISppEDG2wdp99gYxmxYoKKP2AstwBESKud2SMAHT9nUvL5F/gHrelrtGwGXRh90O3hoZxoBhspCf9wbEk45h0vkIF1nBACsRHbmmWdmK0Z3sJXEWDT0GcjIH/V/1rwzs2g5ai3MFXNa6oHzkDuObXega8lAE8PPjVCrl2jmrPwUCTQpAU0STWDkA3e4ByaufeyuHb1EURnjmS+hXZVtyIEeAgZWB6AAACyMvffeO4MG5yjZAWSH+HjwvLkt99xzT7rjjjvyKFOLcZnfYj3SeBe1ZD1k4JCJ7e8UPAaxFOCoJdIS1qoEfMza/VGDOlf9HAEezhSlF4jyKQdrQBNlwQUXbBvvQCBIHqw0lgzfI0cp2QnXDEH8KUhY8GTTJ80TTlB6bbwJ2eqW9S7q0es514vx/3AgYeSb3bIxhDAaTA3yePm7SGBQCfiYkRra98a551sLD78woBE17aAJdkEE+qGJoBwTJ07M1/ivWiGtshlAKj0gEvfAAwio5Pk53AMDYQEazngzJN7/0sAnnwgQAeKNwHlAr0qjyExJXT1mx8Zsu0bxWxVIeW58SsC35AMGEKFYTGhOO21uioHU4r1UYeGXciqTMupBiUFfw33TARaAQPqUX8+I8CWXXDInL8x/wugw+VatFZGEAwu8iuvM4hC3Fg0ZOIIxLxZJMF6ye/8Lw5xwmRcqEmhFApTLxx7t8Op35fvyX9SwraQ/ms9UdcE1/uspY7N80TnpOcjMgSKPcDDLz39VfXUvXsSXFgreBpNxbTjJSUz5I0GoFKhmTQHXwl0jjEDT/mg2ZUrlrkigvgR83FFJicU3QAG0z31voSQ+7F6gagUa+tIuvuleKDoZkU3kQT6snACHABH3Ds/FNX6qabkfTIebsjgAhPH2QbptmEUytRCJ+f/x4gfLONIo5yKBkIBa0LfkHEcong3NfVvGDhnURDHiv3h+PJ4DGMKCIBermpt/Qh8BbzhJI24zcqr3TFPA4UVBLkyaFGODaYv8YOyaa67Jqw/Zvk6cAhzNvJ4SlwR8V4BD+1ozxUdr7oRFsoGG1bs56C0ktccee+Tux/EsOTKhk+SGWGrkZTYuHbziiiuyY5ksW9XJesDRVFMFc14sRuzZMPvss6cJEyakZZddNi211FLp29/+dh7yykQqVCTQrATiI6UAlELTWAVlX1aVlC0W7Xt6xBFHpAcffLDZ5MdsfDpJdiprE9vs8UKGwvyH2q2TQwYOqKYmcIZevMJGkWp78r4ykfT9euGOQkUCrUjA9+V78tEDDopgUqVvimKwNjSXmeDjnVgb5AQkyCrAglzoI9AgM+Hu20kNgcNLlKkzwgRmHDbaNeLM6LRLLrkkm5Ff//rX83TeAhztfEVjM634tpTORx3fjEoJKFAIIyEt4UAp/K/WtNS/BXnnn3/+sSmYJktFjmQVTTuyFEZHhSMyjesmk68bfQBwyMDhZWEAqgVYsDhcOy+00EJ5vUSb5O666665X93cf88CmEJFAoNJIGpMgOCb8dFTAD6z+OZ8T2Fm+ybtr2quR6y4P1ge4+3/kGkAivIb8yK8nTRAw88777ycvr0evEAMeKEBCBjYeOON82ZMt9xySx4QZt8Gu1B52VtssUUf4LST0ZLW2JKA78oiMeZE+LbcAw/g4BvzLWmmWLlbGPDg4zjwwAPzHiHiF6otAbIiQzJy0F/Uzgp9AHCsv/76ORNWhSGqzl4s1DKENdDLpjOWgNfetEiJGsDsP9tBMiXLi639Ukvo/yTg+wAKRiGHjyysDt+bwwcPQHxz1157bfarWeqBVRLKUOQ5UAJ0VvMkfB5hsbVTJwc0VcJEBAhenvtAKhaIa+1Nm8k4m2Jv9t3yyy+fwcJS8JguVCTQSAK+Hd8W0EC+GWG+r7A4fPA+9quvvjrvQRJjOMSPCVuN8hiv/wFWRwAGOZBtO2mAxRGJy8hL8xIRhK++UBN2NFF0j6kRDNCxlKCXC3AKFQk0koAP2/flO/NdVZvFAEVtCUSsRmUnd9+ZIQBqU/8Z3GS9iUIDJWAAmPEuBmxqNZCzo500ADi8tHrkpQEJ5KWZKauP3cu3E7aeFtdebKEigaFIIComcVU4vjEfu2uzYydPnpy7+VVcrFzKYIJloxWthpLvWIhDz6r6ympjnemJMjFQxR7d1+0u75BHjsoYKKgFguEqkPRnrN0I1z/9cj82JQAgmNgcomGVABHfU1iyvjv/Ffof2JID0CC3OJMjeQUwR4uhWZmFzPs/N2TgCCCQUJUpjHqJwhBgibj9Myv3RQKNJBDgoGJy+NbUqM7+E0YR3KvEqrVto3TH6n8hI/JAIb+qvKLswiJehA3lXO+ZpoCjf+bBKAYAiJfqZQovVCTQigR8OyqhsCjqKYe0y3f2uoTpZn9Zvf7v6z7KathQrusBR32HRr9UJeBFObQzY3SfFZwR0Airo9+j5bZIYEgS8G2peIBGfGMe9O35tjRR4kPWlBnvRCZkFrpZlQcQUZmTk0NLoJ00ZOBoZ6YlrSKBIoHelsB/AXh9mx6+g6EOAAAAAElFTkSuQmCC"> <em><strong>A1A1</strong></em></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>use of area = \(\frac{1}{2}ab\,\,{\text{sin}}\,C\) <em><strong>M1</strong></em></p>
<p>\(\frac{1}{2} \times 1 \times 2 \times {\text{sin}}\frac{\pi }{3} + \frac{1}{2} \times 2 \times 4 \times {\text{sin}}\frac{\pi }{3} + \frac{1}{2} \times 4 \times 8 \times {\text{sin}}\frac{\pi }{3}\) <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for \(C = \frac{\pi }{3}\), <em><strong>A1</strong> </em>for correct moduli.</p>
<p>\( = \frac{{21\sqrt 3 }}{2}\) <em><strong> AG</strong></em></p>
<p><strong>Note:</strong> Other methods of splitting the area may receive full marks.</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>\(\frac{1}{2} \times {2^0} \times {2^1} \times {\text{sin}}\frac{\pi }{n} + \frac{1}{2} \times {2^1} \times {2^2} \times {\text{sin}}\frac{\pi }{n} + \frac{1}{2} \times {2^2} \times {2^3} \times {\text{sin}}\frac{\pi }{n} + \, \ldots \, + \frac{1}{2} \times {2^{n - 1}} \times {2^n} \times {\text{sin}}\frac{\pi }{n}\) <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for powers of 2, <em><strong>A1</strong> </em>for any correct expression including both the first and last term.</p>
<p>\( = {\text{sin}}\frac{\pi }{n} \times \left( {{2^0} + {2^2} + {2^4} + \, \ldots \, + {2^{n - 2}}} \right)\)</p>
<p>identifying a geometric series with common ratio 2<sup>2</sup>(= 4) <em><strong>(</strong><strong>M1)A1</strong></em></p>
<p>\( = \frac{{1 - {2^{2n}}}}{{1 - 4}} \times {\text{sin}}\frac{\pi }{n}\) <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for use of formula for sum of geometric series.</p>
<p>\( = \frac{1}{3}\left( {{4^n} - 1} \right){\text{sin}}\frac{\pi }{n}\) <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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="question">
<p>Let \(a = {\text{sin}}\,b,\,\,0 < b < \frac{\pi }{2}\).</p>
<p>Find, in terms of <em>b</em>, the solutions of \({\text{sin}}\,2x = - a,\,\,0 \leqslant x \leqslant \pi \).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\({\text{sin}}\,2x = - {\text{sin}}\,b\)</p>
<p><strong>EITHER</strong></p>
<p>\({\text{sin}}\,2x = {\text{sin}}\left( { - b} \right)\) or \({\text{sin}}\,2x = {\text{sin}}\left( {\pi + b} \right)\) or \({\text{sin}}\,2x = {\text{sin}}\left( {2\pi - b} \right)\) … <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for any one of the above, <em><strong>A1</strong> </em>for having final two.</p>
<p><strong>OR</strong></p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for one of the angles shown with b clearly labelled, <em><strong>A1</strong></em> for both angles shown. Do not award <em><strong>A1</strong></em> if an angle is shown in the second quadrant and subsequent <em><strong>A1</strong></em> marks not awarded.</p>
<p><strong>THEN</strong></p>
<p>\(2x = \pi + b\) or \(2x = 2\pi - b\) <em><strong>(A1)(A1)</strong></em></p>
<p>\(x = \frac{\pi }{2} + \frac{b}{2},\,\,x = \pi - \frac{b}{2}\) <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find all values of <em>x</em> for \(0.1 \leqslant x \leqslant 1\) such that \(\sin (\pi {x^{ - 1}}) = 0\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_{\frac{1}{{n + 1}}}^{\frac{1}{n}} {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}}){\text{d}}x} \), showing that it takes different integer values when <em>n</em> is even and when <em>n</em> is odd.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Evaluate \(\int_{0.1}^1 {\left| {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}})} \right|{\text{d}}x} \).</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin (\pi {x^{ - 1}}) = 0{\text{ }}\frac{\pi }{x} = \pi ,{\text{ }}2\pi ( \ldots )\) <strong><em>(A1)</em></strong> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{9},\frac{1}{{10}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left[ {\cos (\pi {x^{ - 1}})} \right]_{\frac{1}{{n + 1}}}^{\frac{1}{n}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \cos (\pi n) - \cos \left( {\pi (n + 1)} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 2 when <em>n</em> is even and = –2 when <em>n</em> is odd <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_{0.1}^1 {\left| {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}})} \right|{\text{d}}x} = 2 + 2 + \ldots + 2 = 18\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were a pleasing number of candidates who answered part (a) correctly. Fewer were successful with part (b). It was expected by this stage of the paper that candidates would be able to just write down the value of the integral rather than use substitution to evaluate it.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were a pleasing number of candidates who answered part (a) correctly. Fewer were successful with part (b). It was expected by this stage of the paper that candidates would be able to just write down the value of the integral rather than use substitution to evaluate it.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were disappointingly few correct answers to part (c) with candidates not realising that it was necessary to combine the previous two parts in order to write down the answer.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If <em>x</em> satisfies the equation \(\sin \left( {x + \frac{\pi }{3}} \right) = 2\sin x\sin \left( {\frac{\pi }{3}} \right)\), show that \(11\tan x = a + b\sqrt 3 \), where <em>a</em>, <em>b</em> \( \in {\mathbb{Z}^ + }\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin \left( {x + \frac{\pi }{3}} \right) = \sin x\cos \left( {\frac{\pi }{3}} \right) + \cos x\sin \left( {\frac{\pi }{3}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin x\cos \left( {\frac{\pi }{3}} \right) + \cos x\sin \left( {\frac{\pi }{3}} \right) = 2\sin x\sin \left( {\frac{\pi }{3}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x = 2 \times \frac{{\sqrt 3 }}{2}\sin x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">dividing by \(\cos x\) and rearranging <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan x = \frac{{\sqrt 3 }}{{2\sqrt 3 - 1}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">rationalizing the denominator <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(11\tan x = 6 + \sqrt 3 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were able to make a meaningful start to this question, but a significant number were unable to find an appropriate expression for \(\tan x\) or to rationalise the denominator.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f(x) = 1 + \sin x,{\text{ }}0 \leqslant x \leqslant \frac{{3\pi }}{2}\),</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">sketch the graph of \(f\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 31px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">show that \({\left( {f(x)} \right)^2} = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x\);</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the volume of the solid formed when the graph of <em>f</em> is rotated through \(2\pi \) radians about the <em>x</em>-axis.</span></p>
<div class="marks">[4]</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="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><img 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dRT6enpx44dK8JnBgAAXNtDByM6sS1atIg+VfcTzMnJMRgM//73v6UyJYiuRjUzPjWiQgZ21NHAP//8c8+ePdu2bXvuueciIyMTExPbtWtXu3ZtNZMZDAa6QEHqvmstuRJmd4FSmjr0qc/F36Wa3MyZM7ds2VJ0zxAAALiyB2oMyyOS9zjh0bwc6n9xtwYB6vUL4XeBYoSPAVo+yesluc0Ydxqz227r8OHDb7zxxoQJE55//vn69esHBQXx0cVzvyhC8SEnhIiPj581a9bp06fphwwZMoQKZrztvaZpXMSlx3Pjxo3ExES6jl3NjG7LX6Sf0Lhx48jISD8/v/79+6vvN+zWh9IFu/IwAADAvd01GKnbLdPZpU2bNvcuQpQoUUIIQXOfufPFbXeGQOZO1CYRfEhQFwn6Is81lFLabLasrKw9e/akp6f379+/R48er7zyir+/vzrpPiQkhHsLh4eHDx06dPv27fRDOPbRUTdx4kR1BFOn03Xp0kXm9ii22Wx0K9rmy2q1/vXXXzNnznzmmWdSUlLUxQF6vb5EiRJGo5FXZZYqVapz585bt27l+6L1xXzAqxuZO+qZBgAAp3efYER9B6SUZrP53qNCVEWoXbu2Wjaw2xMQgczdWCwWs9mcN5qoB8Y///yTkZHxxhtvjBw5slevXuXLlxe5Y4U8dMgHmJeX16RJk9avX0+3VfvH8ipIq9U6YcIEcfuE/fbt23M1jq7Gx7bMjYb0qP74449u3bpR63/uXkYFMx8fn5IlS9J7j/Lly7/00kunT5+Wt+8uIG9v4VEITyoAALiguwYj3mtZ3j7qdA/cS50u5L0+Apn7sEvkJpOJSkpcFaOS0s2bN6OiosqVK1e7du2UlJRKlSpxGvP09KRDpWzZsuXLl2/btu3777+fmZmpdovlKix3L6PLI0eO5JFHClUBAQFS2QFJTXLqOCMXusaPH08PgzZgFUJERkY2b968Zs2aPj4+Indws2/fvrQ3Of80rMcEAIB8uE8woqLCg0yI4avd41SEQOZu7NbecqNgOlTS0tJiY2OTkpIiIiICAgLUzvvcLSwgIGDEiBGXL1+m3bf45nyYceZTR8lfe+01taCraVr16tVlniKWSs2O/JVNmzY9++yzkZGROp1OHcqk4hkFtVKlSk2YMEHdZIk+qgOyAAAA93avYKSe4dQunfe4Pm8ywxduuzMEMrfBJVKapyWV/cKvXr369ttvN23a1NPTMzQ0tFevXgkJCSK3mYXaDKxDhw47duxQxzdv3bpFF+zatFI3f748a9YsLmKRFi1aqFemC3ZbmOfk5PAP4SM/Ozv79OnTEydOFLkbNOVd4BkeHr5ixQqKmzQSSpexGTkAADyguwYjtaSRt8fYPW5i1/DitjtDIHMn6vHAI5gDBgxo0KBBRESEp6dn1apVO3XqlJCQUKZMGR5e1Ov1FStW7Ny5s1RSEeV7HnBUG/qrhyUftHPmzKHAxNPIYmNjpRIT1bUFdpVdfu9hF/gOHz68cuXKZs2aidubmREvL68+ffrs27ePr2/XUw0AAOAeHrTtxQNOT7abxW93KwQy96FuMM9z3qdNm0YjfRUqVNDr9VWrVqXCWJUqVZKSkijoeHl5ZWRk8A9R5zLe7S5knoHyGTNm8Agj73epTjLja9pNIOMUpX5U/xHOnz8/ZMiQO26jSfWz5OTkRYsWnT9/HmkMAAAenOOCkcViSUhI4NOYw+4XCpvdFvJ2+2jZbLaMjIwRI0ZERETQHPl69eqVLl2a6mHcPV8I4efnN3/+fLpVvneNpBtevnyZcxIdcv7+/ndcDZA/69ate/LJJ8XtHWXpt0tNTW3fvv2MGTMyMzNNJtO5c+f+/PNPeXuPD5K3NQwyHACA2yr0QKZWSvr06cOr3gr7fsExqCbK44Y8d5AnkG3ZsqVGjRoeHh5JSUkcX9TZXUajMTQ0tGfPnmvXruVbyTzbKD04q9V65swZdQ6Zpml16tSRt2/klb+eFGr6PHTo0ODBg728vOi+1J3B2rdvP23atAsXLtCVz549+/vvv9+8eZPnlvHoql3LXAAAcE+OCEZ8yunZsyevTXPA/YKDcX5SM9nQoUMpFfn5+XF9VNO0pKSk1NRUf3//2rVrL1iwwC57qd0oHpbVaj169KhdIGvXrh2/MXiUIpndbmAWi4VGXdVtnegIr1Wr1tSpU3ft2pWZmSmlvHTp0rFjx65du2bXRMbuSQMAAPfkoEBGZ8GQkBCBzcVdDjek4DrZlStXVqxYMWvWrDFjxvj7+3NVjMco9Xq9n59f3759R48efezYMbuopO7W9bAPhmLcwYMH1Tb9QojevXtzTHz06KMuKL548WL//v3pvujX9PPzo0/Dw8M7d+48e/bsAwcO7N+//8MPP1yyZIlUFh/QM5Z37BIAANyNI4IRn1x5BjQ2F3clvGKRwtDJkyenT59eqVKlxMTERo0aUQLz9/enZMYbUHbs2PHIkSN2nV3ViWj5i030GPbu3WsXyMaPH68OC9J+R/n7ffnnqJW2jRs3cm1Mp9N5e3sLIRITE/v27Tts2LApU6ZMnz59+fLlFy5cOH/+/JUrV+y2F8OQJQCAm3PcHDKr1ZqUlMRnrMK+X3AMTlE08evSpUvp6emdOnXy8fEpX758ampq2bJl7bbuHjBgwKFDh6SysyT9qLstmczHQ/rmm2/shiyXLFmi3hdvQJm/ny/zxFAyd+5ctbeteiEuLm7w4MEbNmzYs2fPnj17jhw5wvs15bscCAAALsNxgUxK2bhxY0zqd0lU4Llx48bbb7/NOw4JIcqWLcvtXimadOnSJTs7W52wrzYqs9lsj95M1Ww2f/HFF3aB7Mcff+S7U6f2P+wP59hk9zgtuYKDg+keebqkTqeLjIzs1q3blClTtm3bduDAgX379u3du/fSpUtq/+R8FwUBAMAFOCgY0SkwPDych3Ucc79Q2Lh317lz5yZNmkQ99ykJURCJiIjw8vLy8vJ6/vnnd+7cKfPMi7cpm4LzUKBdQ7sHRw9m/fr1/BjogrqqkReE5u/3lUp8VH8IPfibN28eOnSI793uY9OmTU+dOvXLL798//33GRkZP//88/Xr19UZaQAA4J4cGoyEwpH3CwVCnRRv19Nr9uzZLVu2rFy5spqBfH19a9SoER0dHRsb269fP4dVgCwWy969e4XSEpaONx4cVCeuFeD9qs9JTk5O3bp1eaCWHkNsbGy1atVGjBgxZsyYcePGjR8/fu7cuXv37qXdlqQy21L9FFP+AQDcAQIZPBAqBandTXn22Pbt2x9//PGAgABqVc+jhHq93sfHJyEhYfTo0Q7baZsy1rfffstpjC5cvnxZKtO/CnwGPY1gcv2MUtTYsWO5zQd3KTMajQ0aNHjttdeWLVu2c+fO06dP//bbb1988cWKFStu3LjBj1ANYSieAQC4PAQyeFBcGFPTzKZNm3r37k2Txqg2FhcXFx0dLYQICgpq27Zteno6hyHHLCS02WyffvqpuL1Za3p6utls5jYThXG//PP5Uynl22+/TSsu+ZEEBQVVq1atdu3aFStW7NWr14cffrh3794dO3aMHz/+t99+k7c/z5zPCuMBAwBA8YFABg+Ecwy30T9+/PiMGTM6d+4cHR1NhTGauR8fH+/p6enp6dm1a9erV68WYPevB2S1Wj/++GN1rFDTtFatWtF3OeUU7LQt9UeZTCargrvU6nS6mJgYnU4XEBBQqVKl6OjoWrVqjRs37rPPPvvwww9nzpy5Y8cO/mmPsvIAAACcDgIZPCieI2U2mzMzM+vXr0+7Bvn6+np7e0dHR1etWpV6ogYGBjZp0mTz5s2U3njm+6OvoLwvuq9t27ZROuQiWbt27eTtIaxgg84dG+5zqDp69OioUaMCAgLUtZ9UNmvYsOHkyZPXrFnTq1evl156acWKFf/8849UxohRHgMAcAcIZPCgeEHl+vXrx44dW6JECSGEp6enh4eHh4dHUFBQ2bJly5Qp88wzz6xcufLmzZt0KyoXSYekMZmbh3bt2mXXBuxf//qXVFZH0pXzvZbzbvfLc8gsFguXEmXuU/f6668bDAbeZImbYtSrV+/tt98eN27c888//8ILL7z11ltHjhwxm83Z2dn0w9EzFgDA5SGQwQOhqHH+/PnBgwdXq1YtISGBpkapCwmrVauWkpLCV6ZQohZ4CjAA3YPZbN60aROvr6QL8+bN40WLhdSL1W7lqZQyOztb3RtASjly5Ei75Z/UsE2n040fP/69995bunTpwoULFy9evH//frufBgAALgyBDB7U8ePHk5OTeRCQgwV9GhQUNHbs2I0bN9KV7ab/O3ICmczdyIjnkAkhVq5cqT6kwlhoebfYZFeWe/HFF2ndJfcHMRqNsbGxQohXXnmFxiuPHz/+9ddfHzp0CFsqAQC4iUIPRurZqEyZMtwks7DvF/KNuqfyp9z39dNPP+3Tp0/JkiV1Oh313+e4Exwc/MYbb/DNi+iB//8DPnfuHCdFepCvvPKKegXpqIyo9hLjxQRHjhwpX768ULZXIg0aNFi1atWZM2f2799/+PBh+gl2z6e6MbkDHj8AADiGIypVPEIUFxeHClkxp44qcivXAwcOpKSkxMfHJyUlxcTEqLshaZrm4eExadIkmjhlsVgcMy55N3ywNWrUiApR9AZg+PDh6rpFng/nyMfGK1XNZvPmzZupj65er+fgqNPpKlSoMHTo0KVLl27duvXzzz9/77335O0Dl/TIOds58vEDAEDhKfRgpE6sCQsLQ4Ws+LNarSaTietkn376af369TnZUJ42GAy6XC+88IJd39einfNktVpv3LgRGRmpzp2fOnWqzDNS6bCBVHUmGT85f/zxxxNPPEHPIdca9Xp9WFjY6NGj9+zZc/ny5V9//VV9wBzpUB4DAHAxDh2yDAoK4vN6Yd8v5I/dpKVPP/30ueeeq1WrFicGLy8vTdNKlSql1+sDAwPfffdd3h1cTRtFhVp/Xbx4MSoqio40quR98MEHasMLtYNrYT8eqexqIJVnmGqKGzZsoOZkPKWMYllqaurWrVv530ftFltIWz8BAEARcsTQIZ9CaOYyhiyLOTr322y2JUuWeHl5TZ8+vX79+hRuPD09qdWFECI0NDQjI4OvX0j9vR4W3XtWVtb06dM5RHp4eOzevVst4zlsY031aVEzmdoUY9CgQTRkGRYWFhAQQOsu9Xp9t27dvv766xMnTly8ePHnn3/etm3bxYsX+cEXefYFAIACVOjByJZLSkmjSAhkxZ/VajWbzWlpaUKIiIgIoWwZThPISpcu/dlnn6mxxmq1UhfTol0YaLFYKHjl5OSoE+qp1FQkHfDzboXEex5Qp7Hjx49Pnz5dHRGOiIh47LHH2rdvz0slrl279uOPPx47duzGjRv0leJQjwQAgIJS6MFIrQSULl0aQ5bFHJ3jly9f3qtXL2r6ygsV+W83YsSIo0ePqlGb0F+Z25kWubtVkhxcYbJbtUo4GtJ3rVYrFcaoC0apUqU8PT0jIyPT0tL++usvHrK0Wq03b95UNzIHAADX4IhKFZ/8YmJiUCEr/vr166dO4af5+/yHa9++/dq1a//66y/1Jhy7i8mUc7t+qvTA1C+qlapCxfsT3HE3T3pg9PUzZ840a9ZMHWYVQvj5+U2YMOHQoUPy9mFWhz1+AABwDIcOWdaoUYNHvgr7fuHe7M7l3FJ/6tSpFAj4L6X2u3/ppZfOnTuHHFCweGups2fPDhw4kBdd+vv7G43GOnXq9OvX7+zZs2qU5ARsNwKLRhgAAE7KEYFM5p5yGjRogApZsUJndJ7htGzZsuTkZG5YyoUxTdMiIiKeffbZLVu20AytvOOV8FDUXmhc9zKZTDabrUuXLtSTTAjh4+OTnJxcv3799PT0s2fPXrly5cqVK1evXv3rr7/oynwrtfyGvwsAgNNxaCBr1KgRAlkxwadwKrFcvnx58ODBVapU4QTGfymDwVCpUqXJkycfPHiQb2t3AdNagMEAACAASURBVB6FXXe0U6dOhYaGUiCLj4+vU6eOn59f2bJl+/fv/+677x49ejQ7O/vcuXPcyl9NdergLAAAOBEHBSM6SSQlJSGQFRPqWdxsNr/yyivcs5dbYVGdLDo6+oUXXjh69CgHOHVlZVH+Ds7MrsTIc/bpK9OmTeMhYwpkfn5+sbGxLVq0WLZsGY1Lnjx5Misri8Yu1TJn0f1OAACQfw6a1C+ltNlssbGxfL53wP3CvXF70iVLllBvC6PRWKJECbU8FhsbO3bs2B9++IFuonbVQhp7dOrTqNYd//jjj549e4rcxrYeHh5xcXFxcXGaptWtW3fFihWnTp2iTry8aIBviwoZAIAzclwgk1IikBU3N2/eXLp0KfV9tdsciT5t3rz5hg0bLl26JJXhMHX3BXhEnJ/UmWT06SeffMKTyWhLJbpQv379KVOmnDlz5u+//6b5ZJStqfUaimQAAM7IQXPI6GPt2rUxZFlM0Fl/7ty5NItfp9Pp9Xq7NObt7T1x4kTug0U35GE1ah5blL+DM8u7QNKuYwiNYK5du1YtWPICzIiIiIEDB65ater48ePHjx//888/i+oXAQCAAuGgQEanmdTUVASy4mP79u00LsYne76cmpq6ePHiLVu28JXV0GC3VxLkj91gpTqfTL3aunXrvLy8aHWFGpf1en3btm3XrFmTkZHx5Zdfnj17Vua2vcCfBgDA6TgoGNEZon379hiyLFpc01q9evXYsWOrVatGp3k1jU2YMOH48eO//PLLL7/8cvPmTac7u9MD5g2U1G/ZrS0thtRwxhP29+zZwyuUDQZDSEiIt7d3SEhIzZo1p02btmvXri+//HL9+vVHjx69fv06d9NQJ6Vh9SUAQDHnuFWWUso2bdqgQuZ4nEJ4AviOHTs6duyoaVreNJaUlPTpp5/evHnz+vXr3HKsKB/9wzObzbR9E6UuuyzC7SFkMa4kcW6mCxaL5cCBA8OHD+c/k4eHB62BLVeu3LBhwxYuXPjjjz9mZmYeOnTIbjtzp/vzAQC4J4cGo9atW/PomCPv153ZbR9ktVp37tw5c+bMsmXLCqUBrNFoNBgMzZs3nz179smTJ202G5dnnGuimLrqkKPJrVu3pDLeWpwzil1SpKWUdNlms40bN07k7qpEq2IDAwOHDx/+2WefnT59+tChQ9OmTTtx4oRaGyvOvywAADDHBTKbzcYVMk3THHa/bs5uBd+33377xhtv9OzZk/pcaJpG6/i8vb0bNmw4Y8aMgwcP/vPPP3adxpzupG61Wl944YW1a9fyGwBefsi/SzHPmuosPU6ZN27coDqZXq83GAwGg6F+/frdunUbNmzYggULtm7dSsnsjj8Q4QwAoDhzaIWsS5cuCGSOx7OmTp06lZqaGhUVJZStKulj3bp109PTf/31V/WG3G7UuU7k9Ji5YQQNy0plKhVdrdj+UhwT1RFVzmenT5/m+WQ6nS4kJCQoKIj2tvrkk0+klMePH6c/nF2drNj+vgAAIB08h6x///4YsnQ8nsD+8ccfly5dmkcqiV6v1zTthRdesOv+yudvs9nsXOdyCjR0mNHonsFg4CITp5PiOYHMYrGozzb1FlGXKmdnZ+/evZsWLFPb2MjISFp32b17982bN69Zs+bzzz/n3eKJc/0FAQDckENXWb788stclXHM/QKfiU+dOtW5c2c6hXNHK8orLVu2XL9+/ZUrV2SeOWeyGC9IvBs62Ly9vXl6nBDi2LFjOTk5dr9LMYwpvMrSbjhVHWClX7BHjx5UbK5bt27dunVDQ0P9/PwSExNffvnl1atXm83m4pk4AQDgjhwayMaNG4e2F0XCbDaPGTOGhynVlZXR0dHvvffe1atXr127ZncKv1tnrOKMx2d79eql7j1w+PBhzjrFfF6/2m2fq5VqOwxecsGdyRo0aJCcnFyhQgV/f//mzZt//PHHd/wFi/NvDQDg5hwRjHhe+YYNG/gc6YD7dStqzlB76NtstszMzD59+vAAZVBQUGhoKH3atWvXgwcPytwymAucsCnBXLlyhWuxND7btGlTdX2Dk/6m3AVDSmm1Wr/77rsqVapQnSwkJITqnf7+/unp6Tdv3rS7rdNVOgEA3IrjApmUcuPGjVyxcMD9uhve0FDePo97x44dycnJNFeMJlTRn6Bp06YzZszYvXt3VlaWK52tLRbLH3/8IZTdOWmSnHTaHMa4/kd/ZdpbqU6dOup0QCFEx44dp06d+uWXX9ImpBaLhaamOfuvDwDgwhwUyOh8v379ej5NOuB+3Y1dkYy+cujQoQYNGvDiVl57WLly5bVr1545c+bcuXN0cycal7w3q9W6a9cude2CXq9/44035O2BzLkyqN1aBJvNxpsQ3Lx5s3v37tSWrFKlStHR0XXq1GnWrNlzzz23atWqM2fO8A9xmT8xAIDrcVAwopPfihUrUCErJHyetruwePHikiVLGo1Geto5o4waNYquQLO/1XlLTo1+i3Xr1uly0e87efJku5ZsagvZ4k+dz8fNY9UlmQMGDPDw8PDy8goNDa1WrVq1atWaNm06adKk1atXX716Vd3LHAAAiiEHbS5OHxctWoRVloXHYrFkZWVJpZHV3r17e/fuTSVJ7sil0+k6dOiwadMmmRtKin9rrgfH0Z934Kbffdq0aWoT/GLb9uIeeMwx787uZrP56NGjjz/+OJdCDQaDn59fcnJyt27d1qxZQ1dDJgMAKLYcFMjowieffIJAVkj4eTaZTNS56sUXX0xOTuYxYk9PT+p58eKLL27dulW9ITeAdQ00s4qjCV3gUCJvLzIV3cN8OFzOVNeKqr0tzGbzyZMnx48fr+5PqmlacHBw7969Dx48+PfffxflLwAAAPfk0K2T9uzZw/1IHXa/boKKHzyGdezYsbi4OLtu9UajsUOHDjTRW+ZO/ycuUBtjFovl8OHDHEpo4JIWk3IOsxu+LP7Uxrby9r8XFUR5UHLy5MmtW7fmCf5CiNKlSz/99NPz58/nPz0AABQ3Dl1lef78ecwhKyR8nqYLc+bM4SZVzNfXd8KECVJKq9VKJ/js7Gy1VuREAeVu+Ffg8hh9PH/+vHTmSf1SSm6+n3e9pN3v8sEHH6it5midaWpq6tdff+3QRwwAAA/MEUOWnBX27t3L03oK+37dDQ9pSSlXr15dq1atwMBA2oKaq0RPPfWUdMIg8rBsNtvq1as5htLxpj4/dn1WXdLp06cbNmxYvXr1Fi1a+Pn5CSH69+8/efLk06dPc2blMqELPw8AAM7CoY1hz5w5gzlkhYQ7ux45cuTJJ5/U6/Vq0wdPT8/GjRt/+umnLtMA9m4obJ09ezYyMpKeAfr4448/qjtBOeOk/of11ltvVapUiYujL7744lNPPfX2229zd1kiXfp4AABwFg5dZXn06FH0ISs8NDf/yJEjLVq0sKsPBQUFjR8//ocffjh58qR0gyLZli1beLGhEMLDw+Onn36ib3FvVekGQeSDDz6glRylSpWip6JGjRqrVq3iXv/SJcapAQBcQKEHI/VF/8CBAxiyLDw0Sf+3335r27YtL7KjeuT48eMzMzOllGaz2bn6b+WDzWZbuXIlz52ilQ2nT5+WytsDN8kiZrP5iSee4H86vV5fqlSp6tWrb9y4Ub2ayz8PAADFn4OCEZ0Iz507hyHLQpKdnS2lXLhwYZMmTSiHRUdH09Sxjh07nj17Vt08x4UrZPQ7/vzzzyVLlqQaodFoFEKcOXPGbhdL1y6PqRmrZcuW9E/32GOPhYSECCGSkpLGjh37008/ufaTAADgRBw6ZLl//34EssKzYsUKX19fLodwIpk8ebJUztAU3VyYzWbbvXs35TB6KgwGA3XwUo9G184ivIiBCqLckU6v19epUyc5OTkhIWHmzJnyTms2AQDA8Rw0ZEmv+BkZGTSKhCHLArd7925q1E5BxNPTk57n5OTkPXv20HVMJpPLj1dKKa1W686dO4ODg2n6lNr3zm7qmAsHEV5PSh+/++47boTRqFGjp556qlWrVsOHD//iiy8QyAAAigOH9iH7/vvveVaTA+7XrbRv356e2KCgoODgYKoMaZrWv3//U6dOydwZZup52oUdPXo0NDSUopinp6cQ4sCBA/QtdSPIIn2MhU4tBFoslrNnz1apUoUymYeHR7du3eiY2bVrV1E/UgAAcGAgo62T0Km/QNglqtatW/v6+gYFBfn5+SUkJHBr/kGDBhXVIyxaBw8e5NoYXaCNg3iTKO744MJ4kJrrgnPnzvX396enpUqVKjExMWqXZrvrY6Y/AIAjOTQYLV++HIHsEXGM4Gwxd+5cIURAQEC9evWqV69OOyZReWzVqlVF+mCLhsVi+f3338uWLctDlnq9ft++fbTzusVi4c3XXXsAl7dUkrn1UZvNlp6ezutPKa1qmpaenm53E5vNxs8SAAA4gENXWdKOLghkj0jdYXr9+vXUA9bT07NDhw4pKSllypShE+3SpUuvX79e1A+2CFBm/eGHH95///1evXrR4gaOGpQ2XL48Jm/fvEEqmexf//qXl5eXEMLf3z88PFzTtPj4+PHjx9M6XAqpditSAQCgsDkoGNHr+6JFixDIHh2ninPnzjVq1Iiez3LlynXr1i02Nlan0xmNxr59+xb1wywaPEWMSClzcnJo3rrVauV04g5Rg3Knmj7pQnZ29rFjx+rUqUN9Yrmb/8SJE6US4NwhswIAFB8IZE6Gk4TVal22bFnjxo3Dw8PLli3btWvXmJgYmsDeoEEDmsjvhudUtRG/ukujeoFiinTpWEaFLp49Zjab1TlhFotl9OjRNIEsKCiIeufWqFFj165dlFx5u7OievwAAO4GQ5bOh86XmzZtqlWrlhAiIiIiPDw8ICCAnth69erNnz9funF/KXV0UioLKu0m8rvDwCX9juqvqQbWPn360O5Snp6eAQEBVapUGTp06K5duziQufzzAwBQfGBSv5OhtGEymZ555hl1djZNXTcYDHv27HGTjRrvyGaz0WQpShWcxtTKopvkDFrEIHO3iuIimbqUYfHixVQnI7Vr13755ZcPHz4sXX3FAwBAcYO2F87HZrP95z//8fHxMRgMPAGIdknq16+fm+zSeF/qtHR18aB6BXd4luxyFf/K/FT06NGDm4P4+fl16tTpvffes7sOAAAUNocOWdpsNm4X7pj7dWo8uqQOPtpstjZt2nBVLCwsjDpcaJo2Y8YMusKtW7ckxpvgTnhFiFSOENrjQa/XDxw4cPjw4WlpaWvWrOFDTu1/obbSAACAAuSIYKQuv+dyjgPu1wXYVXEsFsvEiRONRqOvr29CQkKpUqV4vKl8+fLU5MKuJlRUjxyKM7V8SLGMD6QWLVp07949JSVl586danTjY4lGhAEAoGA5IpDxkrcrV65g66QHpJYleMXcrl274uPjecYY1cbo0zVr1tCVueyBNAZ5qWsdpHKQbNy4kTKZTqfr0qVLRETE0KFDN2/erI7zYlYZAEDhceheljt37qQF9hiyfBA8UkknwmvXrr3//vs1atTw9fXljRrp+XzqqaeklDRSyQUMuylTAEyteNF8f4vF8tZbb9FMsipVqvTs2bNHjx7jx48/c+aMVGIcjigAgELioGBkMplsNtvXX3+NOWQPiKsR2dnZUsrMzMz58+enpaVpmlaqVKnExMTo6GghRMmSJfv27ctLC7mJP8HpE+yohTGTyWRXT+3YsSNVXlu2bEm7j0+aNIkPRb4yjisAgALn0MawBw8e5Kkqjrlfp8Y1CYvFMmvWrMjIyJo1a0ZERPB4pcFg6Nmz59WrV7mNJ6U3DC3BPajHlcx9s0QXzp8/n5qaSkdX48aNq1WrlpycPG/ePGqcIRHFAAAKTaEHI3WFYE5ODs95Kuz7dXbqGrddu3a1atWqTJkyvXv3TktL495j9evXz8nJsatb0BPO8/aK6vFD8cQVMjpacnJy9uzZQ9+iZJaZmVmpUiWj0RgSEhIdHV29evXGjRuPHj36xIkTUpkPWmS/AACAi3JEIJNKN04MWT442vFGSrls2bL4+PjIyMjKlSvzRH5N00aNGqVuAaSur5QoZsCd8LZR6tpn7uZPny5dunTu3Lk0maxJkyalS5cODg4eMWLEjRs35O3LTQAAoKA4tEL2/vvv83BbYd+vE8nbwlQ9Oz733HNGo1EIkZiY2Lp166ioKCoxLl++nH8CWo7Bo1P3kjp37hzl/oiIiO7du9MbgFdeeYWa+MvbW2bQgYo3AAAAj8JBwYjegq9du5bKY3q93jH360S4dEEzwCiljRs3LjY2lqaLRURElC1bNigoKDY2duLEidI9dmMEx1APJDoC33nnHR4c56mfr7322uXLl2Xu2ky6Pr+XwFAmAEC+Oa5Tv81m27ZtGw+3OeZ+nQVXGtRq2fz58yMiIqigKITw9fUVQhiNxmeeeYZ6wBKcCKFAmEwmrnXREbVgwQIuadPHvn37fvzxx3wTekvA18doJgBAvjl0yHLbtm08Baqw79dZ8PPDoYp6RO3bt69r164+Pj5CiMDAwPDwcCpUVK9effXq1XxbVMigYNERlZOTQ4dlenq6EMLT01MIkZycHBUVNWjQoEuXLqlDltg+FQDg0TluUr/NZvv+++8RyPJSl0nS5Z9//rlr167VqlWjJhdUGBNCJCUlLVu2TCrPqry9ySdA/nDzC5k7ZGkymei4euyxx2jQnNoRx8TEjB079vTp0zk5OWpBF42IAQAehYOCEY1lnDhxAkOWd6SO+OTk5EyYMCEuLi4hISEmJoZHi4QQw4cPl7nnTjWT4UQIj47HvnmzBzquzp8/r9PpaBpZgwYNdDpdcnLylClT1OuohXAAAMgHx80hk1Lm5OTQ3iwIZIzLY3QWzMzMnDZtWrt27apUqVK+fPnSpUtTFEtKSho5cuThw4f5/Ec3VEeOAB6F3cijeoBJKcuWLavT6Z599lm9Xk8NY0ePHi1vj2LIZAAA+eaIQEaFHxoHQR+yvGgnQTqZffLJJwaDoVy5cklJSf7+/hRey5cv379/f/Vsx4OVEmkMCpQay9T+doMHD6b3BpGRkWXKlKHLL730Et8QaQwA4FE4OhiJXA6+3yLH4Ult3cQ5jL741VdfNWrUiEZ1S5QowcO7oaGh//3vfyWy14PJu9ZPfd4wuJYPtI7y8ccf1zSND0vqHLt37166jjqRkeeTYd0JAMADQiBzHN5iUu3hxKeukydP9uvXLz4+Pjo6mnqP0dQxTdNmzZolkcYejDriRtnrbtPs+FvwIOgJHD9+vDqpsUyZMh07dtywYQO/r1B3K+fnHM8zAMB9IZA5CJ+o1JMWV2uysrKGDBkSHh4eFBQUGhrq7+9Pyyr1ev2UKVO4YSxKO/dFM9PVMpjFYjGbzXd86hAU7stqtfK+Z6RNmzYlSpTQ6XSRuYYOHXry5EnuYaY+1ThiAQAeEAKZg6jzvWgch9OAyWSaMmVK1apV4+LikpKSKlasyK0uunXrRrei6hpObw9CjWJ2O69LDFk+guzsbEpdgwYNotptdHR0cnJyWloabx0hc5929QlHcRcA4L4QyByHKg28ZbjMPVe9++67LVu2fPzxxzt16tS0adPGjRvTTJ3q1asfOnRIXVDJg55wDxR21SiWnZ2tZgKroogeo9Ow63VHcfbMmTN169aljZW8vLxokP3tt98+ceKEOmSZNw0DAMDdIJA5iNlsVjcOt1qt2dnZUspr164NHz68YcOGTz31VEpKire3d8mSJekp+ve//60msKysrKJ68E6En7EKFSrQ02gwGHQ63c2bN7mRKTwUNbxyB/+5c+fy1H6q5g4cOHD37t15sxcaxgIAPAgEMkejExulMbPZPHXq1FatWjVr1qxmzZolSpSgeWM6na5t27Y7duygm3DPdAz9PAiLxfL777/Hx8fTM0m5wW4QjWb04/l8cFwk4/ruqFGj6BnWNC0wMLB58+Zz5sy5desWfdduY1YAALg3BDIHUdts0kez2Tx58mRerSYU7dq1O3XqFN2Qz2oYYnsQ9NyePXu2XLlyXB5Tjzfkg4fFy1HVUGu1Wq9du/bcc89RtzwSExMzevTo7777zu79A55zAID7QiBzELXJAp2oNm7cWKVKFU3TKDFQeqAK2VtvvVWkD9a52Wy2jIwMrjUKIXQ6HX9LKm1HUCF7FHRId+nSRQ2+Qoh+/frt27ePn1tefUmf8pA9nnwAABUCmeOoBYZbt27Nnz+/VatWgYGBCQkJwcHBHh4eQgh/f/+UlJSMjIyifajOi1Lv6dOnK1eurE5ykrlLKNRMhspNvnGt9+DBgwkJCXq9Pjg4mFuUpaSkzJo1Kzs7226Uk1a02IUzAACQCGQOw3UCi8Vy9erVRYsWlSpVioo3XFrQNC04OHjz5s1F/WCdG53v27Vrpz63HAJ4bhPGfx+FOoaekZFBbyfo2U5KSmrQoEHDhg0nTZp04cIFuwoZXaadW/EnAABgCGSOw3WCefPmtWzZsl69epUrV05ISIiJifHw8ChduvTkyZPpCqgcPKIrV64kJydzStDpdJmZmVSbUceOUSHLN7v5YVu3bn366adDQkI0TaMimV6vT01N/fbbb+1uxZMp8eQDAKgQyByET2AbNmx46qmnKlSoQBUynj1WuXLl/fv38wybIn2wTowKYBs2bKAlq1y2OXr0KLXsp6vxxkpF+2idGg/70tP40UcfxcXFaZrm6elZokQJHx8fPz+/SpUqXb16VW0Vy7PHzGYz3ngAADAEMofas2fPmDFj6tatGxYWFhYWxnsCRkVFzZs3D/sxFwiLxUJdsijpUs1m69at9F2e1YQ09ijUHdzpyTx16lSTJk3oaU9LS+vatSs982XLlt2/f79UnnN1xTEAABAEMsf5/fffBw8eXL9+/aioKJ5vHh4e3qNHj3Xr1vHVUB57FNSXf86cObTEkvuQZWRkUIZQ+5QiEzwKdcI+1b1+/vnn+Ph4esJjYmKqVatGh/qgQYO4KobnHADgjhDIHGfSpEkVKlSgvk0eHh46nS48PHzMmDF//fUXXQGVg0dHKeHjjz+mKMax7NKlSzJPCwY81Y/Ibr2kxWLZsmVLvXr16AiPj4/v3bt3jx49BgwYkJGRYbedq8TzDwCgQCArYHmXj9FZZ+LEiTRAWbFiRaqNUdexV155ha6G2WMFxWKxnDx5kgYrhRBGo5GPN14YmHfpHxQIekpfffVVevJ1Op2vr2/Dhg1nzpyZlZVFTzttb4U5fAAAKgSygkftl/jEf/78+TfffLN3797x8fGhoaF6vV7TNFoAWK9evQsXLsjcUxS3Y8Am4vlGufaHH36gw4xCsE6nM5lMdp0XEAgKA70hOXToULt27fjJNxqNFSpUGDBgwPfff89vOeiaAABAEMgKmFp6oXCwYcOGAQMGtG/fXp06RpWbL774QkppMpnUcRzM6H8UFLOWLFlCaYw3QpgzZw5FAdqjXW1+AQWLNnHftGlT69atNU0rVapUUFBQfHx83bp1//Wvf+3du5dzMO3oCgAAEoGs8PBk55deeik6Orp06dKUxiiQRURELFq0SF35z3tWqt0ZIB/MZvPChQu5PEa7+tDmB9x5ga6JClmB4+FIKeXq1asbNGjAf4iQkJDo6Oh69eotXryYrobjHACAIZAVPD7NHD169J133qFZ/GFhYVFRUSVLlixVqlR0dPS0adM4ilksFruIgBNVvtETuHz5cpqixzOZaP8Du02vUYwscGrkPXLkyMiRI3k70cDAwLJly/r6+rZq1Wr79u2oUAIAqBDICh41Vjh8+PDzzz+flJQUHBys1+srVaoUGxubkJDQpk2bV1555eLFi3xNnjFG5TGkhEdBgSA9PZ0OMx4gnj17Nl0BM8oLmzoD8ujRo2PGjKE/QXR09NNPP92iRYsnn3xy4cKFFy9eRCYDAGAIZAWMpir//fffU6dOTU5OjoqKiomJoV/Z09NTCNGpU6fVq1er3bAIt2hCeewRWSyWmTNnGgwGGimjTHb69Gl+wnnxBDJZIVEbi5w/f75///60iqVkyZKBgYHVqlVr1arV3LlzaT4fAABIBLLCYDKZ9uzZM2DAgObNm7du3ZoqZNR8wd/f/6uvvpJ5pjHZLatE5eARbd++PSAgQJ209/PPP6sbWSKNFRKeCqmOXd66dWvUqFH8vx8WFvbEE0906NBh165dRfxwAQCKDQSyfLLrNMaNLunrc+fObdKkSfXq1StXrixyt1MUQixbtkwqU5cQvAqDzWabO3dumTJlhBBUmKHjjXqRYClroeIheFpBaTabT5w4Qc95y5Yt1ZqlTqdr2LDhDz/8IJV/BI7LRfYLAAAUEQSyfMq7OzWdgbZv3z569Oi4uDiR2yaeLhgMhueff57m79tNHYMCRH+U6OhoNY0JpTEsRzHsGVoY6D0JDbvz4Pvhw4enT5++ZMmSGTNmqOsuO3Xq9Ouvv6pzKGWeajEAgJtAICsAJpOJ3tOfPHmyR48e1GdB0zQfHx8aqRRCNGnS5OjRo3R97lWGSkCBo4zVo0cPodDpdFu2bOGuV4hihY1DFT3PWVlZH3300aZNm5YvX56cnCyE8PX1DQwM7NChw4wZM9auXStvr1ni/wIA3BACWf5xDiNWqzUjI6N79+5UmNHpdCkpKU8++aQQIjk5+fPPP5dSZmdn27XCgsLQv39/PtJogGzp0qX0rbyde6EA8dsMLh5znezmzZuzZ8/u3Lkz1S+FEBUrVkxMTKxVq9aHH36olo3xdwEAN4RAlk/c95WHILdu3Tpy5Mhq1arxRHJ/f/+IiIjIyMhvvvnGbDbTiYpPOZhUXhhovIzGiDVNowqlTqdbt26dGhRQniwk6moJPsipNsnbVc2ZM4dnVQYGBgYGBjZr1mzz5s1IyQDgzhDI8o8yGZ0/vvvuu+HDh9NIJaUxnr88ffp0roqpM8+QCQoP/7uZIwAAIABJREFUNyOlP4Svry+3hOXEQDv8FPUjdTV2oUrt4cJ/gu3bt3fv3p3+OjVr1qxataoQIiYmZtGiReoPAQBwKwhk+cQVMpvNtmjRorZt29arV08dI9Pr9f7+/s888wxdnzu+5h3KgQKnjgtzAs7JyeEobLPZ8PwXEnXZBF2gZ55qyVLKU6dO/f77776+vl5eXmFhYZGRkTTtskmTJkeOHCmyxw0AUKQQyB6J1Wr99ttvmzdvLoSoXLkyven39vamfXuqVav2448/SmWOM09gwkyywmM3jYmefHWrUP4UmazA2R3YduOPann4jz/+aNy4MW+lQEWyjz76yPGPGQCgOEAguz+uptidv20229dff920adO4uDgvL6+mTZuWKFGCJ8d4e3uvWrWK++8DuDO1NkkpzWw2HzlyJC0tjevKfn5+lSpVmjBhwtWrV+3+6fjmRfpLAAAUIgSy++O5L+qGMDabbdu2bc8880zdunUDAwNpxhKlMU3TPDw8VqxYQdsoScxTBlBwncxsNi9YsECn0/HCZE9Pz06dOn355ZdSSqvV+s8//0hlNUARP24AgMKEQHYfdnscqTPxq1Sp8thjjyUmJgplix5N04KCgt555x0uA6BIBkDUyf70T/HXX3+1aNGCZ14GBQVFR0d37tx5w4YNfBOkMQBwBwhk96duSs3TksaOHRsdHV2xYsXg4GCdThcdHd2vXz+j0ahp2pw5c3iKEk4kAPIuZWbKZ7t27WrZsiW9LPj6+ur1+qioqEGDBq1evVq9GvfwAwBwSQhk96HOfZG5M1qGDRsWGhqalJQUGRnp7e0dFhbm5+dXsWJFvV6fkpLC+/PwbTFkCW5OrTSrmYw+ZmRkeHt7U6cYmt3fuHHjXr16ffvtt/S/g/2UAMDlIZDdn9VqVc8HGzdujImJ8fT09PPzo4kvFSpUoF/K399/x44dfE06lyCNAcjcLsry9hYw/D/y3HPP8WqYSpUqVa9evV69eh988AHdFm1KAMDlIZDdh9p23Gq1Tpo0qVmzZlFRUUIIo9Go0+mow4Ver9fr9fPmzZO55xt1eAWZDID+g+y+yBtdSCnT09Np63FPT8+AgICYmJixY8deu3ZN5lbIMGQJAC4Mgez+6ERisVhWrVrl6+srhChXrhzvyUPjLBEREadOnco7OsmbxhTRYwcoFrjGzHUyphaSJ02axC8RZcqU6dat28cff3z9+nWJGZkA4OoQyP5HbSJKn6qXpZRWq3X06NF6vZ635aELRqOxQYMG586dk2j0CvDw1NmZ165da9q0aXBwsI+PT3h4eGRkZNWqVf/73//K3J3H8q56RkoDANeAQPY/VquV2oapL/qcyXJycmbPnl29enV68DRkyWv1161bh8aVAPljt4r5p59+atiwoRCiSZMmaWlpUVFRqampO3bsUN/t8Cw0iT1hAcBVIJD9P7WNuF15bOHChYmJif7+/jRjjHetDg8P/+qrryTOCgCPwK7Q9c033wwaNKh37961atUSQhgMhq5du86ePTszM5PeNdFMgKysLL55kTxsAIAChEB2G94yXCoZa8uWLTVr1qxRo0ajRo3owRsMBqqNbdiwAQvyAfKN/sXUfyKKXDNnzgwLC6OZmlSK1ul0c+bM4ZtQJrNarVh9CQCuAYHsf9SG4HZFspEjR/r5+dWsWTM1NZXPEEKIChUq0BIwSy4UyQAeFu9WqW4Av3bt2hYtWoSHhycmJlatWrVkyZKapqWmpm7cuFG9FX3EKmYAcAEIZLexa4/0ww8/jB07tlWrVhEREf7+/j4+PjSXX6/XR0dHf/rpp5Th6ESCNAaQD1yWtuvbl5OT8/TTT9P0gA4dOnTo0KFSpUqNGjV699135e2dyfCvBwAuAIHs/6ltKqWUmZmZgwYNCg4Ojo6Opt4W3OSibNmyu3btUnMY3YQntQDAg6BQRdVozlVms5kuT548mV4ufH19mzVrFhMTI4RISEhYsGAB/wSkMQBwDQhk/6Mu2pJSZmdnf/TRR506dSpZsqQaxXQ6XcuWLS9duqTeik4nNPcFAB4WNydT+8TSxy+++CIhIYFeNLy8vHx9fT09PVu2bHns2DFsSgYArsQdAxlPFFNbGanLKnNycqZOndqoUaO6devGxcXxnGKayL93717MIwYobPwvOWXKlMqVK9PYpaZp9NborbfeUovTUlmLIzGrDACckNsFMnVTI7tZ/Pw2ff369VWqVKHVlEKI0NBQeswlS5b86quvsK0egGNwd8B9+/Y1b96cXz10Op2fn9/06dPtymkmk4n/PbHIBgCci9sFMpnbf5KHSGj9PL+lPnjwYNeuXfV6PS+o9Pb2prn8Xbp0odMD+h4BFCr1XRP9bx45cqR69eo6nc7f379GjRplypRp27btjh075J2yF5rRAIDTcbtApmYpfhHnmLVx48bu3bvHxsbSnpUeHh56vZ7GK1NSUv755x905AdwAHU/JS5gd+3aVdM0TdOioqJCQ0M9PT1Lliw5fPhwqlvTDen9lcSuSgDgbNwukNntbazOC/7kk0+efPLJkJCQ4ODg4OBgmjFG4uPjs7Ky6JoYrwRwAHV7WfqfvXbtWlpaGjdnFrl7lyUmJn755ZfqNpcYrAQAp+N2gUwqG+Hxe2iTyTRp0qRatWqFhoYGBQV5enrSK76Hh4fBYEhNTd23b59UXuiRyQAKW05Ojvp/ym+cOnfuTK8hHh4elMnCwsLeeecdbvrPLc0wcAkATsTtApnaaSwrK4vbgiclJamv8vRRCFGzZs0//vhDKsOU6rIAACgMnKXMZjO//6F3RJcuXWrfvj1N6zQajZqmxcXFvfTSS9u3b1fr30hjAOBc3C6QydwZY//884+U0mQyLV68uE+fPgkJCTRjjNpbhIaGGo3Gxo0b79+/n1dyScweA3AUtZKtvgui/Str1apFmUwIERQU1KBBg3//+98ZGRlSmYeAgUsAcCKuHMgsFov63lpKaTab1bEMq9W6efPmZ5991sfHh9dU6vX6gICANm3ajBo1ihvAgrPIuxspJne7Eh64nDZtWmRkJE8m0+l0cXFxkydPpv9Zdek0b81EN8QMMwAonlw2kPELN73PVk/JnMaWLVvWqFGjpk2bUitwWk1ZsWLFzp07N2vW7NatW3jhdjpms1ldoCcx4c/lUJOLv//+e9y4cTVq1KA3UZqmVapUqX///m+99davv/4qb28Yy6Fc3QkAAKBYcdlARtTqiFo7sdlsv//+e6dOnXQ6Xdu2bel9tqZpXl5emqY1b95806ZNEkMezon/atnZ2Wr3dnAB6iBmdnb26tWradSSS2VhYWHz58/P26kfHfwBoJhz5UDGL74mk4m3K6avnzt3bt68eU8++WRkZGRiYiI9pICAAIPBoNPppkyZYtcdA5wLV0FQHnM96pijxWJ59dVXqbxNFW4hxLhx42TumzG7voOI5gBQbLlsIFM3C+eXb5oOfOHChTlz5rRs2bJ8+fIBAQE6nS40NDQgIIBe0BcvXixzh0Xw8u10eMjSYrHQ6g1s+u56qLEF/4/37NmTGpLRipw6dep8/vnn/F9vN5EUcwoBoHhy2UBG7BZq0Wv02rVrO3fuTLP4aaEW94AdNWoUX1/iXO6E6A+3b98+rpfQXtRF/bigYOSteFmt1q+//jokJIT+o/V6fVhY2LPPPrt48eI///xT5pnFj1FLACieXDaQ0UuwWieTUl6/fn358uXdunUTQtDLN0UxGu9YuHChzH3zLTGBzDnRX+3XX3+ltE1naJ1OV9SPCwoGpytamsOV7JdffpnDd2hoaNeuXcPCwgYPHnzx4kWpVMVQ9gaAYstlA5k6g4TGLC5durR48eIGDRpwSUyXq0qVKitXrpS3N0rANDInZTabV61aRYGbG/wW9YOCgpT3LZPVak1LS+NusRUrVvT09Gzfvv3WrVu5kCZz35jh/xoAiiFXCGTqFLG7fffUqVP+/v7h4eE1atQIDQ2l2omHhwddGD58eAE+HihC9Of+8MMPuY8JHW8fffQR5XJu4I5O7q7EarUeO3asYsWK9Oc2Go21a9cWQvj5+b366qunTp3iqyGNAUDx5AqBjFArSJk78UstcdlstvHjx6elpYWFhRmNxsceeywlJSU+Pp4eSVpaGm2OBK7BarX+97//5fFoKoVev35d3j7OJdEOw7Xk5OSMGzdOr9fz9FAawezRo8eGDRvk/d65AQAULacPZNzlVebpcZCVlSWlzMzMfOONNzw9PWmkkkom1apVo7O1wWC4fPlyQT0YKA4sFsucOXOEggKZzWbjIWm+ZpE+UihI9GeltoLUKlYIER4e3qNHj/fee4+2SlN79wMAFCtOH8hkbjMLPtfSBSqYLVq0qGHDhrwenk/P1ENSr9f//PPPEpUSF0LR/M0336Q/N83oF0KcPXuWq2J2Cz7ABaiLo59++ml+nQkLC3v88ccHDBjw7bffWq1W9f0bAECx4vSBjM+y3OGCT7SXLl3q06ePpmk0cqFpWlpaWvny5SmNJSYmrlu3jl7BMZ3IZdDxMGTIELtApi6vUzfVKbIHCgWNo/aFCxcSExP5DZiPj0+FChXGjBlz9uxZifIYABRXTh/I1NWUVCqTUmZlZb3++ustWrTgdZQUwnx9fenC448/zuvhkcZcjMViSUlJ4WoofZRKy4O8LUPB2dEfl2YHWq3WAwcOqJMIg4ODO3fuvGXLFqlMNgUAKFacPpARrpPRGffNN9/09vbmk7FQWlIJIaKjo3fu3KneBKNXrsRsNleoUIH/6HSBYzf2YHBVdtvJP/fcc/SnDw4Ojo2NrVGjxrhx4/bs2YNABgDFkysEMrumRG3atKFhyqioqODgYHXGmBBi3LhxdE1uACsxh8yF0J8yPDxcp6DjTW15gAjuknjLWpvN9s8//7Rs2VIIkZSUVK5cOSFEzZo1x44de+PGjaJ+mAAAd+BkgYyzlFRSFJ9lR40aVaZMmVKlSnl5eZUpU8bT01NdZyeEmDx5ckH8ElB8Wa3WX375pUSJEhzBycaNG9XCGCK4m+jXr5/6CtC0adM1a9ZQDxQ+Hrh6iiFsAChCThbIpLInknpO/f333wcMGBAbG+vj49OpUye+F9oTSQhhMBhGjx5969atAvgdoHjbuHEjpTGa0U9ooEodr8SMfnfw2muvUYGcl1r36dNn3bp1fAV+PcHxAABFy8kCmdlsttls6mQRmgT2+uuv16pVq379+qmpqTyTl98WlypVaubMmRLnYDdgs9mWL1+udjnh440LIbQcD6OW7iAzM7NZs2b8xiwuLq53794rV678559/KJer8xZwSABAEXKmQMZVMXo7S30uzp0798ILL5QrV87f39/Dw8NoNFIICwgIoDvy9/en5e5IY27iiy++4BMwz+vn7M6HEE7AbmL//v2xsbHUFDowMHDYsGHp6enffPPNqVOneMgSRwIAFDmnCWTc0dFsNnMy++OPP2iAUl1EKXL7dEdHR48YMYI6dHNPSHBtVqt1//79tHmOyN3OktpeSCm5gTBfuYgeJjgI/YmnT5/OrxJVq1Zt1qzZqFGjtmzZYrVaaac1NIwFgCLnNIFMKm9kKY2tWrWqb9++JUuWpBxG52AujURERHz//fcU4HJycjBd132YzeaWLVtyQ2Cinmt5yBInYHdA//uzZ8/mrjcBAQFVq1YdMmTI1q1b1bW3OB4AoAg5TSCjV1Vuxv3ll19GRkbq9XoPDw+h7IxEe1Z6eXlt2rSJRqmk0hcDa+tcHpU6Bg8eLJS+oJqm0f7x6qgluAlO3qtWrdI0jV43hBA+Pj5PPPHEu+++a7fDBwBAkXCaQCZzC2MWi2Xjxo3Jyck1a9akHxUYGFi5cmVKY/SV//znP/xm12QyodmBW7FarW3btuXyGMWyjz76SP3r4xzsJuz2LVXXedA7tzp16rz77rvnz5+XqJABQJHK/+R6LlY9VMnhQQIZnyy5KqbOv54yZUp8fDxPEuJJY7y4HZ243ZzVam3YsKHapl8IcevWLTpK87avA5fH/QsvXrzIGZ0ie5s2bb744ou//vqLr6y+2tg1nQYAKDwPHcj4tYnrVQ81Wf6+gYx+vrrRjXoG7d69e6NGjTp16tS6dWshRJUqVWrUqOHl5RUeHp6YmDhjxgyJV0+3Z7PZ2rRpw2dcOvuq3Q3UtxPg2u6Yv2fNmkXHhpeXV1BQUJ06dUaMGCFv7zjNEx7wBg8AHOOhAxmd2GgckCpYD3Vie/AhS3XKl9VqPX78eKtWragwFhYW5u3tTSdaDw8PWknXtWtX2hQFc8XcGR2f48aNU8tjmqZlZWXRgYFjwz3R5H16yVqwYAEPZ3t7e8fGxjZu3HjdunXUSUfmWT+EoW0AcID8DFmqKxazsrIe6gz3IBUyuze1Fotl5cqVvr6+lMZodJIHpOhV9bHHHvvpp5+orob2Fm7OYrGMHDmS9zClWJaTk2O34xbCmTu42594/fr1Pj4+QojQ0NDo6GhN03r37r1r1y6pvAShhgoAjvRIk+tlnm0l739/D1AhUysZFovlo48+qlWrlsjtKUU39/Lyor4GRqNxwIABf/75pzpDCKMMbovi+IABA7g2ZjAYQkND+Qo2RdE9THAoLnepHS7efvttalhIkb1GjRqjR4/+888/pZRms9mukzAAQGHLz5ClXRn/oer5DxLI+BXwwoULCxYsoMYWpUuX5i4G3t7edCE6OvrNN9/kB0ClO6QxN2ez2V5++WWdTmc0GumY8fHxQQ5zT2o5ny/TMbB79+7GjRsLITw9PatXrx4dHd2iRYsPP/yQMj1nOEw3BADHyGeDVrpATa4LdsiSfqDZbN61a1efPn0SEhKEEDT2xINQ9DEkJOSHH37IuyRTYs6HG+Omwe3atatVq1ZISEhqauqYMWPsJpBhO0v3Qe8h1dVCNpuNLmzfvr1UqVJCiLi4OG9v75CQkHbt2n388cc3b96UuQGOXuUAAArbQwcynl1hs9mqVKkihEhJSSnYIUsp5U8//dSjR4/AwEDa90adoE0FM2oJSy+y9IrJg6d4O+vmuMdB3nSOIpm7yfuywHUyymQhISE8uq3T6fz8/Bo2bLh8+XL1ZQTBHQAcIJ+NYadOnUqvYtSL9SHuL5ddw0auur3xxhtt27blBMa1MRp74gWV33///a1bt5C9ACAfKJZRE//U1FTOZCVKlIiJiZk3b97Jkyd/++03dY6s7U6K+NcAABeSnwpZr169hNKL1WAwPPg7SLVCxq9o586dmzdvXseOHRs0aMDto3iJnNFo9PPz8/DwMBgMTZo02bBhg8QCKAB4NGazmV5GPvjgg6CgIHXn08TExGnTpq1Zs0ZKSc0y+MoSvYUBoHDkc8hSSmm1Wo1GI72zzPeQ5a1bt5YvX965c2chRExMTFJSEv1ASmNGo5FeJemLfn5+M2fO5LaN8vYZuwAAD8juPeSSJUto7bZOp9Pr9VFRUT179hw4cODOnTvtXtxsSh9/BDIAKED5bHtB82R5YPEh7i/XypUrhwwZ0rhx4yZNmgQFBdGgZNWqVUNCQqjfGC2A8vLyoppZQkLCqlWr6IfQpDGkMQDIH85SvNdIREQEvzpFRUUlJyeHh4enpaVdv379bgu3MbcMAApQ/rdOklLy7K58DFnGx8f7+vpGR0d36NChQoUKVA8LDw9v1qwZ1cPKli1LPz81NXXgwIHbt2+ne+fZZnh7CgD5Y7fyIysra8uWLdQqhd4BRkZG0tjlV199deLECXUxk8SLDwAUgvxUyHiiKw0sav/H3pnHVVV1///cewFBBhkEBUFwQgaVUUGRxAEURXIAAXNA1JzQRzSH7FFRRFMzHLMsy9SvPk7lWJLhUGapOaSmT5n5kGampYnKcIezf3+sH+u1OeeCiJd7Gdb7j/s6HM49Z59z99n7s9deey2l8jmuVw6wOECpVILzvpmZGZx82LBh+/btk7jW0sCUIIgXBEMqYgzYSZMmWVlZgXdseHi4r69vYGBgYmLi7t27Hz16xMo2PtQQEQRhWF7IQoZusM9xvVIwngWIMD7RjbOzc48ePaZOnfrll1/Ct/iGDwqg0WioNSQIosrws5b4540bN2BpkY2NDUxitm3bdubMmefOnYOJSwpfRxBENVFFHzLI1Au6SqVSVd6AzwsygYv4ivuXLl3K5yznPWop3S9BENUHzEjysQ9huNioUaNXXnklNze3qKhIcrypikoQRN2j6rks0amft5CBTpLEttbpdE+ePFmwYMGGDRsk8kvO/fv3MeY+n56SIAjCCKxatQqN92jLb9u2bXJy8po1a1hpK4fOrJJmilQaQRBVw8CCjJUufrx+/fqhQ4c2bty4aNGi6dOnjxo1qkOHDm+++SY/WakXXDuJHh4kyAiCMBrQ4HTs2FGpVKJXhrm5eVRUVFZWFh/RWuLbCl+HYLMmKz1BELUWQwoy1FIQSvHdd9+dOXNmampqXFzc8OHDU1NTz507h277lbkEQRCESfjpp5+sra1x3aUgCPb29gMGDDhw4AArzbcLR6Iso1aLIIgXwWCCDHP3giUfZi0fP358+/btK1eu3Lhx4/fff2eMgW1MpVJVcHLyDyMIwlRoNBpovv7zn/+MHj0anclCQkJiY2MzMjLkoWJhA4NiEARBVAFDWshEUQQ1ht4VrHQFAHrio2fGM8/POOM/NXMEQRgHGBAWFhZeunQpOzsb1i05OTmlpKSkpaWFhobCaqSgoCBIdgnAdynHJUEQVcbAFjLGGfN5KcZKJzSVSiVMAVTmQnhO8skgCMIIQDMFES60Wu2GDRsCAwNxGGlhYeHk5ATbffr0uX79OnyL/MYIgnhxDGkhgyQk+F9o2mC4if4Wz4wlq3d8SY0dQRBGAPPCYczYVatWeXp68guPIHLsgAEDzp8/zycOwSEoWcgIgqgCBhNkfBskyTIJbRYMOiszZclfiJ8OIAiCMA4Y9VCn0124cKFHjx6wPFylUmEMxbS0tNu3b0P7Rs0UQRAvyHMLMhhBQusjlEZ2ZVyk1oqt93ojZRAEQdRkCgoKBC6GIiQXUSgU06ZNO3v27IEDB3D1pXzRJe8Ua6ryEwRR86m6hUyj0YAaMzc3xz0SWSY33ZMgIwii1iGK4uzZs0GTYZI3kGXR0dHQ7t26dQsORida+JOMZwRBVIbnFkZoyV+yZAnGsEhLS8NIiRVHcyVBRhBEbeTWrVtBQUGgyWBxEnhfREVFMcbUavXkyZMlSUoA8rsgCKIyVGXKcvz48bxzK44Ut23bxrjWR68mI0FGEEStA1xgV65caWFhIXHwV6lUvXv3njJlSv/+/TMyMjIzMyUNIPrCkrM/QRAVUBVhhD77Go0GfSZwTSVvIZN49zMSZARB1E6Ki4uvX78+b948QRAaNGggcFngWrduHRgY2LNnz2PHjp05c4aV+pCRCCMIovJUxUKGkazlgVvVajXmeiMLGUEQdQMcW65Zs6ZHjx64nkmlUnl4eAwZMsTLy0sQhJSUlCVLlvzzzz/4RZqsJAiiklRdGBUVFcEGiDN+LFjBqiISZARB1EagWTt06NDUqVN585hSqQRx5ubmNnr06AYNGqxevRqbPlxuSfHJCIKomCoKI3Tt59cT8WNBnMqUXo8EGUEQtQ0UWBD/AuMpYiCMZs2aDR06tFOnTjY2Nu+9915BQQE2kngSEmQEQVSAsYURCTKiWuHHBuDjyCj+E2EI0D1j8ODBgiB06tRJpVKBLFOpVNiyWVtbZ2dn79q16+uvv2ZlPW5NWXqCIGo8JMiIuoZWq0UFpjcMAUE8L7y5y9fXF21j8An5eUGcCYLw0ksvLViwYO3atTdv3mScOwf5kxEEUQEkyIg6hU6nw/Sp0P9htkGCqDL8svFr16516tQJopGhGgPAsczMzMze3t7Pz2/dunV//fUXxSEjCKIykCAj6hTY82GWaEaWCcIQoNl17969lpaW0I716NHDysoKZi1BoqGdzMLCIjIyMicnJzc394cffqB5c4IgKoYEGVHXACkGG2AeI2dq4sURRfHevXtbt269d+/eV199Be3YG2+8MW3atE2bNnl7e2NKJZVK5evr27x5c4VC4efn99FHHz1+/JhGBQRBVAwJMqJOAf46oig+ffp09+7dEydODAwMHDdunKnLRdRudDod+CP27dt33LhxDg4OkJ5EEARPT88HDx506dIFgpPBzubNm7dp08bCwiIoKGjlypVffPHF0aNHaWBAEEQFkCAj6ho7duzg3XrAs8fUhSJqN6Clrl+/Pnfu3IsXL6alpYHwCg0N9fb2vnbtWlZW1vvvv//ee+8FBgbCrCV8RkdHjxo1yt/ff8aMGeDjTxAEoRcSZESdQqfTpaWl8d7WsG3qchG1G0yFtGbNml27dkGcC/Abi42NZYzNnj178uTJ+/fvx6xK8GljYxMSEmJubj5mzJgrV66Y+j4Igqi5kCAj6hqzZ89Gx2rsF1nZtPe0AJOoAjqd7vLlyxcvXvT29sZI/RMnTmSMabXay5cvl5SUXL58uUWLFjgeAMcyb2/vTp06JScnf/3111D3SkpKMHY/uZcRBMFIkBF1DJ1Od/ToUX7Vm3zKkl+ASRCVoaSkBNwTDxw48MEHH7Rr1w6MrwqFYvTo0XwC3+vXr7dq1QrsZCDIXF1dW7dunZiYGBcX98Ybb1y6dAkVGA0JCIJASJARdQpRFB88eIAzlRCGQKFQaLVa9PeHI2GPaUtL1CKg5rz55pvDhw/39PSUT4jrdLp79+79/vvvWVlZzs7OMK2pVCqtra3btWsXFBRkZ2fn6uo6ZcqUI0eO5OfnY45LqocEQTASZETdY/PmzZIpS76+oQ6jjpCoPFhbFi5cGBAQYGZmhnXM0dFRMgn+8OHD7du3h4SEYCW0tbVF9RYTE/POO+989dVXBQUFjOYrCYIohQQZUdcYM2YMTlmCIFOpVBhI2M4pAAAgAElEQVTBn3GpbAii8oB5dcOGDd7e3tCIKZVKlUrVokULFGQlJSUQHePkyZOjRo1q27atlZUVuJGBpVapVHbt2nXKlCnLly8/evTo/fv3GSX4IgiCMUaCjKh7nD59mo8IpVKp7OzsGGNarRYTPJMPGfG8gKDfsWNH3759oXaBxWvWrFmsNKsS70x28uTJ8ePHu7m5YT20srJyd3cPCAjw8/Pr2rXr9OnT8/LySI0RBAGQICPqGqIo4kwlKDM7O7uioiLGLbSkPDZEFdBqtdu2bZszZ47ApbD08/OTi3vYc+zYsaioKL422traOjk52djYeHl5DR48eMWKFXl5eThOIAiiPkOCjKhTiKL47rvvoj81zBapVKrvv/++uLiYldow+E+CqAw6ne7ixYtr1qzZuHGjs7Ozj49Ps2bNBEGwt7dnXF2SVKpz584NGzbMzc0NZy1tbW3d3d19fHyGDBkyZ86cRYsWgTMZQRD1HBJkRJ0CUtyACMNPpVJ56NAhTHAp2SCISvLll18uXrw4ISEBnPSbNGkiCMLo0aPhv7hYhP+TMfbhhx/27t0bFwEAKpUqNDTUz89vy5YtN27cwIBk+HWaUieI+gYJMqKuodFocJIIwl4IgnD69GnGzVTSEkviuQCP/hMnTmRmZnbs2BGXTAqCMGPGDD7mMByPPv6Msfz8/LVr10ZFRTVs2BDVmEKhaNmyZfv27TMyMt5///2LFy/CweB/RmqMIOohJMiIOsjs2bN5U4QgCN9++y0ra3WgPo94LjQajSiKR48eHT16NGaAEAQhKysL9T3IKa1WixFfIczKN998s379+u7duzs5OTVs2JC3likUCg8Pj2nTpn3zzTeMWxZgwjslCMIkkCAj6hSiKN64cWPu3LkqlQq7PaVSeerUKbRhyCeVCKKSfPrpp2PHjoVA/GDomjZtGivrlQgVDEQV7Pn999/PnDnTuXNnbACtra1x28HBITk5+eDBg/BFfsEmQRD1BxJkRJ1CFMUzZ874+PigD7UgCA0bNrx69Srj1BjNChHPBcij27dvL1y4cMCAAYGBgZiVa/DgwYwxrVbL537gY6yA/aygoGDEiBFgrzUzM3NycjIzM7OwsLC3t2/YsKGVlZW/v/+qVat42xitviSIegUJMqKu8ddffzVu3JifsrS1tf3xxx/5RM4oy0xcVqJW8cMPP6SmpkZFRXXp0gX8+gVBiI2NlVQtdCnjK5tGo9m5c+err7760ksvOTk5KRQKCwsL3hfN3Nw8MjJy3759OFogQUYQ9QoSZESdAgwMTZo0Qb9+hUJhb29/6dIlnELi01masqxE7QFE0v79+/38/BwcHOzt7V1cXBo2bGhmZvbqq6/KpxclggxnIX/99dc9e/YkJSV5enq6u7vD3GWLFi18fX3btWuXkJCwbds2jUZDSccJoh5Cgoyoa2i12tDQUN724O3tfefOHcYlskQbBkFUnl9//TU5OTkwMLBjx46+vr59+/Z1c3Nbs2YNKxX6kgxd/HcxOURxcfGXX345Z86cXr16NWjQwN3dvV27dh4eHhAzLzo6et++fYxRgi+CqHeQICPqFKIolpSUpKSk4CI4hULx2muvmbpcRO0GZFZ2dratrS3EUmnatClkT/r8888lUWEr0Pr4r+zs7I4dO9rZ2aEvmpubm7+/f9euXUeNGpWTk3PmzBk4kl/CiSH0qvNeCYIwDSTIiDqIRqM5f/78gQMHjh8/fvz4cVMXh6j1wJRl7969BUEA3y/4FATBzs4ODa6V8frSaDQ6ne7y5cvz5s1LSkqysrICTWZpaenq6tqyZcsuXboMGzZs06ZN9+7dQ3sbBN2o5CUIgqiNkCAj6hQ4KYleOBSRnzAIjx498vHxgRSW8Im5uRhjOp0OcnNVHHOY/29+fv6xY8fi4+P9/Pysra2bNm0KoWK7dOkSGxu7YcOGX3/9lTGm0WiwApOFjCDqMCTIiLoG9HkU0okwIKIoFhcXJyYmmpmZKRQK0GFAw4YNeZfEZ1Y2SayyFStWxMXF2djYYMw8pVJpYWExefLk8+fP8x5pFKuFIOo2JMiIugakoGGUQZwwHKIoqtXqb775ZvHixZApHMRT8+bNL1y4wEotsoWFhc88lUS63bt378MPP+zevXvjxo0xt5IgCL6+vuPHj8/Jydm5c2dRURFjDCxwuDiAIIg6Bgkyok6BIoyyVRIGR6fTaTSa+/fvjxkz5uDBg40aNdqzZw8rG9auYsEEVi604D59+lSr1R46dGjChAmxsbHdu3cPDAwMDAz09/d3cHCAUGfOzs6bNm0CwUdVmiDqMCTIiDqF3rgDFNWJeEFg9S5s661OYEKrzKkkmcivXLmyZ8+eN954Y+zYsX369GnZsiU4qEGIMpjBTExMPHHiBH7dMLdEEEQNgwQZUadACxl4j8FcD03xEC8IH4Uf9ujNwQWDgQoWkfBfx7NBYqXvvvtu2bJlEELP3NwcImtAwLPo6Og5c+acO3fumecnCKL2QoKMIAiiUqBxCySRAS2voijeunVr9+7dU6dOhQj+GFOjc+fOcXFxUVFR8+bNe/r0KcS/gALAqINm5wmibkCCjCAIorJgUBWDm6ngzFeuXFmzZk1SUhKuG1CpVD4+Pm3btu3SpUt2djYrNc7BFCrGPyMzMEHUdkiQEQRBPBvMhQqRXZlB3blwSvTixYvLli0bMWJEq1atsLW0srKysLBo0qRJQkJCbm6uVqsF4xwsIKh8TFqCIGoyJMgIgiCeAe8rhhuGnSgEwffo0aMjR45kZmZGR0cHBASYmZlhSlYHB4dx48bNnDnzzz//hK+gNISIGARB1GpIkBEEQTwDdB0DNaZWq3HRpUEAtzCMZnz9+vVt27YtWbLEz89PEAQMRRsZGdm1a9c333yTlV1AIIoizVoSRG2HBBlBEESlgPhh8rgqLw5MOJaUlKCuKi4uPnToUFRUlLm5uaura/Pmzc3NzR0cHJo3b96zZ8+dO3eiDxmv5AxVHoIgjA8JMoIgiGeDcgd0mGEtZKxs0lVwESspKVm2bFmPHj369u3bpUsXR0dHS0tLT0/PkJCQIUOG7Ny5E7/IKNgeQdR+SJARBEE8A4ytD2rszz///P777w04RYhqDJcOoPlt7dq1CQkJjRs3hpazefPm3t7eHh4eAQEBH330ERzDR8EgCKKWQoKMIAji2fAiafXq1ePGjXvy5Amv0vgQZQbkxo0bb7/9NjqTKZXKwMBADw8PyHf5999/8yFq+RwAvMmNIIiaDwkygiCIZ4AWMtA906dPVyqVhw8fZmVnDMFGZUDfMlg+WVhYePz48dTU1NatWwuC0KtXrxEjRnh4eNja2mZmZrKytjGMUibZIAiihkOCjCAIolJotdqvvvpq4MCBSqVSoVBAU7Z79274lySrkqFA89u2bdv69OmjUqnc3NwiIyO9vLwUCoWTk9O8efMePXrESlUjakdGk5gEUasgQUYQBPEMYO6vc+fOKpWKj6EPygx1j1arNfgsISyi1Gq1ubm5Y8eOjYyMhOzjWJKAgID58+fzE6ZarRZWa4Iyo4lLgqgVkCAjCIJ4Np988okgCCiGIDAYKDOwS2HECoNnlsSclfn5+V999VWPHj0gWiyEjbWysoqKilq0aBHGy4Bv0UwlQdQuSJARBEE8m8ePH3fq1Ik3TYGRzN/fn3ExY5lB4+bjfCUuKVCr1Z999llMTAwG8TczM3NwcBAEYc6cOfAtiFtbTRkFCIKoJkiQEQRBPANwzNq/f3/Tpk3Rewz0UExMDONET2FhocGvLpl51Gq1hw8f9vHxgZKArU6hUKhUqnnz5v3222+s1KhWHVOoBEFUEyTICIIgno0oih9//HGTJk1ABoEaUyqVM2bMYKWCDGSQAWO0ykNa4GdSUhLa6hQKBZTK0dExKyvrzp07jIvTQRYygqgVmFiQYUvBG/wJ4kXA0Jrwp1qtpqpFvCCiKB45csTV1dXS0hJkEOb8VqlUcAwfA6y60el0xcXFBw8efOWVV1q1auXk5ATlady4cURExJo1a8BQJ0mIzseeZbQGkyBqGCYTZHzQQkmrQRAvCMYgwIVmBPEiqNXqK1euxMTEuLm5oRQDxo4dK5lSNE6VKyws3L17d1xcnFKpxCWfIBCtrKxmzZp19+5dfpEBLL3EQho89RNBEC+IKS1k1ZGjl6jn8B4zGo0GczabrkREHaGkpCQnJyc8PBxCXYCdTKVSDRs2jHFzi8YpDM6KXrt2bcKECba2tjiFCqWyt7ffsmWLRqPBkQlukFcZQdRMTCbI+EbByG0ZUbfR6XQwA46r0kxdIqLWA8p+8+bNECufp0uXLij9jTy2fPr0KWPswoULI0aM4N3aYKNDhw7ffvst43IJYHAyeCmoySWIGoXJBNnt27f5/cZ0vyDqMP/880/btm0h5Z+Dg4O1tXXr1q0llY0gnhdooNasWQOKBxY2QjSyESNG8EfiSMA4RYKxx/vvv9+kSRMom4WFhbOzMzSz4eHhhw8f5tda4uwq2ckIoqZhMkF24cIF3nWsmvLyEvUNURQ3btyIgQCgi0pOTjZ1uYhaj06nW7duHdqfMPhF27ZtISiGMS398vnH1atXY2SyXr169enTx93dXalUJiUlPXz4kJVG7tBqtWQeI4iaickE2bVr15hsspKcyQiDgGGiwNGna9eupi4RUbuBQePatWvt7e1BikHIftjmjzRyI4Za8OzZs3FxcVDhra2tGzRoAMVzdHTMyMi4desWKzt9z6e8JAiiJmAyQQaDNkwJwsg8RhgC6DhnzpwJRjIwGPTp08fU5SLqAm+88QavxkDxxMXFMU7fGGdhL28b44MH7du3LyUlxcrKCiOTwdTqyJEjnzx5wsoGhSFBRhA1CpMJss2bN8MesooRBgSq09KlS7GmKZXK2NhYeWgV6o2IygMCKDs7W1EKmsfeeOMNxlhxcXFNiML6448/zpgxw9LSEtaBdu3atX379jA4CQ0NvX79Or/cUvIK4B5qkwnCJJhMkB05cgT2UAdJGBZRFMH5Gi1kUVFRJMiIFwE0yt69e1u2bCmUprMEZfbKK6+gV1ZNkDKnT59OSEiA+m9hYQHlhBdh2rRp9+/fZ6Vzl7z8wtehqKjIdGUniHqNyQTZ5cuXcSfZzwkDotFodu/e3bBhQ7Rh+Pr6mrpQRO0GFjOeOXNm8ODBoHIwxsScOXN4fyzTtmMQEvmXX34ZNWqUtbU1zK6CcFQqlZaWlqNHjz537hzjWl25Cy81xQRhEkwmyK5cuQJ7eBO6kQtD1FVOnjzp6emJbmROTk6SOkaVjXhetFrtZ599NnHiRG9vb1hrCSrn008/ZYwVFxfjYSYt5v/XWFeuXJk8eXLHjh3Rngeh/B0dHWGOlXH+ZMXFxTAnS00xQZgQkwmy7777DvZQE0AYELBVfPbZZy4uLhieQKlUYn8JUH0jqsCqVasGDBgQHBxsaWmJ9tf169fXNE95rVb76NGjrKysoKAg9Ou3tbUNDg52dHRMSEh49OgRHMmvuyQHMoIwLaYXZNCWUbB+woC8++67KpUK45AJglBQUAD/oppGVAEwOy1dujQ8PLxp06a8a39qaiofpt/kggZmVxlj+/fvHzp0aK9evaCcdnZ23bp1c3R0bNmy5dq1a/H4kpISOB4GLfRqEISpMJkgu3nzprxrpLaAeEFgBnzp0qWS6J3gy8xqkhmDqF3odLqNGzcOGjTI3d1dEARLS0uoY0OHDmWMYVYi0woyGN9iCsvLly8fP358/Pjx8Ba0bNnS0dFREIRmzZqtW7fu5MmT8C1I9grWZZMLSoKot5hMkBUVFcl9SKmbJF4QqEKrV692dnZWcNy5cwcO4C2yBFFJoMLs3Llz9uzZvXr1UqlU4JglCMLUqVNZqdA3SUZLCXy7CuU5depUWFgYOPh7eXlhSNvu3btv3LgRDqbw/QRhckwmyL7++mvYw9v5qTkgXhCdTvf06dMpU6a4ubnhrKVCofjjjz8YF9acUWUjnp/XXnutT58+MTExHh4eoaGhrVu3ViqVaWlpkgT2Jqxa8gkHrVZbUFAwb948W1tbMOmpVConJ6eYmJhFixbdv3//4cOHaFEDAWfydQkEUT8xvQ8Z9Y6EYdHpdPHx8dD3AAqF4tdff5XHvaTZGaLyPH36dNSoUe3atQsMDGzWrBms4RUEISIigpWKGFiuaMJC8uMNLAmUbcuWLT179rSzs1OpVK6urm5ubhYWFtOnTz99+jQrm0mJ3guCMAkmE2QXL17EncZJNkLUB6AvSU9Px7idUN+uXbum0+kksZeo1hGVp7CwMC4uzt7e3tnZWRCEVq1agSYzMzODA2qIUz/ClwT01nvvvQeFF7jYtoMGDdq+fTuj2XyCMDUmE2SXLl1iZB4jDA2ork8++cTOzg6CRUF9O3v2LJ/+j8YAxHMBtSUrKyssLAz84iFkv0KhaN++PU5Zok+9KctaCnjo81X91q1b4AAnCEJAQECzZs3i4uLatWsXHR2N38JxC0EQRsZkguz7779nFPmGqAZEUfzoo4/s7OxwlaVSqdy7d69kERzVOqLyQEv14MGDTz/9dMaMGW3atOEXjDNOx8CKRROWE0oiWSmFsuzixYvJycleXl7Ozs4WFhYBAQHwmsyYMePevXuM3guCMB0mE2R5eXm4k5oAwlBArxMfH4+TlcB7770H1guM0sTIeZl4HiDJI1i/Dhw4sGbNmrfeeuunn35ijKnVap1Oh+FhTW581Wvlgj1qtfrMmTPTpk1LTEz09/ePjY0NCwtr06ZNz549ly5dCkfSe0EQJsFkgmzPnj24k8JeEIYCep3IyEhcYgls3LgRTQXYV1HHQ1QSFFu8o5gkyEVNiEOG6E1Midtr167t1KmTvb09xMJo27attbW1q6srZk8iCML4mEyQbd68GfbwLg4kyIgXR6PRTJkyxdHRkY8Nu3fvXsY591CvQzwXEqdDuQmKd080asnKIi8Glpz32T9+/Hj37t0jIiLAn8zNzS04OLh169bXrl0zUcEJgjCdIPvss88Y+ZAR1UBJScngwYNBjSmVSgsLC0EQrl69ymSdZQ1xviaI6oYP9IjKbP78+SDILCwsunXrlpiYOG7cuPfff//u3bus9O3A9QrUShNEdWMyQXbw4EF59AGykBEvjlarnThxIugwCEquVCr1CjKCqD/wxjMQW/n5+RMnTmzSpAmMXjw8PJKSkqZPn378+HFJvG70vzRd8Qmi7mMyQbZ//370fiVBRhiQ4uLinj17ghpTKBRgA5DMxVBNI+obkG0TDV1g+lKr1YsWLcLXJCYmplu3bgMGDFi3bt3du3d5JzlSYwRR3ZhMkO3bt09iIaM+kjAUffv2VSgU6EOmUCiuX78uqWw1YTUcQRgHycoDxphWq4UVxwUFBTNmzHBxcfH09AwNDX3ppZf8/Pxatmw5fPjwCxcuoJs/ze8TRHVjSkGGO0mQEYZl5syZMGWpVCotLS0FQfj9999r1CI4gjAmvPcYH/kFePToUU5OTkhIiCAI0dHRL7/8crdu3bp27bpy5Uo4oKSkhGIpE0R1YzJBBqveABJkhKEAu9fo0aPBewyWWHp6ej59+hQPMG0JCcIkSEJawAwm+IeJovj48eNx48ZBsJhu3bpNnTp1yJAhGRkZP//8M9nGCMI4mEyQffLJJ7iTBBlhWHJycpKTk7t169asWbP33nvv+PHjkgMkKWUIom6DtR3j+MtHJpcvX540aZKFhYWHh8dLL700YMCAfv36TZ8+/fbt26xsAD+CIKoDkwmy3bt3S7pD6h0Jg8Dre/Bchh6In6mhroWob/BOk5hzU5JqSa1Wb9y4MSIiwt7ePigoKCQkxM3NLSMj48mTJ6YsOkHUD4wtyGAKSaFQgA+ZJLEgdZMEQdRA+OEiRu1nZVeI19LV4vL0A6+//rogCA0aNPD29u7YseO//vWvI0eOXLhwAY6XD6Sp3SYIg2AMQcY3VbDwTRCE7du3y80V9GITBFED4Z3iGecdzxjDLJa1ceku3+TijZw+fXrEiBFt27Zt0aJFRETEtGnTPvvss23btv3111+86KTVlwRhWIwkyLCdwgyDW7Zs4f+L20YoD0EQxPOCjZhWqwUVghFWgdrolShJ8IqK848//pgwYYKXl5eHh0fPnj2TkpISEhJWr17966+/Mk7GFRcX4ydBEC+I8SxkKMjAQrZ161bGmf1rY1tGEEQ9AR0QMZUQbjDZtF3tsvTj6ks+kZ1Wq/3mm29iYmLMzc3NzMysra29vb0jIyPnz5//008/McZKSkoYmccIwqBUuyCTyCz0Idu4cSMjQUYQRO0BLWS80xUqM9jPC7WaDz/HiuoKN/bt2+fj4wNDaDc3N19f3+Tk5I0bN169epUCxhKEwTGqIBNFEd5thULx9ttv435KykEQRE0GZEdxcfGFCxdOnDhx7ty5JUuWJCYmTp8+HVYgQqA7UGNgPaoV8G0vH6gMV1++9tprtra2giCEh4cHBAQMHDhw/vz5q1ev3rFjx9OnT8mjnyAMiDEEGa/J0EK2dOlS3EmvNEEQNRloxJYuXRoSEtK8eXM3NzdBEMAB44MPPkBPLNQxpi5vZeGDk7FSfcbPXf7222+TJ0/28PDo3Lmzv79/7969Fy1alJWVNXPmzO3bt5P3GEEYENOsskQLGd9ykZ2MIIgai06nW7RoUevWrS0tLUGQAZs3b+abuFqkxgBehElyK8G9/Pnnn7Nnz3Z3d3d2do6MjMzMzMzJycnIyFi8ePE333xD3iYEYSiMF4cMXloMezFz5kxWdijGamFbRhBEfQBaqpkzZ3bu3BnSpDZt2hTGlhs2bJCn7q4DYBQMnU6XlpbWq1evwMBAiEwWGxubkpIyY8aMVatWSWxs/Jp6GmMTxHNh1MCwvA/Z1KlT+f2sbrVlBEHUPf71r3/5+fnBqLJt27bQmr355pusbPNVxwaWoMz+85//TJgwwd/f38PDIyQkJD4+fvz48a+99trx48cfP37MGNNoNBgNhI/+TxBEJTHqlCUKMqVSOXnyZIkOo7eXIIiaCRh7ZsyYERUV1bBhQ6VSqVQqLSwsbG1tjx49ipHJJOH7ay8ajQbuBRcoaDSab775JjEx0cfHJzg4uHfv3v369UtJSRkyZMiiRYv++9//wmEUVJIgqoxpfMiUSuWsWbMk/6UXmCCImglYiTIzM0ePHu3r69ukSRMnJycnJ6dmzZpBXC48zISFNCzYIJeUlOB9nThxYu7cuQMGDPD39w8JCQkJCQkPD4+Li3v//fcfPnyIX6QmnSCqgLF9yCwsLGBp0pIlS5hsspJeYIIgaiZarTYrK2vs2LEhISGurq79+vVzc3NTqVQ5OTkwSVf3InJptVoI5MGnirp3797ixYv9/f0bNWrUrFmzwYMHp6amTp8+fdeuXb/88gvMVFJLThBVwNiCrEGDBmZmZoIgrFq1iskEWV0aXxIEUWeA+A5Lly6dO3du3759fXx8fHx8IIjP3Llz+fFknZEjvEsc+ofBc/jxxx9TU1ObN2/u4uIyaNCgV155JT09ffLkyZs2bfrtt9/gW+TRTxDPi7EFmZmZGcxa5uTk8PvJQkYQRA1nwYIF2dnZAwcObN++vZOTE6xPmjt3LtrG6owPGXrlyyMTQfzYU6dODRs2LCAgwN3dvU+fPklJSb169ZowYcK3337LKLslQVQJowoyURQxMCwsTWJ1qAkjCKIOU1BQMHPmzPT09KCgIEEQYmNjwfsiPj6elR1S1j3jkDwiv06n27JlS3x8vCAI3bt3j4uL8/LyatWq1eLFi3Nzcz///HNq0gnieTGSIMPl0BgYds2aNYy8xwiCqA2Axlq6dGlcXFxISAikEurfv79CoUhKSsIDoKGre4JMAtzg1atXR4wY0bdv3w4dOjRp0qRNmzZdunQZPXr06tWrjx49+vnnnxcVFcHxGo1GkkPPNOUmiJqN8VZZwktoaWkJkS+WLl0qd7mgF5UgiJqJTqdbvnx5amrqwIEDmzVr5urq2q5dO0EQmjdvzhvG6kkjBnOaly5dGjZsWIsWLWxsbCIjI1NTU9PS0oYOHTpmzJg33ngjLy8PtSmkXcdgswRByDGGIANbN3za2NjArOXChQvhv7xcqydtGUEQtQ5RFNesWZOenj5+/PiYmBgvLy/064cD6sOaJMm6S8bYnTt3Vq9eHRcX17179/bt2zdt2jQwMPDVV1/NzMxcv3791atX5YtP695yVIIwCEaykLHS1srKygpmLRcsWMDKJk2qD80ZQRC1EWjEtmzZkpGRMX78+NjY2LCwMF9fXxRkfPNVt5syyHQpicW/du3arl27urm5BQQEDBgwYObMmZMmTRoxYsSSJUuuX7+OY3IyjxFEBVS7IEM1hrksQZDNmDGDkSAjCKKWIIrinj17JkyY0L9//+bNm3fr1q158+aCIDg6OtYfAz8E7uc/QWbl5eXFxcX17dt3+vTpkyZNioiI6Nevn5+fX6dOnVauXPn777+jFJP4kxEEgRhJkOHbCGH6BUEYNWoU70NGryhBEDUZURT37ds3ZMiQ9u3bC4IQGRnp6OgoCML48eMlThd1e2zJu8phzii1Wv3hhx+6ublZWlr6+/t37do1JSUlNDQ0LCxs0qRJX3755V9//VVUVARu/nX7+RBElTHSKktssGCJpUKhGDJkCAa8oPeTIIiaDLRRBw4ciIuL8/T0FAShdevWZmZmCoXi1KlTrGws+7o9MQd3B4YxXoDqdLqkpCSYw42Ojg4LC4uIiOjVq1dSUtKiRYu++OKLCxcunD17llp7gigPYzv1Yxyy5ORkPgIZGckIgqjJiKK4c+fOmJiYZs2aCYIAGUcEQbhz5w7fxNX5dkyvFRDH1ZmZmQ0bNoQIbUql0s3NrU2bNgkJCevXrz99+vSZM2cwWzlBEBKM59T//+lKQPAAACAASURBVK8nCPCuJiUlSSLT1O1hJUEQtRcQHJ9//vmKFSs6duwIUkyhUDRq1OjIkSPYlEGE+nq4ihCfgEaj2bJli7e3N7TzgiB07do1LS0tNTV148aN8i+S9zBBIEaykOGgCt9SX19fRnmTCIKoJYiiuHbtWj8/P4VC0aBBA3SHvXDhgiiKGo0GhpQQFaIeUlJSAtJKq9UeO3YsMTERmnpnZ+fu3btHRkb269cvMzPzl19++eeff3SlwHfJcYUgmHGc+lFv6XQ6oRQzMzNWavcm2xhBEDWfCxcuDB8+HNONCILQuHHjR48e4WASRUl9Ax3LYEZSo9Hs3LkTnO0EQYiIiIiOjh44cOCwYcMOHjwIX1Gr1Y8fPwb9SmNygmBGW2WJ202bNgXfi0aNGuF+9L2gF5IgiBoIBmtAT3ZWmmablcoR9I6qh5qMzxkFt3/nzp0pU6YIgmBpadmlS5dhw4alpaVlZGSkp6d/9913kuzj9SrJAUGUh1GnLLVabXBwMAiysLAwiScsH2aQIAiiRgGZf2DcyHumwza0XfXQewwBP2CYvQW716NHjxYvXty/f/+AgAAnJyc7Ozs3NzdBELy9vZcuXXru3Ln79+8/fPiQ1BhBAEZ16tdqtfb29uDXHx0dLVkiTi8kQRA1Exwu8mYwvskCucbqcTvG2wW1Wi0+qI8++ig2NhYmec3NzaH9t7OzGzt27DvvvHPw4ME///yT1ePnRhCIkeKQsdL2C9wvVCqVt7c3jDXroXmfIIhaCk5T6jgk4bjqoVMsyinMqsRKE4ozxk6dOjVp0iQMFAK0bt160KBBq1at+vnnnyU+/gRRPzGeDxk0UkFBQbBACRPA8W8yDZIIgqiBQNOEKyjLc3uth1IMgelaidMwUlxcnJSUhPHbIKZut27dNm/efPv2bZq1JAhmBEEGLRe+q/7+/rA6qVOnTtic0cCIMBT8kl7GmEaj2bRpU25urqnLRRD1EVxEWVBQEB0djXEolUqlt7d3r169EhMT/+///u/x48eMsadPn8L0JSsbZZeEGlFPMF7qJMaYTqcLDg4GZwJPT09WNiiGcUpC1HmgKV+5cqVCoYDW38XFxdSFIoh6Ci50OHz4cEJCAriRgTLz9PTs2bPniBEjVqxYcfHixfz8fMalnyL3YqK+YTwfMtBe/v7+MDxq2LAh7EdjNb11xIsD4Qny8vKg3VcqlWCRNXW5CKI+gqIKp0Fef/311q1bgzNx8+bNo6Oj4+Pj4+Pj58+fn5eXJ1npVZ+ngIl6iPFWWcIL2bJlS7CQ2dnZYSwM/jCCeEGePn06ceJEGxsbzG8THR1t6kIRRH2EdxHG9RDvvPNOQECAjY2Nu7t7QEBAdHT04MGDExISli9ffvny5QcPHty5c+fhw4c6nU7ul0YQdRijRupnjAUGBoLFonXr1hLHWHrriBcHYkQlJiZaWloqFAqobB07djR1uQiinsIvO4U2v6ioaNmyZT4+PmAns7Oz69Sp00svvTR06NDMzMw9e/b8/PPPGDmWjGRE/cFIggyNZD179oQ+skOHDmjEJqd+wlCIolhUVJSYmGhnZ4cL7ENCQkxdLoKop0DzrlarId0n9AUFBQWzZs1ycnKCRZc2NjYtW7YMDAzs16/fnDlzvvvuO8ZFzSCIeoKRwl5gWJpOnTrBLFLLli0lC+IYGckIQ/DHH3+Eh4d7eHiAhczMzKxfv36mLhRB1FN4C5nEVLZ169b4+HhPT0+IheTu7t69e/fRo0d/8MEHv//+O9nGiPqGkQQZDoxcXFxAkDk4OMABEIqsnge5JgzIH3/84eXlhemfFQqFv7+/qQtFEPUXiNrPr9/CqODff//90KFDoV9wdXWNj48fPnz4tGnTPv744//+978PHz7Mz8+vzwmpiHqF8Zz64ROc+mHtG+brpSWWhAHR6XTDhw8H9xQQZHFxcaYuFFG7wQYK3cwljhZ8Zl7ywagM+AwvXLjQq1cvhULh6urq5ubWokWLIUOGZGVl7dix46+//uJ7B36+hboMou5h1Dhkoih6e3vDRJKNjQ0r+4IRhEHQaDRJSUlgHgNZ5uXlZepCEbUbnHe7f//+33///fjx46KiosePH9vb269YsYJvx9RqNWmFSoL9wsqVK7t16+bn5wfxyYKCghITExcuXHjy5EnG2NOnT1mpFCZrGVGHMZIgg6ZKq9W2atUK+kgrKysmG+uQMiNeHFEUJ0+e7ODgoFAooLI1bdrU1IUi6gKvvfaak5MTWF4dHR3B/mpjYwNajc/haOqS1nQkcyNPnjzZs2dPdHQ0vLOYXqldu3bz589fuHDh/v37GfdgcXaFIOoSRrWQabXaxo0b41wSOpbhMdSQES8I1KicnJwWLVqAkczc3JymLAmDMHHiRE9PT3Nzc3RPFAShXbt2rNRyg5mCTFzQ2gA8K1SxarX6/PnzQUFBaNiGx5uSkpKYmLhgwYKHDx/CF8EjjZHwJeocxk6dBPHTYdaSN4xRE0YYkJycHGdnZ9D9lpaWkydPNnWJiNoNZPLJysoKDg62tLRUKpVKpTIiIkIQhODgYF0poihSpIbKwNu6+J3vv//+wIEDvb29QZM1adKkd+/ewcHBPXv2XLFixY8//lhYWMgoOBlRRzGeUz/AJ7TRewBBvCBarXb27NlYzZRKJYW9IF6ckpKSjIwMOzs7lUoFcsHa2loQBG9vbzwG5AVZbioJ6CqIAQu9wJ07d5YtWxYZGSkIgoWFBSYjB+E7a9as3Nxc8jwm6irGFmRNmjRBi7TEMEZrLQmDoNPpUJBBTWvRooWpC0XUbkBprV+/vl27dpCVS1nK8OHDweUfgzuQ/eaZQBolnU5XVFSEzT48tzt37mzdurVPnz4wZalSqRQKRaNGjQICAkJCQjp06DB79uzjx48zGskTdQ4jCTJ8c3x9fdE5gD+A0dtFGAIYN7/11lu2trYwsFapVK6urqYuF1EXSE9Ph0oFXufAJ598gouT+CCoRAXgMi9W1v0O9v/6669bt27t27cvxCeD9ErQcYDZzNzc/MKFCya9A4IwPNUuyPgXTxRFfMF4QUYQhgJkfe/evV1dXXGyw8HBAXIb88cQxHNRUlKSk5Pz8ssvg3kMNJlCoXjrrbdYqckHjiRN9uIUFxcvWbIkPDwcB/AIhPVXKBRRUVHwLmOoEclCMR5aNEbUfIw6ZanT6VxdXfHtMsKlifrJrFmzsClXKpUwZYnruRi1zsRzArVl9erVw4YNQylmZmZmbm6+c+dOJksKRFQNPnHLxYsX161bFxgY2KRJE+g1eMMkyOLs7GycgalAjQE0EiNqOMZQRRibRxRF/nUywqWJ+gZUtqFDhyqVSnS+dnd3RxstP7tk4rIStQdQ8/PmzUtISEDLK6wa2bt3L9/EMdJkLwD/AOGdVavVEyZMaNOmDTiTYcBn/BW8vLxEUSwuLkZZxtvCSYQRtQhj5LLEd0wUxUaNGpEgI6oPnU73999/h4aGoq8PeF7/8ccffD00dTGJWoZOp/vpp59iY2M7dOjArxcRBGHTpk24uJJXEkQVwJcUpoDhSV69evX999+fPHkyPnMwlYFEEwRh1KhReAaMUsa4vJnGvxGCqAJGSi6OQ8YBAwbQlCVRfUBNGzJkCB9gRRCEH3/8URK0k1pq4rnYv3+/l5eXi4sL2GlAHKhUquXLlzMunhapsRcE3D35jkOj0Tx58mTo0KGenp7u7u7e3t7Jycm+vr4w7lKpVOPGjbt69WphYSE4k/Gh4EiTEbUIozr1M8ZGjhxJFjKiWtFoNCkpKd26dYNJjQYNGjg6Ol6/fl2SpZjmlYjnYseOHV5eXq1atXJwcMBGTKFQQNhhmiYzCLj4BgUuvqdbtmz597//bW9vj+YxpVIZGhoaGhrat2/f/v37Z2Rk7NixQ+856RchagXGEGTwPsB7FRMTQ4KMqFZKSkpwHRYYyZo1a/brr7/iATRoJp4XnU6Xm5vboUOHwMBAZ2dnPnX9yJEjJU7llAC7yoAaKyoqYqXGbNjGjU8//TQ8PNzR0ZFfejlkyJApU6bk5OTcvHnz6tWrjPz3idqJ8Zz64Q2JjY1FPwAjXJqob0BfKAgCTFmCkczFxeXSpUs41KbwBEQVOHfuXHBwsJ+fH8h9nLKcMmUKHkNC/8WRy1ne7xOe8OjRozGCEjonCIIQHBw8bty4jz/+mFxFidqIUcNeiKI4b948fHmMcGmifoLmMRRk//vf//i0NtRxEs/L2rVrfX19R44c6eXlhfEXFArFkSNHWFmTDM2GGxw+dhJj7MmTJxs2bGjdujUfzV8QhIiIiMDAwKioqOzs7IKCApwAhexMTGYdp3aAqFEYO3XShAkTBFmkfoIwFNAEgw7DSY1x48ZBTmImW2VCEJUkNzd34cKF8+bNgxSWmCn18OHD/JQldfDVBHq/MMZEUbx+/fqWLVt69OjRpEkTSMvRoEEDPz8/WHVha2s7b968R48eMW7dJSYD4F9/kQsZTRCmxdiCrE+fPiTIiOoDzGDFxcX8xAcfD5ZxHv0ky4jKc+DAAV9fXwyFBVYZZ2fnU6dOwQHyOXHCgMjd/LVabV5e3siRI52dnWHiEszhELM3ODh4w4YNV65cwRyjfE8EOkyj0ZCAJmoOxlNFUO/9/f0p7AVRfYiiiFOTxcXFfKxOHBnz4WFNW1qiFrFr1y4MNQwx+gVBcHZ2/uqrrzDUAoZpMHVh6xr8smiJiWvlypURERFgFIdfB7oYZ2fnbt26TZ069dSpUxqNBqxlTPbWk4WMqDkYW5B5eHiQICOqFWheJY2sxGuEpBjxvBw/fnzChAljxoxp2LAhupObmZlt3bqV6cuQTVQH/LPFOeIbN24sWLCgcePG4NuHs8mCIHh6eo4ZM+aLL76Ijo7et28fvv7k9U/UQIwtyLy8vEiQEdUKBIfEZM8SZcZPX1KLTFQSURS//vrrhISEuLg4hUJhYWGBxpiVK1fyh5Eaqw7Q2UDyzqLvwd9//52SkiIIgoWFBYbDgG2VSuXo6Ojh4fHxxx/zJ+TFmZFvhyD0YiRVhK9T+/btSZAR1Qcsp8JJDbRboEcwv1yLGmKi8vz000/Jyck+Pj7Y36tUKpVKBUEW+AlxU5e0DiLxAMOsSqz0HRdFccOGDcOHD1+0aJGrqysmgMcQsmFhYZs2bcIz0KpYogZibEEWFhZGgoyoVtApBMa+mEeFZi2JKoOeiKdOnTI3N8dclmZmZujUz7iGzmQFrdPIX1tUY7hn1apV3bt3x6Ak0N14e3tbWlqmpKRs2LAhLy/v7t27cDCfiZwgTI5RBRljLC0tDdMLGufSBEEQRJ1EElyQMaZWq69cuTJo0CDoZTCngrm5eWho6ODBgydOnHj06FF+nbXEtEnuZYSpMF7YC/gcOHAghb0gCIIgXhyJroINtVp98ODBsLAwgQPml0GcOTg4zJw58+zZs5iXSW4no3lMwvgYQxVBXYf63atXL3xDjHBpgiAIoq4iCSiIS3kKCwtfe+013ocM+x3YbtOmzdKlS2/dunXt2jXGWQ00Gg2lViNMhbEF2csvv0yCjCAIgjAUfCwxXL5z4cKFyZMnN2vWDJdbmpmZoZFMqVQGBATMnTv3448/zs/PZ2XXX9P6DMIkGDtS/+TJk+GVIEFGEARBGAR+3SXuYYxdu3Zt0qRJNjY2aCSDMHLg7+/m5paenn7mzBn4CizHZhQThzARRlVFoiguX76cBBlBEARhEDD9EVi2+HA2sML65s2b//73v/n5Slwha2lpaWFhkZ6e/sMPP8D6AMyyQJqMMD7VropwuThsv/feezhSqe5LEwRBEHUbnU4nz1WFmgwSWf72229+fn42NjboUoaaDD5Hjx6dl5cHUQwJwlQYSZBhWL8PPviAfMgIgiCIF0cixcCBDG1m8Cd8/u9///v888/T09Mx8xVGxFQqlXZ2dqmpqbm5uQ8fPkQfMnLqJ4yMsX3IPvjgAwoMS1QrkgzilOmZMAjQPaMRhQ9bxSgkbC1BFMXLly+npKRgOEwwkkGw3/j4+KysrM2bN+NCNMkqTh5JrA1Gvz7xwhjDQsa3XDk5OWQhI6oPdATRG/KRIKoMn6tHbxofSS9O1Fh++umnESNGoDczbgiC0LNnz5EjR549e5ZvOjDzByv1S9ObVZN+d+IFMWqkfp1Ot3DhQvSpNM6liXoFBCKCllFuzCCIKsOv40Ptxefj4mUZUTPBXElqtTo9PV2lUnl6evL+/h4eHhYWFoMHD16xYsWDBw9Y2UEdbGM4WVaqwzD+GUG8CMZ26idBRlQ3GK2bkRQjDARmRC0uLkbPJMaYKIrgNo5Q4p0ajlarBTt6fn7+7NmzY2Nj0avMwsICwmFAD5Wdnf3NN99s2rRpx44dubm5Fy5c0Gg0UBNIfhHVgZGmLMGpXxTF9evX05QlUa3A+BXXkVDTSRiE4uJi0F5Yo3BmHDMqom4jaiaFhYWwAcK6uLg4JycnOjoaRBgmvoTPhg0btmzZ0sfHZ8WKFdeuXTt37tzNmzfXrVsH1QBWD8DZyIOQMAjGm7KEynrs2DESZES1wjeL1EQSBgRnrKDPjoqKOnnypGRFntxmRtQoYMCGqy91Ot0PP/zQv3//3r17gz+ZmZkZ5lxSKBRWVlYxMTGLFy/eu3fvd999l5KScujQIcmcNZlFCYNgPFUE9fXPP/8EmzAJMqI64E0UtHadMCAwquzWrRsOKaEpmzVrFq4g0Wq1FMuqJoMOpmDcQhV17NixpUuXdujQARMrmZubC4JgbW3dqFEjpVJpa2s7a9as3NzcrKys99577/z589DUSFZ4EMSLUO2qCL0dNRoNdI1kISOqlYKCggULFowZM2b79u2iKI4fP/7LL780daGIWs/BgwdBjYGnEbZjQUFBTJZO0aQlJZ6BJGIFmrs2btyoVCqjo6Nx9SX+0G3atFmyZMmOHTtmzZo1duzYdevW3blzR6vV3r1798GDBxRbhzAIxlBFfACCdevWUaR+ovqA0SouZUe/EFbWz4PCYRDPy/fffw8xqxQKhZmZGVYtCwuL+/fvs7INHVHr0Ol0a9euDQ8Pt7e3d3BwgB8XPs3NzWFj7NixN2/efPvtt7Ozs99+++309PS33nrr2rVr165du379uk6nKygoePLkCZ6QcSEz9LY2Ev8KXCzCylYkmgSvJxhplSUOH3fs2EEWMqJauXv3Lq5j59W/JDIZqTGi8kBtCQwMlAevio+Pl/h3kyardeA0TkFBwbvvvjtmzJjAwED0rkFPf9DfgiA0aNCgV69eMTEx3bt3nzBhQmZmZk5Ozl9//fXPP//cvHmTlfpO6PWaEEuBP+G/ILmgFqnVaox5Jol4Z4xnQZgOo6ZOEkXx0qVLJMiI6kOtVo8ZM0YoDcANTSdvIeMHoKTJiEqi0+nu379vZ2cHNQrNYyqVauLEiZLOlah1SHJfMsZmzZplbW2NvRU4+8M2bEBOTEEQvLy81q5de+TIkYsXLy5fvjw9Pf3rr7+G5Zw4COTj1fFqjNfxxcXFvEca5ugkHVZ/MF7YC/jz77//JkFGVCtz587F6QbwzBUEARtBPoIUtXRE5bl165ajoyMf1R3648jISN6eQZqsloLrgWAZZn5+/rBhwxo2bMj7C/JeZSDLoA6kp6fn5eXt2rUrOTk5Pj4+Ly/vwYMHN2/ehBUeGo0GIqRI1BjjTPXYFknqDyo2GkDWB4znQwaQhYyoVrRaLSSw50MKeXp6Mn1tH0FUHp1OFxUV1apVK+iVsWOGJSNoWSFBVhtBwYS51xhjBQUF2dnZ0IygPQwbFj6ErCAILi4u48ePX758+fbt22/dunXq1KlVq1Z98cUXGLsOB4S8tQw2JAs/YawIFQl2UiaAeoLxkovDrOXWrVtJkBHVSn5+PgxeYbJSEISwsDCJMwe1bsTzUlRUFB4e3qNHD6hUqMm2bNnCSts3MmPUanhJhEsvb9y4kZmZGRERAdoLTaT8tDVue3h4pKen5+fn3759e9++fZcuXfrzzz/1msSQkpIS3KNWq8+cOXPt2rUDBw48evQI1SGNJOsJxlNFUL83bdpEgoyoPnQ63dChQ7HdBGXWpEmT27dvs7JDUpqyJCqPWq2+e/du586dExMT+V5ZpVLt2rULZqZovrL2Anoa01wyfcqpXbt28IujiRRDyPICXaVSzZgx4/Dhw/v37//hhx+++uqr//3vfxIfMtjmzWZqtRqdxsaNGweCLygoiJz66xXGc+qHP/Py8rAGV/eliXoIzilAl4mV7eTJk1AVebdZMmYQlefSpUtt2rTx8fHh+2BBEL7++mvoZcvryIlaBG+LkswzLlu2zMbGBn53NInxrhE4g6lQKOzt7d3d3adPnw4BOOVaSqvVFhUVQcAUXsfrdDqsYCqVipWd0CTqNsZOLp6fn4+VmHE9Irg9wrbel4Fx4wPehIs79Q5BJCXRO6HAvyoYMwa3MRwzfxL+0pIRDysN2C0ZBqEZHE7LX+iZz43J3kbUFpKTSKRGxefnD5Y/RvldMJljqWE7HkkFwBJKrA7yJJX8zfJtGWyYm5unpaWxsslt5F/EA/A3Kg9J8SoIacbv1Gq1UJfKS3coykIQSZZcsbKGPb7e4pGST72eK/iccY09FkDyQOQbkmeoNzyS3ovyN6i3a0GnGSyP5Ab5YshrPl82vTVEvtJNcnXJU8Uz8D8BTIJjrkOoYCNHjuTLXzF8bcFJMX4/bvDTZ6xs9aj4vZMkWJRUKkmNkvxAyDNvpF6Bv+xff/21bt26uXPndu3alW9kJG0O1BP4MyIiIiMj46effmKMff/993v37s3Nzd26deuRI0e+//57PvnSL7/8Mm3aNFZ2VFl97S3Bw7cesMEHLpG8U/y2YdPXGi+XJXyeP3+eF2QA3hI/ksBWle8D9DblYmnyclZWMEHLAnZg/iSibGZB0t/wzaLewwBdKUxfZwOl1SuMxEpMlkn0HGxLulI8Uq1Ww6OTdyd6zyzpsCX9rqRr4U8LVKuHqVwP6RVefCH5/QIX8ALmFARBaNWq1e+//w73y4f2wQ3+3auY8tQGK32eCP+rYb3iNa4crLGS4knuER8Uk6lVvCJ/j7xAkdQcya0xrnLyj1f+vmBebcbVRvkwA+9IMtThO35JqeBxyV8oeZsoF6ZyTab3afOvAF+d8MbllxZF8fHjx5MmTUKnC6xdMTExrOzPUXEbzYstiRqTFK+8PyswlvA/Cq/t+IcmyfajtyZXUP76hqT5hUHI+vXrO3fu7Onp2bJlS4gSLKkVoMnQitalS5fVq1dv3Lhx7969n3/++eHDh/mfG2LJ4ssCa8PhE47BOik+JxXc1HMdX7eRLLAVuXS0fPfHNzV83DgDPjfjOfVDF/Xll19CBQXtj00Dr5kkEkpvg4uLYoqLiyUtr1iOJwff7vOmLDitpJHi+w+9xiF598A37vIN/BM0QWUemuTMfNuNHg96laV8uC85Bh6FPAub/E++L5f/Igb0mNFrvsK74CWOTmal4x8yWsisrKysrKxglOnu7p6fn48nh2uh3w9/R7pn+ZbJf27cWV6LJnLCV3JmvV+RBB9C0xorK2VwW6IR5YXhL8ofgG8Zr+Ylz5M/XlsKXxhdWUsq09c84UmgyvHn5IcE8hEF44SFXJfzLz7/X2wB9JZE/gPx/+XtUvgnvnRxcXHg34PrRRQKhZeXF56hYsEN8LcJD0QySOBro+TH1VVu0QAegx253iqt44z3zzxnvYWvvbx0LiwsvHnz5q5du0JCQhQcqKX4DTCpglbr0qVLamrqsGHDVq5ceejQoU2bNmVmZs6ePXvixImDBg0KDQ1Fe5u5uTkj17HqBNsKbNlKSkpAn7Rp0waXuOJ/+TcLX3YDGsmMF/YCSn/8+HEcN/DDUPgvNE/ykTe8CZJek5U/pmRlW17+vxIRI2nNGTcLw69/xiPx6xKNhcdIOhWdvokSvQpPjiiz/In6THf8+SUXkv+rgj8ln/wxFbTpBkTklDTfsUkKI3/g/F1g88evSFepVBCnUSxro8KbVavVvDmzgt9F8pT4n1JeTv5PXlRJ7LUVfEuuQuRFkmgR+cMRS2WTlltIr/e+JLJY7+8ucjBZHcNXQJTBKqzzurImT97ghN/i2wpJ+4g3KJE7fNn4xyt5rfiSwLckOcLhcjExMa+//jr0stCIderU6cSJE5Kp24oFPd/usbKjGr6Jk9gy5VJVL/wzlLeTkiZUb92r+Pz1E71DXHzUH330Udu2bYOCgjw9PdE8L8iyt/HKDD5dXFx40QbfbdOmDU59KhQKUV+HSBgKbCGxPRFFEaKcCKXrZ8FsxmSNM2zjfw2C8SxkcDPXrl3DbpLvfSVzFnqnbPnRPOO6H/kzYrJnx++Br6MHLv9L6Mpa0fAM/EX5E/IdhvxyeB5Jlym5kF70Gp/0jvv5Rpa/tPisNV9QMF7zyVtnvd/CW6jg5M+LyGlZ/tIS/SR/kvy96HQ6aOmwyzQ3N4dmTmLyRGkirzwVd0iSe+fLhiXhzyDpFPmZPr03IpYKI76uyh9CxdfV+yQlPQpu8JVErmn4W+brOW8uFcu3SEnulC+nZFgiLxgr/xcpr/D87UtsXRWUSvKT8S8mX8gDBw4IguDr68tPRXl7e8+bN49xr1slh8uSAYCk8Fg2yfQu/1jKQzJdK5aFP21lylnPkYghbDDx90J3kaysLOjX3Nzc4uLiJDEy+DWY/DoADPovMaehIMCfjNRYdSBp0OBTKOsninPHfLPDN6cG7AqNl8sSPh8+fIjjBsYYTjiysnm75PVPrjb0zvLyx7ByVIVEkMm7GY1Gww958b8SkzUiWV0lUTkSgxzfSlb8Q0Kngh2e5EHxPb28D5MXsjzkXaxEH/D95TMnp6oMXwzsSuWWJL09NG9c4dtBfvqSlY33w8qarFhZE3TF9yW/Oiv/ndT79CQGFf6B83NkHoGz7AAAIABJREFU/C1LKrmOC2IpyiyLkkenky0FgMPALoiFlEzoazlHLkmlwlNhabE+V2DIkSgVsax9UVdqN+J7Pv5b/BQzDoQkryQ0C1h+ybX4sunKzrTihvy7/CNatGjRnDlzsIsVBMHOzu5f//pX5ftLuFn0WeTrAy/R9A6lKni25R2GyIUv4wZ4Eolv2LFWbQd+F0nTh/UEn6FWq926dWujRo0EQQgJCfH09BQEwdraGqWVUDZ3iMRyhgoAQAM/416BKmgysRwMdXwdQKvVgpULf2J+6SH8XhYWFqCkV69eDd+SvLyGwqiCjDGmVqtxYbC81cOv8FKU/xP7IZzu5asL36dKCsBknToArbxkMhguJ9dkUMjCwkK++8HWE/0B8Su8nQ+viyVnFb5gfG8n6c75W8BJHNzP982sfAd8PImOmw/iOy3JNr8hclKgvPJXGbwWylz5hSTdDP8VfsqSH6HyfY+u/FUL8LtUYFkUORlRnhSDX0E+hIK7QAfeZz4EyapM/gfiXxAm0xP88XyZsQbyw0Gcm5PfGo6RJJfjKxVWBonpXiebL8MqJ3KjKfwXlpmV/gpYLRm32FlXCr8TC8nbmVjpS4QXlfxS/NwxX2y8R7w1vlXRaDR8zhzJM4eH+cw2mp87ljQFfEXFK0qkWAWVB1WsfHoaVSxqX1YJxwmC/+n5CsPPZOEPtGPHDgcHBysrq6ZNm6LSglYI/Q6F0mBmvJ1Mkv5BKOvYQxK5mpDMxeGLAw+c18qgzGA7Pj6ebxgN+AYZL+wF3jBWuKioqC5duuzcufPy5csZGRkLFy7cv3//9u3b09LS0tLStm3b9umnn7755ptvvvnmhx9+eOjQoVOnTt28eRPPXFJScuLEiXfffffYsWO5ubkzZsyIjY0dNWpUdnb28uXL58yZk5qaGhMTEx0dPXz48FmzZmVlZS0qZeHChdnZ2YsWLVqyZMknn3zyySefbN68efPmzbt3787NzT1//vzPP/+8Y8eOQ4cOHT58eNu2bW+//fb69etPnjz54MGDS5cu5efn37lz58aNG7dv3y4pKdFoNPfu3fvhhx/u3bsHvc7Tp09/++2327dvi6JYVFR07ty5PXv27Nu3Lzc399ChQ19++eXp06d5i2B5z02r1Z44cWLbtm3Hjx8/d+7czZs3//nnH+gzHj9+fP369YsXLz59+pTJumSNRvPw4cOHDx9KEtby8GJXo9Hk5+dfvnz5t99+KywsVKvVt27dunz58o0bNyBgNH+GBw8e3L1717DtuEQN3L179+zZs1evXv3777/BjAq9eGFhobxtwu759u3bgiwOGbxCp0+f1mq1O3fu3L59+4ABA2bPnr1t27Z9+/bl5eWdOnXq5MmTx44d27Nnz86dO2/dulVxUaGE9+/f/+23327duvXjjz9eunTp/Pnz33777YkTJ44fP56Xl3fkyJETJ06cOXPms88+e/3110eMGLFw4cJvv/327t27Bw4c4Je7y5/Df//733379n344YerV69etWrVsGHD+vTpExgY6OXl5ebm1rx58w4dOnTr1i0lJWXChAlvvPFGZmYmVPXBgwcvXrz42LFjZ8+ePXfu3KFDh/bv33/kyJH9+/dv27Zt69atH3/88QcffLB69eq33nprwYIFCxcunDlzZnh4eHJy8quvvhobG2trawuNjoeHx7Bhw8aPHz916tSZM2eOHj26b9++/v7+7u7uQUFBkZGRaWlpy5cvz8nJmThxYnx8fFRUFDxna2tra2trCwsLgQOWnvHrzszNzUNCQqKiovr06dOvX7+OHTu6ubnBr9apU6fOnTt36NChZcuWjo6ONjY2MDwNCgpyd3eX/KzOzs4CFwWqadOmnTp1atq0qUKhaNu2bXh4eO/evfv37x8fHx8dHd2lS5egoCAfH58WLVrY2triqSS9IOotOO3ChQsnT54cFRU1f/78cePGvf322wI3ZSmUzW8IG7wjkV6wM3Z2djYzM+vQocOyZcsuXrz4ww8/7Nix49SpU1A3bt++ffz48c8//3zXrl1Lly6Ni4tr2rRpxSfH6bCuXbv269evcePG+HxcXV0HDx48e/bsf//73xMmTHj55ZcHDhyYkpIyZMiQ/v37R0REtGrVys7OThLKgUATF9Q6tGzBDwH9NGxDXZJnOwXw67if12R4PP7JXwu/pXhOyrup5z2+riJJhyXZgy+43sOUSuX69esr9gt6XowU9oI3Qpjs2RMEQQiCoE+HSf6L3WHFHZVk4knveXAnHxyhvJJU4S6quys1NzeX3xf2WM+8tKSECm61jeQwvfsF7nFVsKG3MHrLCdsSucP3uIIswBivkwTZY8eSo8ySiKpnlhNFNp6H12qSSc/ynjkfq5bXExU/McnjquC5gRkPS4WDLnmR5DcreVwV1x/5qXBPeTWkYsqrA5J/4U60XyrKvr+SkuAv+8UXXxhKKRlDkGF2VcaYVqut2jMlCIIwINja8g0ub+7CwypzKslh2Fhji1/rLE+SjtbCwkKhTzPp3flMeE8piWaVJ4sUyiokhSxQvrzYFZSzvB+U96/nFZK8PvAaRSjrni+XYnzhJcILNQ3vWMZfRaVSmZmZoRuT3jvlv8hfHcsgqYrwO/LPQaFQYCg1+YOFq/NmwgYNGuAxcp1awXMTZIqKrwASx1/5/eq9EH9yoewqCv4nkEsxoewPJ/n5hLKR5CQn5E+Smppam6YseX8sKDcJMoIgag5mZmYuLi7BwcExMTG9e/f29fV1c3Nr2bKlp6enpaWlUOEUJG8OEQTBwsKiUaNGLi4ubm5u3t7eYWFh3bt379u3b//+/QMCApRlkYzLnwv5GSRWnBdEwVl9FAqFlZVVq1atPD09bWxswGBWyWJjx2xubm5nZ9esWbM2bdr4+/t37NgxIiIiMjIyNDTU19fXx8cnNDS0R48eXbt29fX1tbOzs7a2bt26defOnaOiomJiYsLCwjw9PVu0aNGrV69XXnll0KBBffr04SejQ0NDg4OD27dv37p1a3d39/LKCTfVoEEDmP13dXUNCgpKSkpauXLlhg0b5s2bl5yc3Lt37/DwcKVS6eLi0rZtWxcXF+jmra2tvby8goODIyIievTo8dJLL/n7+/v5+UFCrYyMjDFjxsTFxcXHxyckJCQnJw8aNKhfv34DBw4cMWJEv379oIMHnYG6E4RIgwYN4EGhMVJSbAsLC19f38DAwODg4JCQkE6dOoWFhYWGhrZr187Ly6tp06Zubm5t2rSJiYlJTk4eMWJEQkJCeHh4cHBwly5devbs+dJLL8GjDg0NjYuLS0pKGjJkSHJy8siRI9PT02fPnj1t2rTExMSwsDAXFxd3d/cOHTpER0eHhoa2bdt2yJAh4NsQFxfXrVu3tLS0V155JS4uLjMzMzs7e86cOaNGjerZs2doaGhISEhQUJC3t7ebm5uPj0/Hjh2DgoLatGkTHh6empo6adKkoUOH9unTp1evXt7e3nBf7u7ugYGB7du3Dw0NHTRoUGxsrI+Pj4eHR3h4eFxcXEBAgLe3N8QT6dGjR2JiYt++fYcOHRoVFdW9e/fp06fn5OTMnj07ISEhJiYmKioqLCysY8eOHTt29PHxgbuIi4vr169fSEhIixYt/P39o6Ki4uLiunbt6u3t3aFDhz59+qSmpo4dOzY5Oblfv37JyclTpkyJiYmZOXNmZmamIAjvvPOOIAigVuVKkckWir041S7I+FVjuFylui9KEARRARU3o2I5sUXkyBffyVczlLc6uybDO5loS0MT41xH5aNmyNdR8UtMJPv1rnuVPEz51UUOyS3IyylZHsEfiX9KVnrJi8q4tSz8qhpJ3GPGOU/j2fBCGHSzMtUMnr9kZZLeW5OsjwH4NddMXyB0vQ9H5MIB4q8midMpAcuG8I8R/eUlK6AZN4HGX1fHrZljXJwm/CL/6PD2JUuO5MewstVbsnqPVyz29vY4yMFQSmPGjGGy1VSGwniBYdmz8pkQBEEYE75dFrmgX3hAeYuUJejVBPwqTrFyodpqFPxyM0RefrFyYTIqEB/8r8DKilrG/S6SwyQnZLLet+ICsNLel79NXj9hYfjjoT7gSYqKivTmIpPXBL4kOi4IpfzpyUVqef+VnFNeAF3ZReiYlEzvA5GURCwbTVpSBv6hyR+s3p9PfhL5jyUZBrCyQg2BtZCo5ORDIB2XCQ3+xCV0/HV5lawrG6wKXRfQbS4wMPDx48f80+Z9sQyFMVZZSupKJds4giCIakKtVku6n/Ja84oFh15bhciFvJGEApZ01VVG70kMqPZ46wjskeuzylxOrjbKUy0it85a7wHl7ZQYRZisi6n4Wem4CACsbCAhPEASUAm7bUzqgN+SSPzyFI9encHfndx4wceuQ70l0SISQSkxO0mOKe+3gG15gHS9uSUkJZfIsortx/zxGHwHgyNKIjIyffJU8rThpZZUKr33iEJTFEW4NC+seVc8kGXyAvCq2oDB+o23ypICDRMEUXMQS7NzViARqiBxsN/lrQu6sunVX+T8RkYSvYynMuXX+0jL+5ZEgkj2692Wm9YqLgNeQu+R+HthPMLyLgf7S0pKJLOBTGYTkuuS8m6cye5dIjj0PhlAW5o/QP4MUdNIvq73hHI1KdciKOwYJxAlT0nyzPVaYcSymhImUrG08jiU/POXm/r03pfIwUpNZXgjfLYeLLNarcYVBiEhIfLKw7jYkBXHrqoCxhBk/Kw5q1VGe4Ig6iQVqAT5lFkVLPqSzhu7nAqKUaPAfpG3tUhCVbPyNZAEvbf5TEEg2ahYneg9VWXKiZ29pLvFTldSPURuCpXpC12Oh0muKJkp4s2Bck0jUVS8lQstuJJbllyO720lxdNbOfnzSJJ24DF8mGUme4n0moolN8L/q7zC4/1KLKZ6b1N+Kr1XL6+2SF5MfFyffvop0/e4cKe2bApHvQWoAs8tyCQDJslMLdP3m+FN6nVKIAiCIIi6TXme7M8UHESNAuWXRCXDTn7iGz0U+dUYFau3KgoyzGciKZzEFxJn1iW52wiCIAiinoD2Kug0JXYjEmG1BX75Jyt/KSv/g6JkqozXVlWmLCGZI++1CtthYWEQtAP84HiTKZq+yZOMIAiCqIfw5hNW6tLEuOlR3pBB1Fi0XKJbfkabj6AhsUnBzvJcF5DnFmQSaYU1CSPYnjlzBpYnmJmZTZgwQeITQBYygiAIoh7C28kkVjHspAsLC01QMqJy4GpQVjbcPWNMqVQyxrRa7aNHj2D/jRs3BEFYtmwZb06r2Ce1ilOWOAUJl4E1om+99RbjbHoQwwNLQA5kBEEQRD0E5yWhF4euWuBSEjHO2EFmi5pMcXExrnsA8cPnVGClEigqKgrTLcAXK+P+X8VVluiNqNVqIcsExOrgffyxiPLQuiTLCIIgiPqDKIqgxi5fvqxUKiGTEjj5QG4uVn6ECKKGwAe8YKXaa9GiRZh5PSMjgzH2yiuvYAoyhUKh0+nAtKbXOMrzQmEvoHrBVRs3bgw7cV6cMQZx1WAbQ4zQBDlBEARRf+AXVPJZNflPHx8fOJgsZDUZPlQyv3wSFdh3330nCIJkCa1YGnStAgcyVgVBxkdmY4xBFDUMZYuF4GsemuyYLNQeQRAEQdRtcDrS2dkZJiixJ+3UqZNQmsCeVejxTZgcXufwbvE6nQ6EEGjrlStX8r5ivPt/xeevioWMDzDm7OyM85WMk/a8ZuQFGUEYBN7Zgg+AV0EAQEnAPBobEHJ0ZaOT68qmpkE3Xnlry8oJK0UQiE6ny87OPnr0KNYfqFdoXNHKUnHzqbf49k2S3IlxbR0FYK8+sMfBh4xu/gcPHlQoFBBlgpW1jUlCTBhylaWOy6Wl1WotLS2hMsljImu1WnR2e96rEMQz4XO7Yo2XtEoSJ1lJO0gQepEoe34NEys7S4Dr2GEDvTUIggdrFPoSMS54JyZPxIOxcYPKxtcrPq+2WDa7ot74WIQB4YdbGOQCnzYayfgQrZLxf8VNRBWnLLES4BIDVjbXBCNBRlQn/JhDWwr8i+8vJTv53IU0NUD8v/bOJzSOMgzj387sbqJIqN4sqBeLl3oJgqAXC4JSBJFSFEHUgwcP0hyKoiKFVrw0/jm1oGiVSgtSbKmWgiBFivWgICol1ZOI2kpBkEqS2Z2Zz8PDPrz7TVKTze7OzuT5HcL+yW5mky/v93zv3yLF0ZYwu1hpZ86c+fvvv/ft2/fee+998cUX3vu5ubkXXniBzRe9NJlYHZ4GeXTEukLI8tdff7XuWK7DTqdz8uTJTz/99Nq1aydOnMCDeHZ+fv7bb789dOgQHwwY9yesO/Z3a4ege+/TNI3jGNn9VkYnScKW/eA6JmJAqUSzhZLdRqNhPRMsQLjpppskyMTooMdrenqanfBYbBxFERanPTYARZfEatDC2nadXEKtVgvlVFhvrVaLT8Vx/Oeff3r5J8RKIKHbzkDEJv3bb79hCdl6OPLSSy81m0260LD2uPxo5d555x2+RIJsdATHePuffujQIevpxO8/GEP+v9HkQaSSbVPLlWETy3zPMbtjxw4JMjEi6DFuNBrYF1HVC5vVbrfpoIUDAykarHORh0wUse0SuauhzbXdDp0ZSUJBFkXRlStXbOtIISzYHNH6lbsy1o83+UmMPD7++ONcb1yE1sTB4jnn3nzzTR0vx0PQpt+b4tlnnnkG+njnzp3FxAbevc6BbRCpZKOWXC5MdLXny3a7LUEmRgQOH/fccw83y0ajQT+Z3Sl5mwcGqTFxHZgj673/+eefoeZ5BIVBS5JkZmYGS465PkFGhxCWYu8nOFlhtfCIzT2CWUvT9NSpU1h4v//+u/f+ySefxJHg0Ucf9d6fO3cu7xWgyCs2UrDp2Bw+3HXO3Xrrrd57uAaccziYnTx50hcCMkP2kPW9vrfVeWOGkiRZre2FEEPBWjRGzLnYgr0TLC0t4Sl6QXSmFAEMgtON+uCDD951113MWWw2m0eOHEEO0OHDh+Gr8P0Z1toURRHrTfHenz9/nidGhrw7nQ630W+++QbTeHwv8T+OY6wxKAC3Uuo2QIBM9m0UMMby9ddf4ze8f/9+bjTOObT89d4nSYKwjF9z6euAUsluhxRkvr/MTW0vxKjJ8/yhhx6yx5Qoirgpcu0FQ3xVaCmuDyW73dXyPL969Wqj0bh06RK+4e6772Z8oNgFQwhLsSuV9/7+++93vcxX+smswzU4efLdYOus68UWAYztQ2026IM8ffq0zVXYsWMHzMXBgwcZluEfNOhDNvxO/XTW2Z8aFOgGgoyRVG2EYijYVDA7NIJGit6LvL8TzFr+MYSwYOWcPXvWHnmZwWNtmuybWCN0hm3fvj3IB/f9C+nSpUtxHB84cAAvgYyLokhusLLYunUrpyxs376dj1+5cgWPx3G8e/fu9b7tgMPFfU+TwUJhZXjTohPPst4N5ox6XzkWYoMwFYwOZCRfcykWHWa+f8691zoU6wEGjf5+CDLkAPn+PizSZKJIUPdm85BguyDIgl7EeHbfvn2uNyfRe3/LLbcE3hcxNihyoihqtVq7d+/Oex1McAMRzF27dg3w5gN6yFjg5ntuiT179mAlFT1kv/zyiy9E0IXYIMFcsDRNbXzcesiAHQerNv1iXdgkxSRJ7F3m+wOpfLEiHPPg+3MnEKBkrMnGH+m/wEng9OnTeHB2djY4G5TzkTY3rLUkaZoy63+w/WXdgmx5ednWqWVZNjs7azdC6MQsy1BiybldtFOKcIuNw3Y+IEmSv/76K4qi22+/nXXFztST2/gm6l90NhDrhW4J69VoNpu56fUj+yZWw7rzg68ffPCB9bZaQdbtdhkZh9Xau3cvK391sBwn9CvZBmPe/CnTNEX12GBFY4N4yNhoh+MdIMgOHDjge2vo+PHjUGNcLjxWaiMUG4cJi/zfwCK0fZNR8GKdYYEnQ0tRrJE8z7dt24ZDpjedhxAlxyNzc3P85hIvVUwydKvYIYSdTufll18OCiexr6dpeuHChWaz2Wq16Dx7++23od7wyL///lvmR9pM2BLs3DTrD3KxBnaTD5hDxtAkxNkPP/yA9hsYJ9LpdNgOyp4XbTRzsMsVgtDRFfS8sB4ySrSHH36Yydd04pZ59aJqIAKOuSjYR21lnPfeOff6669rXYkiuRlOHzwIYKxarRZ2SbvB29q4LMuSJPnuu++wGo8ePZplGfxqYmww3lKc+JLnOaKIg0Ut1y3Igtpahi+PHTvGElA6zPjNdKoVw65CDIB16SN/H7ujN/kZdnSSc67dbvv+7A2l+4g1wjr2vXv30lFx4sQJ59z09DQXmIybWBEGsObm5rZs2YIRW9jRsyybn593vT5kvj/DFfHKOI6npqZ4+PRmaCFcISV9rE0HW/vagRzFNiXj85D5fmFog6l5np86derLL7/0Zq6zHSDgTYXCYJcrBEA2BszTP//8g1Qe5xzPCciEtUMt6dwdOONSbFqYLhYMp4ejAoeBxx57TOtKrAbWBnsiOOcuX7585MgRloZQ6NuScLZWeeKJJ7i6kL/IoyYStcXYoN4KamZtxcaYkvqFmBzwj3Hw4EEYpqeeegqP45/hlVdeoWOD/S8YNVBoSawdrKKbb77Z96+cbdu2YYu94YYbeFCW51WsBuUXQ0mNRuOBBx6gGrNZsL43W8n1D1bKsmznzp186uOPPy7p04ghI0EmqgrjlWmaNpvNrVu3BoUteZ4fPXrUOffqq6/aKktGCqTJxNphZIDHYpvDy6c0XFysBk6DFy9e/Oyzz/bs2YMHkRbm+1PFWQ/e6XQuXLjge/0NWKdpa5Jkx2qDBJmoJEHsm3Uu3hRg4i5v2IJzFjqN9aJF9YHesj4wG0jCDa0rsSIQW9BVTN7HaqHnPsjSBjbhh6sr6OYjaoAEmagk9FLYzS84NTLFstvtsisjzRxNoRD/y4oDHmx3fiww2/xTCEJ7xWYW3sgsmikW8OJVweDwoFssngoqN0WlkSATVcUGiWyDRDr8g4Qe3lhcXPRyY4h1YrulUNlnZgC5dcEKYbHJYVZy0emFPp1BizL7DraDPypL6CGTKasNEmSiwtiNEEaKtgmqy3oseBL1aqcu1gnrp2zrgSRJmMi4wZEpot6slvhFEUalBd1PN5i1aZzM443uV8uCOiFBJqrNioXHvt9RgX2UaWdBKxYh1oKNDa3oDGOvH2kyUSSYomvzX3ED6YlW7tvX0lhZUzamSxfjQoJMVBWbS2E9XsF2yKq3oLJSakysF+vMsFFy6/aQk0wUCRpZFz30PE/axVM8ZAZSLNfEkXohQSYmHVtV5AcVUsGrVJ0khBgRtuSoKLCC5ilD/KHeqDcZtyoiQSYmGqqxTqezYjhyjTAHiJmwcvgLIUaHndJGAtuVDwnrms1NccAYP64YAhJkogLYUNHAZd62Imkj0y2EEGJF8v5hzbhrXWK2jnKIP5dmrVg9ICqEBJmYaIKpYRtxa+GtlpaWKOnUwkcIMURs3JC2y/WYmppqt9tbtmxpNpucRDlEWq3WuXPnmN0oTVY5JMhEBWBja/TnHMBUcdB4o9GI41hzk4QQIwLdK7z3WZa99tprrjdNHPMrp6amYIiazeawpBjeHz/CS4pVFgkyMdFY3zs7vm7EbEGW+UJwQQghNkhRCSVJQuEFQdZutzcswFam1Wq99dZbzM1QSkblkCATkw6ClUPJjcD7yE4JIUYBs+y9MVa2Ttx+w9AJevqo/XXlkCATEw1riPwwysXz/spwIYQYBXayVvBU0MdnKNg+PvkG2gOJcpEgE0IIISYIHkEDH5uoNxJkQgghxGRRHAUh137tkSATQgghJhFWMkmNbQYkyIQQQogJwvZf5A0FLmuPBJkQQggxWaDnou/vwlj2RYnRIkEmhBBCTBAcFqdiyU2FBJkQQggxWSwvL3vTv1rKbDMgQSaEEEJMEMePH5+fn//++++96e+qvP7aI0Emqo1Net21axeGk3BspbIuhBATCx1gHOPGCUsYv9tqtTjqrcwLFWNBgkxUEpgn2zURIgy2DGB8bxzHsmVCiEkGRsw5Nz09TfOFGzBiTz/9dNnXKEaOBJmoJHR94QbH9zrnZmdnYdQozr766qtSL1YIIVaGs+Aows6fP49HbrzxxjiO8bgOlpsBCTJRVWCeut3umTNnILx8L/W12+0ePnwYtoxPCSHEpMFsfXjCfH8axocffggnmXPup59+Ku0qxVjQRiWqSpqmsGWUXKhLSpLEe59l2bvvvks/WbmXKoQQRdjudWFhIYqiIOcViWVII3PO3XfffeVcpRgX2qhEVaH8uvfeey9evMjcWOvYd84xK1YIISYKKDAcLD///HMUVHY6HXaFzfP8xx9/bDabOlhuBvQHFpWEPax9fwNrqDFGAZB+EUVRGdcohBDXA/Yqz3P49X2/NcuyLMuyhYUFRC3n5+fLuUoxLiTIRFVJ09Sm9vM2m1xnWYbcCwkyIcRkYj36Qdo+Sy+RDjvuKxNjR39jUTcYtex2u8i9eP7558u+KCGEWAG687Mevn+4OBqS3XHHHaqyrD0SZKJWIAmj2+3afH9NHRFCTCBWjcG1T02Gr8vLy8i7YOdYUWMkyETdYDYG4pWffPJJudcjhBDXgcORut1ukHoBNfb+++9r6MhmQIJM1JBut3vbbbcFzcmEEGLSgNJaXFzEXXZS9N6/+OKLURQ1Gg08Rd0m6ooEmagVnU7Htr3Oe5R9XUIIEcJIpTe2Cz7+xcVF+Pi9aVcm6o0Emagh8I0xCaPsyxFCiBWAjWJhuO95yNI0ZeMxVikxGUPUFQkyUTdsB0WcLHW4FEJMIKyspA7DV3RPpGMsyzLFKzcDEmSiVrTbbefcH3/84b3PskzBSiHEJMPjIhxg8I3FcWxHj1hHmqgxEmSikhS78/ueb+zq1avsQ8ZvfuONN+gqw7OSa0KICQFqLM9zG6lh+PK0AAABv0lEQVT0vUR+eM6OHTt2+fJlbwKd1qCJGiBBJqoKNRmz+BmppNjK8zxN0/379zvnrl27xteq7lIIUTq25ViapjBizz33nPc+z3N2Jsvz/JFHHnHOTU9Pe++TJKEUUyizTkiQiUoCpUV7NDMzE0URDpcoFMcN59zU1BS1GuuY8CrllgkhyoUJZCgMB9ZwYW4SnvXG8W/lWonXL4aIBJmoJMirgElqt9sUYfhK+4W7rVaLzjNasTKvXgghjLqCpYL2Qg4ZBorjVIkcfzuTF33LkIYhQVYbJMhEhUnT9KOPPoLeouWCXWs0GjBqVGN0py0tLXlTTC6EEGWRpmmWZWfPnrXHSCgwPHLnnXc+++yzMzMzCwsLSDXjeVIHy5ohQSYqCdVVt9stZlEkSYKYZpArRgUGIzieSxVCiBWxOfu+P7cVFgy3baMy+5XfOa7rFaNFgkxUlSzL0PQ1qLi03fnhBmNZJV+IGxxXIoQQpQAbBVPGhAorxWy1uPc+z3M2yCjnisXIkCATlYTGyDq9cINGrWjI+HJr8oQQohTsYdJOsbTPonrJnirZmazoLROVRoJMCCGEEKJkJMiEEEIIIUrmP/RzAEoTKiLFAAAAAElFTkSuQmCC" alt><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"> <strong><em>A1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><strong><em>[1 mark]</em></strong></span></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(1 + \sin x)^2} = 1 + 2\sin x + {\sin ^2}x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 + 2\sin x + \frac{1}{2}(1 - \cos 2x)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(V = \pi \int_0^{\frac{{3\pi }}{2}} {{{(1 + \sin x)}^2}{\text{d}}x} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \pi \int_0^{\frac{{3\pi }}{2}} {\left( {\frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x} \right){\text{d}}x} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \pi \left[ {\frac{3}{2}x - 2\cos x - \frac{{\sin 2x}}{4}} \right]_0^{\frac{{3\pi }}{2}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{9{\pi ^2}}}{4} + 2\pi \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were almost invariably correctly answered by candidates. In (c), most errors involved the integration of \(\cos (2x)\) and the insertion of the limits.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were almost invariably correctly answered by candidates. In (c), most errors involved the integration of \(\cos (2x)\) and the insertion of the limits.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were almost invariably correctly answered by candidates. In (c), most errors involved the integration of \(\cos (2x)\) and the insertion of the limits.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined on the domain \(\left[ {0,\,\frac{{3\pi }}{2}} \right]\) by \(f(x) = {e^{ - x}}\cos x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the two zeros of <em>f </em>.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of <em>f </em>.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The region bounded by the graph, the <em>x</em>-axis and the <em>y</em>-axis is denoted by <em>A </em>and the region bounded by the graph and the <em>x</em>-axis is denoted by <em>B </em>. Show that the ratio of the area of <em>A </em>to the area of <em>B </em>is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{e^\pi }\left( {{e^{\frac{\pi }{2}}} + 1} \right)}}{{{e^\pi } + 1}}.\]</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({e^{ - x}}\cos x = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow x = \frac{\pi }{2},{\text{ }}\frac{{3\pi }}{2}\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[1 mark]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em><img 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" alt> A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em><strong> </strong></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><strong>Note: </strong></strong>Accept any form of concavity for \(x \in \left[ {0,\frac{\pi }{2}} \right]\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalize unmarked zeros if given in part (a).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em><strong> </strong></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Zeros written on diagram can be used to allow the mark in part (a) to be awarded retrospectively.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt at integration by parts <strong><em>M1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(I = \int {{{\text{e}}^{ - x}}\cos x{\text{d}}x = - {{\text{e}}^{ - x}}\cos x{\text{d}}x - \int {{{\text{e}}^{ - x}}\sin x{\text{d}}x} } \) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow I = - {{\text{e}}^{ - x}}\cos x{\text{d}}x - \left[ { - {{\text{e}}^{ - x}}\sin x + \int {{{\text{e}}^{ - x}}\cos x{\text{d}}x} } \right]\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow I = \frac{{{{\text{e}}^{ - x}}}}{2}(\sin x - \cos x) + C\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalize absence of <em>C</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(I = \int {{{\text{e}}^{ - x}}\cos x{\text{d}}x = {{\text{e}}^{ - x}}\sin x + \int {{{\text{e}}^{ - x}}\sin x{\text{d}}x} } \) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(I = {{\text{e}}^{ - x}}\sin x - {{\text{e}}^{ - x}}\cos x - \int {{{\text{e}}^{ - x}}\cos x{\text{d}}x} \) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow I = \frac{{{{\text{e}}^{ - x}}}}{2}(\sin x - \cos x) + C\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalize absence of <em>C</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^{\frac{\pi }{2}} {{{\text{e}}^{ - x}}\cos x{\text{d}}x = \left[ {\frac{{{{\text{e}}^{ - x}}}}{2}(\sin x - \cos x)} \right]} _0^{\frac{\pi }{2}} = \frac{{{{\text{e}}^{ - \frac{\pi }{2}}}}}{2} + \frac{1}{2}\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {{{\text{e}}^{ - x}}\cos x{\text{d}}x = \left[ {\frac{{{{\text{e}}^{ - x}}}}{2}(\sin x - \cos x)} \right]_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} = - \frac{{{{\text{e}}^{ - \frac{{3\pi }}{2}}}}}{2} - \frac{{{{\text{e}}^{ - \frac{\pi }{2}}}}}{2}} \) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">ratio of <em>A</em>:<em>B </em>is \(\frac{{\frac{{{{\text{e}}^{ - \frac{\pi }{2}}}}}{2} + \frac{1}{2}}}{{\frac{{{{\text{e}}^{ - \frac{{3\pi }}{2}}}}}{2} + \frac{{{{\text{e}}^{ - \frac{\pi }{2}}}}}{2}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{\text{e}}^{\frac{{3\pi }}{2}}}\left( {{{\text{e}}^{ - \frac{\pi }{2}}} + 1} \right)}}{{{{\text{e}}^{\frac{{3\pi }}{2}}}\left( {{{\text{e}}^{ - \frac{{3\pi }}{2}}} + {{\text{e}}^{ - \frac{\pi }{2}}}} \right)}}\) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{{\text{e}}^\pi }\left( {{{\text{e}}^{\frac{\pi }{2}}} + 1} \right)}}{{{{\text{e}}^\pi } + 1}}\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks] </em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates stated the two zeros </span><span style="font-family: 'times new roman', times; font-size: medium;">of </span><em style="font-family: 'times new roman', times; font-size: medium;">f</em><span style="font-family: 'times new roman', times; font-size: medium;"> correctly but the graph of </span><em style="font-family: 'times new roman', times; font-size: medium;">f</em><span style="font-family: 'times new roman', times; font-size: medium;"> was often incorrectly drawn. In (c), many candidates failed to realise that integration by parts had to be used twice here and even those who did that often made algebraic errors, usually due to the frequent changes of sign.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates stated the two zeros of <em>f</em> correctly but the graph of <em>f</em> was often incorrectly drawn. In (c), many candidates failed to realise that integration by parts had to be used twice here and even those who did that often made algebraic errors, usually due to the frequent changes of sign.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates stated the two zeros of <em>f</em> correctly but the graph </span><span style="font-family: 'times new roman', times; font-size: medium;">of </span><em style="font-family: 'times new roman', times; font-size: medium;">f</em><span style="font-family: 'times new roman', times; font-size: medium;"> was often incorrectly drawn. In (c), many candidates failed to realise that integration by parts had to be used twice here and even those who did that often made algebraic errors, usually due to the frequent changes of sign.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = \frac{1}{{4{x^2} - 4x + 5}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Express \(4{x^2} - 4x + 5\) in the form \(a{(x - h)^2} + k\) where <em>a</em>, <em>h</em>, \(k \in \mathbb{Q}\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(y = {x^2}\) is transformed onto the graph of \(y = 4{x^2} - 4x + 5\). Describe a sequence of transformations that does this, making the order of transformations clear.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = f(x)\).</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the range of <em>f</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By using a suitable substitution show that \(\int {f(x){\text{d}}x = \frac{1}{4}\int {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{\pi }{{16}}} \).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4{(x - 0.5)^2} + 4\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for two correct parameters, </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A2</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for all three correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">translation \(\left( {\begin{array}{*{20}{c}}<br> {0.5} \\ <br> 0 <br>\end{array}} \right)\) (allow “0.5 to the right”) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">stretch parallel to <em>y</em>-axis, scale factor 4 (allow vertical stretch or similar) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">translation \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 4 <br>\end{array}} \right)\) (allow “4 up”) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> All transformations must state magnitude and direction.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> First two transformations can be in either order.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It could be a stretch followed by a single translation of </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {0.5} \\ <br> 4 <br>\end{array}} \right)\)</span>. If the vertical translation is before the stretch it is \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 1 <br>\end{array}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">general shape (including asymptote and single maximum in first quadrant), <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">intercept \(\left( {0,\frac{1}{5}} \right)\) or maximum \(\left( {\frac{1}{2},\frac{1}{4}} \right)\) shown <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0 < f(x) \leqslant \frac{1}{4}\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \( \leqslant \frac{1}{4}\), </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \(0 < \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(u = x - \frac{1}{2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}u}}{{{\text{d}}x}} = 1\,\,\,\,\,{\text{(or d}}u = {\text{d}}x)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \int {\frac{1}{{4{{\left( {x - \frac{1}{2}} \right)}^2} + 4}}{\text{d}}x} } \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{1}{{4{u^2} + 4}}{\text{d}}u = \frac{1}{4}\int {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> If following through an incorrect answer to part (a), do not award final </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> mark.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{1}{4}\int_{0.5}^3 {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for correct change of limits. Award also if they do not change limits but go back to </span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;"> values when substituting the limit (even if there is an error in the integral).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{4}\left[ {\arctan (u)} \right]_{0.5}^3\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{4}\left( {\arctan (3) - \arctan \left( {\frac{1}{2}} \right)} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let the integral = <em>I</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan 4I = \tan \left( {\arctan (3) - \arctan \left( {\frac{1}{2}} \right)} \right)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{3 - 0.5}}{{1 + 3 \times 0.5}} = \frac{{2.5}}{{2.5}} = 1\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(4I = \frac{\pi }{4} \Rightarrow I = \frac{\pi }{{16}}\) <strong><em>A1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Exam technique would have helped those candidates who could not get part (a) correct as any solution of the form given in the question could have led to full marks in part (b). Several candidates obtained expressions which were not of this form in (a) and so were unable to receive any marks in (b) Many missed the fact that if a vertical translation is performed before the vertical stretch it has a different magnitude to if it is done afterwards. Though on this occasion the markscheme was fairly flexible in the words it allowed to be used by candidates to describe the transformations it would be less risky to use the correct expressions.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Generally the sketches were poor. The general rule for all sketch questions should be that any asymptotes or intercepts should be clearly labelled. Sketches do not need to be done on graph paper, but a ruler should be used, particularly when asymptotes are involved.<br></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e).</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) and (f) were well done up to the final part of (f), in which candidates did not realise they needed to use the compound angle formula.<br></span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question covered many syllabus areas, completing the square, transformations of graphs, range, integration by substitution and compound angle formulae. There were many good solutions to parts (a) – (e) but the following points caused some difficulties.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) and (f) were well done up to the final part of (f), in which candidates did not realise they needed to use the compound angle formula.<br></span></p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the graphs of \(y = \sin x\) and \(y = \sin 2x\) , on the same set of axes, for \(0 \leqslant x \leqslant \frac{\pi }{2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the x-coordinates of the points of intersection of the graphs in the domain \(0 \leqslant x \leqslant \frac{\pi }{2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the area enclosed by the graphs.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\int_0^1 {\sqrt {\frac{x}{{4 - x}}} }{{\text{d}}x} \) using the substitution \(x = 4{\sin ^2}\theta \) .</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The increasing function <em>f</em> satisfies \(f(0) = 0\) and \(f(a) = b\) , where \(a > 0\) and \(b > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) By reference to a sketch, show that \(\int_0^a {f(x){\text{d}}x = ab - \int_0^b {{f^{ - 1}}(x){\text{d}}x} } \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>Hence</strong> find the value of \(\int_0^2 {\arcsin \left( {\frac{x}{4}} \right){\text{d}}x} \) .</span></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="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><img 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" alt><span class="Apple-style-span" style="font-family: 'times new roman', times; font-size: medium; display: inline; float: none;"> <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica; min-height: 24.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for correct \(\sin x\) , <strong><em>A1</em></strong> for correct \(\sin 2x\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1A0</em></strong> for two correct shapes with \(\frac{\pi }{2}\) and/or 1 missing.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Condone graph outside the domain.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica; min-height: 24.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\sin 2x = \sin x\) , \(0 \leqslant x \leqslant \frac{\pi }{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2\sin x\cos x - \sin x = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sin x(2\cos x - 1) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 0,\frac{\pi }{3}\) <strong><em>A1A1</em></strong> <strong><em>N1N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) area \( = \int_0^{\frac{\pi }{3}} {(\sin 2x - \sin x){\text{d}}x} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1</em></strong> for an integral that contains limits, not necessarily correct, with \(\sin x\) and \(\sin 2x\) subtracted in either order.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica; min-height: 24.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left[ { - \frac{1}{2}\cos 2x + \cos x} \right]_0^{\frac{\pi }{3}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( { - \frac{1}{2}\cos \frac{{2\pi }}{3} + \cos \frac{\pi }{3}} \right) - \left( { - \frac{1}{2}\cos 0 + \cos 0} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{3}{4} - \frac{1}{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^1 {\sqrt {\frac{x}{{4 - x}}} } {\text{d}}x = \int_0^{\frac{\pi }{6}} {\sqrt {\frac{{4{{\sin }^2}\theta }}{{4 - 4{{\sin }^2}\theta }}} \times 8\sin \theta \cos \theta {\text{d}}\theta } \) <strong><em>M1A1A1</em></strong></span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1</em></strong> for substitution and reasonable attempt at finding expression for d<em>x</em> in terms of \({\text{d}}\theta \) , first <strong><em>A1</em></strong> for correct limits, second <strong><em>A1</em></strong> for correct substitution for d<em>x</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica; min-height: 24.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^{\frac{\pi }{6}} {8{{\sin }^2}\theta {\text{d}}\theta } \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^{\frac{\pi }{6}} {4 - 4\cos 2\theta {\text{d}}\theta } \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = [4\theta - 2\sin 2\theta ]_0^{\frac{\pi }{6}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {\frac{{2\pi }}{3} - 2\sin \frac{\pi }{3}} \right) - 0\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{2\pi }}{3} - \sqrt 3 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="font-family: 'times new roman', times; font-size: medium; display: inline; float: none; line-height: normal;"><img 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" alt> <strong><em>M1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">from the diagram above</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the shaded area \( = \int_0^a {f(x){\text{d}}x = ab - \int_0^b {{f^{ - 1}}(y){\text{d}}y} } \) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({ = ab - \int_0^b {{f^{ - 1}}(x){\text{d}}x} }\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(f(x) = \arcsin \frac{x}{4} \Rightarrow {f^{ - 1}}(x) = 4\sin x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^2 {\arcsin \left( {\frac{x}{4}} \right){\text{d}}x = \frac{\pi }{3} - \int_0^{\frac{\pi }{6}} {4\sin x{\text{d}}x} } \) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for the limit \(\frac{\pi }{6}\) seen anywhere, <strong><em>A1</em></strong> for all else correct.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{\pi }{3} - [ - 4\cos x]_0^{\frac{\pi }{6}}\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{\pi }{3} - 4 + 2\sqrt 3 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award no marks for methods using integration by parts.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A significant number of candidates did not seem to have the time required to attempt this question satisfactorily.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was done quite well by most but a number found sketching the functions difficult, the most common error being poor labelling of the axes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (ii) was done well by most the most common error being to divide the equation by \(\sin x\) and so omit the <em>x</em> = 0 value. Many recognised the value from the graph and corrected this in their final solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The final part was done well by many candidates.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates found (b) challenging. Few were able to substitute the <em>dx</em> expression correctly and many did not even seem to recognise the need for this term. Those that did tended to be able to find the integral correctly. Most saw the need for the double angle expression although many did not change the limits successfully.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Few candidates attempted part c). Those who did get this far managed the sketch well and were able to explain the relationship required. Among those who gave a response to this many were able to get the result although a number made errors in giving the inverse function. On the whole those who got this far did it well.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A significant number of candidates did not seem to have the time required to attempt this question satisfactorily.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was done quite well by most but a number found sketching the functions difficult, the most common error being poor labelling of the axes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (ii) was done well by most the most common error being to divide the equation by \(\sin x\) and so omit the <em>x</em> = 0 value. Many recognised the value from the graph and corrected this in their final solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The final part was done well by many candidates.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates found (b) challenging. Few were able to substitute the <em>dx</em> expression correctly and many did not even seem to recognise the need for this term. Those that did tended to be able to find the integral correctly. Most saw the need for the double angle expression although many did not change the limits successfully.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Few candidates attempted part c). Those who did get this far managed the sketch well and were able to explain the relationship required. Among those who gave a response to this many were able to get the result although a number made errors in giving the inverse function. On the whole those who got this far did it well.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A significant number of candidates did not seem to have the time required to attempt this question satisfactorily.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was done quite well by most but a number found sketching the functions difficult, the most common error being poor labelling of the axes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (ii) was done well by most the most common error being to divide the equation by \(\sin x\) and so omit the <em>x</em> = 0 value. Many recognised the value from the graph and corrected this in their final solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The final part was done well by many candidates.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates found (b) challenging. Few were able to substitute the <em>dx</em> expression correctly and many did not even seem to recognise the need for this term. Those that did tended to be able to find the integral correctly. Most saw the need for the double angle expression although many did not change the limits successfully.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Few candidates attempted part c). Those who did get this far managed the sketch well and were able to explain the relationship required. Among those who gave a response to this many were able to get the result although a number made errors in giving the inverse function. On the whole those who got this far did it well.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(\arctan \left( {\frac{1}{2}} \right) + \arctan \left( {\frac{1}{3}} \right) = \frac{\pi }{4}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence, or otherwise, find the value of \(\arctan (2) + \arctan (3)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(x = \arctan \frac{1}{2} \Rightarrow \tan x = \frac{1}{2}\) and \(y = \arctan \frac{1}{3} \Rightarrow \tan y = \frac{1}{3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan (x + y) = \frac{{\tan x + \tan y}}{{1 - \tan x\tan y}} = \frac{{\frac{1}{2} + \frac{1}{3}}}{{1 - \frac{1}{2} \times \frac{1}{3}}} = 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so, \(x + y = \arctan 1 = \frac{\pi }{4}\) <strong><em>A1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for \(x,{\text{ }}y > 0\) , \(\arctan x + \arctan y = \arctan \left( {\frac{{x + y}}{{1 - xy}}} \right)\) if \(xy < 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so, \(\arctan \frac{1}{2} + \arctan \frac{1}{3} = \arctan \left( {\frac{{\frac{1}{2} + \frac{1}{3}}}{{1 - \frac{1}{2} \times \frac{1}{3}}}} \right) = \frac{\pi }{4}\) <strong><em>A1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">an appropriate sketch <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>e.g.</em> </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correct reasoning leading to \(\frac{\pi }{4}\) <strong><em>R1AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arctan (2) + \arctan (3) = \frac{\pi }{2} - \arctan \left( {\frac{1}{2}} \right) + \frac{\pi }{2} - \arctan \left( {\frac{1}{3}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \pi - \left( {\arctan \left( {\frac{1}{2}} \right) + \arctan \left( {\frac{1}{3}} \right)} \right)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only one of the previous two marks may be implied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \pi - \frac{\pi }{4} = \frac{{3\pi }}{4}\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \(x = \arctan 2 \Rightarrow \tan x = 2\) and \(y = \arctan 3 \Rightarrow \tan y = 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan (x + y) = \frac{{\tan x + \tan y}}{{1 - \tan x\tan y}} = \frac{{2 + 3}}{{1 - 2 \times 3}} = - 1\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">as \(\frac{\pi }{4} < x < \frac{\pi }{2}\,\,\,\,\,\left( {{\text{accept }}0 < x < \frac{\pi }{2}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and \(\frac{\pi }{4} < y < \frac{\pi }{2}\,\,\,\,\,\left( {{\text{accept }}0 < y < \frac{\pi }{2}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{\pi }{2} < x + y < \pi \,\,\,\,\,{\text{(accept }}0 < x + y < \pi )\) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only one of the previous two marks may be implied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so, \(x + y = \frac{{3\pi }}{4}\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for \(x,{\text{ }}y > 0\) , \(\arctan x + \arctan y = \arctan \left( {\frac{{x + y}}{{1 - xy}}} \right) + \pi {\text{ if }}xy > 1\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so, \(\arctan 2 + \arctan 3 = \arctan \left( {\frac{{2 + 3}}{{1 - 2 \times 3}}} \right) + \pi \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Only one of the previous two marks may be implied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{3\pi }}{4}\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 4</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">an appropriate sketch <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>e.g.</em> </span><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correct reasoning leading to \(\frac{{3\pi }}{4}\) <strong><em>R1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates had difficulties with this question due to a number of misconceptions, including \(\arctan x = {\tan ^{ - 1}}x = \frac{{\cos x}}{{\sin x}}\) and \(\arctan x = \frac{{\arcsin x}}{{\arccos x}}\), showing that, although candidates were familiar with the notation, they did not understand its meaning. Part (a) was done well among candidates who recognized arctan as the inverse of the tangent function but just a few were able to identify the relationship between parts (a) and (b). Very few candidates attempted a geometrical approach to this question.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Sketch the curve \(f(x) = \sin 2x\) , \(0 \leqslant x \leqslant \pi \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence sketch on a separate diagram the graph of \(g(x) = \csc 2x\) , \(0 \leqslant x \leqslant \pi \) , clearly stating the coordinates of any local maximum or minimum points and the equations of any asymptotes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that tan \(x + \cot x \equiv 2\csc 2x\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Hence or otherwise, find the coordinates of the local maximum and local minimum points on the graph of \(y = \tan 2x + \cot 2x\) , \(0 \leqslant x \leqslant \frac{\pi }{2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Find the solution of the equation \(\csc 2x = 1.5\tan x - 0.5\) , \(0 \leqslant x \leqslant \frac{\pi }{2}\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) </span><img 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" alt><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="font-family: 'times new roman', times; font-size: medium; display: inline; float: none; line-height: normal;"> <strong><em>A2</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica; min-height: 30.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for shape.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A1</em></strong> for scales given on each axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) </span><img 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" alt><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="font-family: 'times new roman', times; font-size: medium; display: inline; float: none; line-height: normal;"> <strong><em>A5</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Asymptotes \(x = 0,{\text{ }}x = \frac{\pi }{2},{\text{ }}x = \pi \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Max }}\left( {\frac{{3\pi }}{4}, - 1} \right)\) , \({\text{Min }}\left( {\frac{\pi }{4},1} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for shape</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A2</em></strong> for asymptotes, <strong><em>A1</em></strong> for one error, <strong><em>A0</em></strong> otherwise.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A1</em></strong> for max.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A1</em></strong> for min.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\tan x + \cot x \equiv \frac{{\sin x}}{{\cos x}} + \frac{{\cos x}}{{\sin x}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \equiv \frac{{{{\sin }^2}x + {{\cos }^2}x}}{{\sin x\cos x}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \equiv \frac{1}{{\frac{1}{2}\sin 2x}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \equiv 2\csc 2x\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \(\tan 2x + \cot 2x \equiv 2\csc 4x\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Max is at \(\left( {\frac{{3\pi }}{8}, - 2} \right)\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Min is at \(\left( {\frac{\pi }{8},2} \right)\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) \(\csc 2x = 1.5\tan x - 0.5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{2}\tan x + \frac{1}{2}\cot x = \frac{3}{2}\tan x - \frac{1}{2}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan x + \cot x = 3\tan x - 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2\tan x - \frac{1}{{\tan x}} - 1 = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2{\tan ^2}x - \tan x - 1 = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\((2\tan x + 1)(\tan x - 1) = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan x = - \frac{1}{2}{\text{ or 1}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{x = }}\frac{\pi }{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A0</em></strong> for answer in degrees or if more than one value given for <em>x</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span></strong></em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">Total [21 marks]</span></strong></em></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Although the better candidates scored well on this question, it was disappointing to see that a number of candidates did not appear to be well prepared and made little progress. It was disappointing that a small minority of candidates were unable to sketch \(y = \sin 2x\) . Most candidates who completed part (a) attempted part (b), although not always successfully. In many cases the coordinates of the local maximum and minimum points and the equations of the asymptotes were not clearly stated. Part (c) was attempted by the vast majority of candidates. The responses to part (d) were disappointing with a significant number of candidates ignoring the hence and attempting differentiation which more often than not resulted in either arithmetic or algebraic errors. A reasonable number of candidates gained the correct answer to part (e), but a number tried to solve the equation is terms of sin <em>x </em>and cos <em>x </em>and made little progress.</span></p>
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