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</div><h2>HL Paper 2</h2><div class="specification">
<p>A function is defined by \(f(x) = A\sin (Bx) + C,{\text{ }} - \pi \le x \le \pi \), where \(A,{\text{ }}B,{\text{ }}C \in \mathbb{Z}\). The following diagram represents the graph of \(y = f(x)\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-29_om_10.14.17.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(A\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(B\);</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>\(C\).</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">Solve \(f(x) = 3\) for \(0 \le x \le \pi \).</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>(i) \(A = - 3\) <strong><em>A1</em></strong></p>
<p>(ii) period \( = \frac{\pi }{B}\) <strong><em>(M1)</em></strong></p>
<p>\(B = 2\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award as above for \(A = 3\) and \(B = - 2\).</p>
<p> </p>
<p>(iii) \(C = 2\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(x = 1.74,{\text{ }}2.97\;\;\;\left( {x = \frac{1}{2}\left( {\pi + \arcsin \frac{1}{3}} \right),{\text{ }}\frac{1}{2}\left( {2\pi - \arcsin \frac{1}{3}} \right)} \right)\) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1)A0 </em></strong>if extra correct solutions <em>eg </em>\(( - 1.40,{\text{ }} - 0.170)\) are given outside the domain \(0 \le x \le \pi \). Do not award <strong><em>FT </em></strong>in (b).</p>
<p><em><strong>[2 marks]</strong></em></p>
<p><em><strong>Total [6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function \(f\) defined by \(f(x) = 3x\arccos (x)\) where \( - 1 \leqslant x \leqslant 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Sketch the graph of \(f\) </span>indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.</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">State the range of \(f\).</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">Solve the inequality \(\left| {3x\arccos (x)} \right| > 1\).</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"><img src="images/Schermafbeelding_2017-03-01_om_06.12.12.png" alt="N16/5/MATHL/HP2/ENG/TZ0/05.a/M"></p>
<p class="p2">correct shape passing through the origin and correct domain <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p3"> </p>
<p class="p2"><strong>Note: </strong>Endpoint coordinates are not required. The domain can be indicated by \( - 1\) and 1 marked on the axis.</p>
<p class="p2"><span class="Apple-converted-space">\((0.652,{\text{ }}1.68)\) </span><strong><em>A1</em></strong></p>
<p class="p2">two correct intercepts (coordinates not required) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p3"> </p>
<p class="p2"><strong>Note: </strong>A graph passing through the origin is sufficient for \((0,{\text{ }}0)\).</p>
<p class="p3"> </p>
<p class="p2"><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">\([-9.42,{\text{ }}1.68]{\text{ }}({\text{or }} - 3\pi ,{\text{ }}1.68])\) </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A1A0 </em></strong>for open or semi-open intervals with correct endpoints. Award <strong><em>A1A0 </em></strong>for closed intervals with one correct endpoint.</p>
<p class="p2"> </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">attempting to solve either \(\left| {3x\arccos (x)} \right| > 1\) (or equivalent) or \(\left| {3x\arccos (x)} \right| = 1\) (or equivalent) (<em>eg</em>. graphically) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><img src="images/Schermafbeelding_2017-03-01_om_06.22.47.png" alt="N16/5/MATHL/HP2/ENG/TZ0/05.c/M"></p>
<p class="p1"><span class="Apple-converted-space">\(x = - 0.189,{\text{ }}0.254,{\text{ }}0.937\) </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( - 1 \leqslant x < - 0.189{\text{ or }}0.254 < x < 0.937\) </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A0 </em></strong>for \(x < - 0.189\).</p>
<p class="p2"> </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;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Given \(\Delta \)</span><span style="font-family: times new roman,times; font-size: medium;">ABC, with lengths shown in the diagram below, find the length of the line segment [CD].</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{\sin C}}{7} = \frac{{\sin 40}}{5}\)</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>M1(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{B}}\hat {\text{C}}{\text{D}} = 64.14...^\circ \) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{CD}} = 2 \times 5\cos 64.14...\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Also allow use of sine or cosine rule.</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;"><span style="font-family: times new roman,times; font-size: medium;">\({\text{CD}} = 4.36\)</span> <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">let </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\text{AC}} = x\)</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">cosine rule</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({5^2} = {7^2} + {x^2} - 2 \times 7 \times x\cos 40\) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({x^2} - 10.7 ... x + 24 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = \frac{{10.7... \pm \sqrt {{{\left( {10.7...} \right)}^2} - 4 \times 24} }}{2}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 7.54\); \(3.18\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">CD is the difference in these two values \(= 4.36\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Other methods may be seen.</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;">[5 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This was an accessible question to most candidates although care was required when calculating the angles. Candidates who did not annotate the diagram or did not take care with the notation for the angles and sides often had difficulty recognizing when an angle was acute or obtuse. This prevented the candidate from obtaining a correct solution. There were many examples of candidates rounding answers prematurely and thus arriving at a final answer that was to the correct degree of accuracy but incorrect.</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;">Consider triangle ABC with \({\rm{B}}\hat {\rm{A}}{\rm{C}} = 37.8^\circ \) , AB = 8.75 and BC = 6 .</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;">Find 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: 23.0px 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: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Attempting to use the cosine rule <em>i.e.</em> \({\text{B}}{{\text{C}}^2} = {\text{A}}{{\text{B}}^2} + {\text{A}}{{\text{C}}^2} - 2 \times {\text{AB}} \times {\text{AC}} \times \cos {\rm{B\hat AC}}\) <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;">\({6^2} = {8.75^2} + {\text{A}}{{\text{C}}^2} - 2 \times 8.75 \times {\text{AC}} \times \cos 37.8^\circ \) (or equivalent) <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;">Attempting to solve the quadratic in AC <em>e.g.</em> graphically, numerically or with quadratic formula <strong><em>M1A1</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;">Evidence from a sketch graph or their quadratic formula (AC = …) that there are two values of AC to determine. <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;">AC = 9.60 or AC = 4.22 <strong><em>A1A1</em></strong> <strong><em>N4</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> Award <strong><em>(M1)A1M1A1(A0)A1A0</em></strong> for one correct value of AC.</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>[7 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: 23.0px 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: 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;">Attempting to use the sine rule <em>i.e.</em> \(\frac{{{\rm{BC}}}}{{\sin {\rm{B\hat AC}}}} = \frac{{{\rm{AB}}}}{{\sin {\rm{A\hat CB}}}}\) <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;">\(\sin C = \frac{{8.75\sin 37.8^\circ }}{6}\,\,\,\,\,{\text{( = 0.8938…)}}\) <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;"><em>C</em> = 63.3576…° <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;"><em>C</em> = 116.6423…° <strong>and</strong> <em>B</em> = 78.842…° or <em>B</em> = 25.5576…° <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>EITHER</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;">Attempting to solve \(\frac{{{\text{AC}}}}{{\sin 78.842...^\circ }} = \frac{6}{{\sin 37.8^\circ }}{\text{ or }}\frac{{{\text{AC}}}}{{\sin 25.5576...^\circ }} = \frac{6}{{\sin 37.8^\circ }}\) <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;"><strong>OR</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;">Attempting to solve \({\text{A}}{{\text{C}}^2} = {8.75^2} + {6^2} - 2 \times 8.75 \times 6 \times \cos 25.5576...^\circ {\text{ or}}\)</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;">\({\text{A}}{{\text{C}}^2} = {8.75^2} + {6^2} - 2 \times {8.75^2} \times 6 \times \cos 78.842...^\circ \) <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;">\({\text{AC}} = 9.60{\text{ or AC}} = 4.22\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1A1</em></strong> <strong><em>N4</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> Award <strong><em>(M1)(A1)A1A0M1A1A0</em></strong> for one correct value of AC.</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>[7 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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A large proportion of candidates did not identify the ambiguous case and hence they only obtained one correct value of AC. A number of candidates prematurely rounded intermediate results (angles) causing inaccurate final answers.</span></p>
</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;">The diagram below shows a semi-circle of diameter 20 cm, centre O and two points A and B such that \({\rm{A\hat OB}} = \theta \), where \(\theta \) is in radians.</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="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-17_om_06.17.13.png" alt></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;">Show that the shaded area can be expressed as \(50\theta - 50\sin \theta \).</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 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 the value of \(\theta \) for which the shaded area is equal to half that of the unshaded area, giving your answer correct to four significant figures.</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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(A = \frac{1}{2} \times {10^2} \times \theta - \frac{1}{2} \times {10^2} \times \sin \theta \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for use of area of segment = area of sector – area of triangle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 50\theta - 50\sin \theta \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><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;"><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;">unshaded area \( = \frac{{\pi \times {{10}^2}}}{2} - 50(\theta - \sin \theta )\)</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;">(or equivalent <em>eg</em> \(50\pi - 50\theta + 50\sin \theta )\) <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;">\(50\theta - 50\sin \theta = \frac{1}{2}(50\pi - 50\theta + 50\sin \theta )\) <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;">\(3\theta - 3\sin \theta - \pi = 0\)</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;">\( \Rightarrow \theta = 1.969{\text{ (rad)}}\) <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 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;">\(50\theta - 50\sin \theta = \frac{1}{3}\left( {\frac{{\pi \times {{10}^2}}}{2}} \right)\) <strong><em>(M1)(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;">\(3\theta - 3\sin \theta - \pi = 0\)</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;">\( \Rightarrow \theta = 1.969{\text{ (rad)}}\) <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">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;">Part (a) was very well done. Most candidates knew how to calculate the area of a segment. A few candidates used \(r = 20\).</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;">Part (b) challenged a large proportion of candidates. A common error was to equate the unshaded area and the shaded area. Some candidates expressed their final answer correct to three significant figures rather than to the four significant figures specified.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In triangle \({\text{PQR, PR}} = 12{\text{ cm, QR}} = p{\text{ cm, PQ}} = r{\text{ cm}}\) and \({\rm{Q\hat PR}} = 30^\circ \).</p>
</div>
<div class="specification">
<p>Consider the possible triangles with \({\text{QR}} = 8{\text{ cm}}\).</p>
</div>
<div class="specification">
<p>Consider the case where \(p\), the length of QR is not fixed at 8 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the cosine rule to show that \({r^2} - 12\sqrt 3 r + 144 - {p^2} = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the two corresponding values of PQ.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area of the smaller triangle.</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>Determine the range of values of \(p\) for which it is possible to form two triangles.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({p^2} = {12^2} + {r^2} - 2 \times 12 \times r \times \cos (30^\circ )\) <strong><em>M1A1</em></strong></p>
<p>\({r^2} - 12\sqrt 3 r + 144 - {p^2} = 0\) <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>\({r^2} - 12\sqrt 3 r + 80 = 0\) <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p>using the sine rule <strong><em>(M1)</em></strong></p>
<p><strong>THEN</strong></p>
<p>\({\text{PQ}} = 5.10{\text{ }}({\text{cm}})\) or <strong><em>A1</em></strong></p>
<p>\({\text{PQ}} = 15.7{\text{ }}({\text{cm}})\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{area}} = \frac{1}{2} \times 12 \times 5.1008 \ldots \times \sin (30^\circ )\) <strong><em>M1A1</em></strong></p>
<p>\( = 15.3{\text{ }}({\text{c}}{{\text{m}}^2})\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p>\({r^2} - 12\sqrt 3 r + 144 - {p^2} = 0\)</p>
<p>discriminant \( = {\left( {12\sqrt 3 } \right)^2} - 4 \times (144 - {p^2})\) <strong><em>M1</em></strong></p>
<p>\( = 4({p^2} - 36)\) <strong><em>A1</em></strong></p>
<p>\(({p^2} - 36) > 0\) <strong><em>M1</em></strong></p>
<p>\(p > 6\) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>construction of a right angle triangle <strong><em>(M1)</em></strong></p>
<p>\(12\sin 30^\circ = 6\) <strong><em>M1(A1)</em></strong></p>
<p>hence for two triangles \(p > 6\) <strong><em>R1</em></strong></p>
<p><strong>THEN</strong></p>
<p>\(p < 12\) <strong><em>A1</em></strong></p>
<p>\(144 - {p^2} > 0\) to ensure two positive solutions or valid geometric argument <strong><em>R1</em></strong></p>
<p>\(\therefore 6 < p < 12\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>diagram showing two triangles <strong><em>(M1)</em></strong></p>
<p>\(12\sin 30^\circ = 6\) <strong><em>M1A1</em></strong></p>
<p>one right angled triangle when \(p = 6\) <strong><em>(A1)</em></strong></p>
<p>\(\therefore p > 6\) for two triangles <strong><em>R1</em></strong></p>
<p>\(p < 12\) for two triangles <strong><em>A1</em></strong></p>
<p>\(6 < p < 12\) <strong><em>A1</em></strong></p>
<p><strong><em>[7 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>Consider the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPoAAADgCAYAAAAqoXvAAAAgAElEQVR4Ae1dC3BVRbbd4AdERBzgaVAHVCQUIshPHMQ3CBrw+RwVFAd9ihJ9+gQ/OE78RcryMahIqePoG0dMdFBRBPk5U2MiiQH5iBNABIFEQEBIBgULERFUklerw46Hm3tv7r2nzznd5+yuSvXN+Xav7n26e+3dezepra2tJUmCgCAQagSahrp2UjlBQBBQCIigS0cQBCKAgAh6BBpZqigIiKBHrQ/UbKFZuX1oSOF6qola3SNcXxH0iDV+zYYS+kvhciqevoQ2iKRHpvVF0CPT1Kjoblo571O68MkxlFX8Li3asD9StY9yZUXQo9T6e1bQW2Vn0bCbh9H1WTMov2AJ7YlS/SNcVxH0yDT+fqqc+RbRPb+hzq2705Dre1P1q/OpfI/M36PQBUTQo9DKqGPNZlq0uCuNOPcXRNSWBubeTjnVxVRU/nVUEIh0PUXQI9H8NbSnbBrNOj+Herasa/KmnfrTNTlV9OrrC2m7DOqh7wVNxAQ29G1MRDup9L6hNHjS8jiVvZoKKqbS6M7N45yTQ2FBQEb0sLRkknrUVM6jp2gSfVNbS9jawH8Ht71No7MW0fRFm0WnngS/MJwSQQ9DKyatA1RqFTQstz+1irmu6cn/Ttdd31506jG4hPFfEfQwtmp9nX6g6tI/0r1PHUEdTzq6/ujPP1rSKdmnExXn063571G1rNV/hiZkv2SNHrIG/bk6+6my8AbKzp1x6FDsWjz2PBFlPUAl6yfQoFby/f8Zx3D8EkEPRztKLQSBpAjIpzspPHJSEAgHAiLo4WhHqYUgkBQBEfSk8MhJQSAcCIigh6MdpRaCQFIERNCTwiMnBYFwICCCHo52lFoIAkkREEFPCo+cFATCgcCR4aiG1KKyslKBsG/fPtq0aVM9IFVVVbR169b6/3HdnXfeSYMGDao/Jj/Cj4AYzFjUxtu2baMtW7bQjh07aP369YT///znPyetQfv27aljx47qmt27d9PatWvV7127dtEvfoG96ZKigIAIuqGtzEK9cuVKWrJkCb3xxhuHlTQ7O5vatGlD7dq1o2bNmtGxxx5LLVq0UNfgeLw0e/ZsWr16tTr1+9//niZNmhTvMjkWQgRE0A1p1K+//prKy8tp8eLFNGvWLFqzZk19yfr3768EGiMwhDmRINffEOfHF198QS+//DJ17tyZ+vTpQ9OmTaOKigr1f5zL5VDIEBBBD7BBP/74Y1qxYgW9+eab9N5776mSYKrdpUsXysrKUsLdqlXs5tL0C/zTTz/RK6+8Qj179qSuXbvSunXr1PQf75ozZ076D5Q7rENABN3nJoNwl5SUqNH1008/VW/HiH3aaadpE+zYKq1atYrmzp2rRnDMFIYPH67KMHjwYCoqKqKcnJzYW+T/kCEgrLsPDQqm+5133jlMuMF6n3vuuWrkPvJI75rh+++/V0L+u9/9Tk3TwcojnXLKKXTzzTdTfn4+XXDBBXTMMcf4gIS8IigEZET3CHkI2AcffEBPPPEElZaWqrdAuDt06OC5cDurNH/+fEXmMcsOLgBr/Lfffpu6detGIPX+9Kc/0dixY523ye+QIeDdUBIyoFKtDtjy6dOn07333qtugSBdd911SsC9HLnjlQ/CDcYeQs2qNOR9+/ZV6rlhw4bRH/7wB7rjjjvoiiuuUKN8vOfIMfsRkBFdUxti7f3iiy/W67UvueQSOv300zNiyDUVSX1wmjdvTkuXLj1sap6Xl0dYToCIwwg/dOhQGjhwoKjbdAFv4HNE0F02CtRh48ePV9NzsNgDBgxQo3fQa96NGzfS66+/TosWLaLzzz//sFpCfQdCDut1lJP/h87+nHPOOexa+SccCIigZ9iOTgHH9LxHjx7UqVMn8nt6Hq/4UKdNnTqVLr74YiosLGxwCWYfULWxHh18wsiRI9V1om5rAFcoDsimljSbEVPeq6++Wo3c27dvp5tuuomuueYapfs2QchRHYzM4Aruv//+uLX75S9/qY6zUQ5GdVjKQQWH0V1S+BAQQU+xTbGWvfXWWxVLDYJrxIgRSsBPPfXUFJ/gz2V79uyhf/zjH4pkgxVcvOQk5Pg8pvcQ9scff1yt2/m45OFAQAS9kXbEtPb5559XpBrINgj4jTfeqEbwRm4N5PRHH32klhG33XZb0veDfMO1zgS9+j//+U9lHus8Lr/tR0DUa0nacMGCBQSBwU4x6MB79+59GHud5NZATv3rX/9qoE5LVJDzzjuPnnzyScKHjIlDzACgU4e6DdZyiWYEiZ4px81FQEb0OG3D03SMeohTNmbMGLUmZ4GIc0vgh0DA4cN06aWXElR7jSWo/pCw2cWZcnNzlZ79pZdech6W35YjICN6TAOCdYa5KJw3YJqODSY2pA0bNigWvaCgoH6ETlZuJyHnHLnxMQOJB/Xb5Zdf3kA1l+yZcs5cBGREP9Q2GMWx9r7yyivp+OOPp7vvvtsaIcf0GwQhPlCxOvNEXS8eIcfXwmIOQs5Tez4uub0IyIhOpAioa6+9ljAq2jSKc7eDmgzqNHyc0knxCDm+/5FHHlG6djD4EHxJdiMQ6REdIyFsvbGL7IgjjrBqFOduB3t2CCNINOxISyeBkIPuHDjEJljIQd2GKTxmO5LsRiCygo4R8Le//a3apgnyCh1ah5MHv7sDjGNglQcSLd2UiJDj58CJJNILL7zAhyS3FIFICjp0xb/+9a+V6yZYtmE3lylWben0IzDmWJvD1j4TjYCTkIv3XswQ/vrXv9JDDz2kNsHEu0aO2YFA5AQdHRdT9aOOOkqtx02zbEu120CdBvdTUKdluoZORshxOWDuiw8hiDlJ9iIQGUHHOvThhx9WzDpcN9k6VeeuBuIQy48JEybwoYzyZIQcHoiZAt4BvXpxcXFG75CbgkcgErvXIORYj8+bN08Zk2CEsjmhPhhhdbhs5i2qvGU1ES5wTIFgEDDKyWSZkOi5ctwfBEI/omPUw9QWQs7rcX+g9e4ty5cvVw9PtDstnTc3Rsjxs+ADHtzGjBkz+JDkFiEQakH//PPPFen2ySefKCG3dT3u7E9Qp8EHHbgGrLHdpsYIOX4+rOegihw1apSo2xgUi/LQCjqEHBtR9u/fbzXpFtuXwLL36tVL7YmPPZfJ/6kQcvxc3hGHrayS7EIglILOQv7DDz8oIbdRPx6vG8E9FPTmjz32mNZ1cmOEHJcFHwU4mgQ/AAcckuxBIHRkXFiFnN1DwQUU+AadKVVCDu8EEQgbBInyorMFvH9WqEb0sAo5ugGiuoBYnDx5svZekSohhxeDcX/66afF7ZT2VvD2gaER9DALOUZR2KSDDHNuKdXVNVIl5Ph92CEHbzRYq6NsksxHIBSCHmYhRxeCx1kkJsN0d6t0CDl+N3T4ULdh/7sk8xGwXtCxswrsetiIN+468aKt8DmdeaqEHL8TMwt2O4UlhSSzEbBa0DFtRLijzZs3h4pdd3aZ999/X6nTUnEP5bwv3d/Jtqwmehb28MPK8Nlnn010iRw3BAGrBR0Wb++++64yhgmLCs3ZL+CUcu3atUqQvDY7TYeQ4zJiyg/rPKjbEBRCkrkIWCvo2KACIceIHgaLt9guAnXahx9+qEivVN1DxT4jnf/TJeT42ZhpwO0UPNIIMceomJdbKejTpk1TO6rQyc444wzzUNVQIhjGbN26VW1c0fC4Rh+RCSGHh2KmAWIOWgF4upFkJgLWCToYaIziMByxfRdaoi7B0VZAdnmhTkv03nQJOX6ORHlhJMzNrbKMQ4CCX/3qV9SsWTO1n9xGrzCpdIX58+crG304lsBI61dKx0Iutkxg3rGEwsdp7Nixsafl/4ARsGZEx/oP20zBsGPKHlYhZ/dQcN/kp5CjH2ZCyHH/hdspVreJHTyjYk5ujaBPnDgx1Aw7ugQIuGXLltFll12WUrQV3d0oU0KOyyFRXhgJ83IrBL2srKyefAsjw87dAu6hoE579NFHte5O4+c3lmdKyPFzQcyxuo2t+fic5MEiYLygw/INU/bs7GxFwAULl3dvx9IE0U3BYMOnelApU0KOywvbBqjbxo0bJ+o2BsWA3HhBR4DDsK/L0Q8QbQX1ZF/qQfWNTCzkYssKnTrs4EXdFotMcP8bLejQl7/55puhNW/lZncTbYWfoSt3Q8hxGSTKCyNhTm6soEOV9sADD6jpui0RTTNtVkzZ4R4qk2grmb4z0X1uCTl+LjuulCgvjEiwubGCjjUeLMPgzSTMCeo0THOhTvPanj0VHN0ScvwOPEeivDAawedGCnpUpuxQp5WUlCh1WqbRVrzoQm4JOS4TR3nJy8vjQ5IHhIBxlnFg2Xv37k0nnHCCEoCAcPHltdid9tZbb1FFRYWvpq6NVc6NhVzssxHdZciQIVRUVEQ5OTmxp+V/nxAwbkTHFBbsc9in7FCnQcihTvPTnj2VfqWDkOP3QLjhdio/P1/UbQxKALlRgo4dWyBvoIcN4/5yZ/vqjLbifK6O37oIOS4Lu52SKC+MiP+5UYIOn2hwI3zWWWf5j4SPb9QdbUV30XURclwuzFgkygujEUxujKCDgIOaCeu5sG5Y4SaGe6g+ffpoi7bCz9WZ6yLkuEzs2FKivDAi/uZGCDrWq/AYgz3mYbZlR9Mi2grs2THCmaBOS9TddFjIOZ+NWQJHeRG3U05k/PlthKBPmTKFNm3aRIhbHuYEdRo2e4CcMp2B1knIcZtiezGchcBEVpK/CAQu6FCn3XXXXUrI27Rp42/tfX4boq1AowByyvSkm5BDfTGDkSgvwbR84IKOhkc699xzg0HAp7fCPZSX0VZ0V0M3Icflc7qdwpJNkj8IBCroGM0nTJignCyEXZ0G1SESk1L+NK+7t+gm5Lg0WLpIlBdGw588UEHn0bxbt27+1Dagt2CDzoIFCxQZhZHSlqSbkON6S5QXRsK/PDBBd47mJrPPOppi6dKlgbmHclN+Lwg5Lo9EeWEk/MkDE/SojOawZ1+9ejXdd999RqvT4nU3Lwg5fg9mNuJ2itHwPg9E0KMymkOdVl5erlh2P6Kt6O4uXhFyXE6O8oKQTkLMMSre5IEI+vTp01VteGroTdWCfyoIONgHgHyyNXlFyAEPLNmgU5coL973Dt8FHV/uyZMnh15v7nQPZdrutHS6lVeEHJeB3U7BNBYzPUneIOC7oP/tb39To1zYmXY4e4Q9O0gnmxPPuuAJx6sEh5hQt2G/gyRvEPBd0EHCwXXzSSed5E2NDHgqhALqNPi8s0mdFg86Lwk5fp9EeWEkvMt9FXR8taFqwkgX1sQEHPbUm+QeKlO8vSbkuFwc5QXEnCT9CPgq6FOnTlX7zTt06KC/JoY8EdFWoE4L08YNLwk5bjYQc1C3vfTSS2rjDx+XXA8CvvmMA9GCTSu8g0lP8c16CojG2bNn05VXXkmTJk0yq3AuSqPTh1xjxbjiiiuoqqpKLX3CbkjVGBY6z/s2oqOzIGF9HtYE91AY0YOOtqIbXz8IOS4zPpBY4kmUF0ZET+6boBcWFirHEmHdvMLuoRA6GORSmJIfhBzjBVUktvEOHz5c1G0MiobcF0HHPmyQcF27dtVQZDMfsWLFCkUymhBtRTdCfhFyXG6O8iJupxgR97kvgo74aUhhJeGgTsOHzHT3UG66ix+EHJcPHxZEeQEDX1lZyYcld4GAL2TcmWeeSZj+DRgwwEVRzbwV6jR8yM4++2yaM2eOmYXUUCo/CTkUF8QmfPvDK3CYcdXQNCk9wvMRHcQKCKqwjuaoG+zZw8Syx+s5fhJyeD8YdxhXwQ4e0V4kuUPAc0GHWSO+ymH07opRx9RoK+66RcO7/STk+O3Y8SdRXhgNd7nngv7MM89Qjx493JXS0Lvhhx6JySNDi6mlWH4TclxoifLCSLjLj3R3e/K7MW1HMm00r921kqY9/w5tbFD8f6PuF11APbI70WltmjU46zwAdRrs2UEaQQiikPwk5BhPZ5SXQYMGhU51yfX0Ovd0RJ83b54qv2kbWJq06UnXjX+QxlzWleiMy2jMw+Np/PiH6f4xA6nlZ8X06tRiWrnzQFLsFy5cqHyUIzRwVJLXW1YT4cgONZ999tlEl8jxRhDwVNCxfsVX2I7UhI5u04UG/+dAOuPblVT2cRUlEnVEW4E9O8iiKJlp+k3Icb/BjEmivDAameWeCTr0n/izjW1v0rI1nXgc0bc7dtPe2oagQp0GnTlIIhvdQzWsUepHgiDkuHTsdipMm4W4bn7kngk62ypnZWX5UQ9t76jdu5t2fEt03ImtqWWTho9l91A2RFtpWHp3R4Ii5FBqzJyAOdRtvG/CXW2idbdngl5aWqps242PjPrlNtq6q26SXrv3C1r2wTLaSGdSv7OyKJaOQ7QVfMBgAWezeyg3XTwIQo7LK1FeGIn0c08s43hL6ogRI6hLly7pl8qXO36iXSvn0PPvrD3sbcd1v4gu6dedsrNaUuyADpb9u+++o6Kiosgw7YeBQ6RGU2w42bdvXyD8BJaD2AGJzUNjx46NLZ78nwABT9RrIKqQ2rVrl+C1Bh0G635tT2oTK9UxRbQ12kpMNVz/6yTkgpjV4J0Q8jvuuENFpA2iDK5BDOABnkzdy8rKlDVcmKKjYq853EOBFIpyCpKQY9w5ygu80UhKDQFPBP29994zeMqeGjDOqxBtBYIOMihK6jQnBvw7SELOWQaJ8sJopJZrF3SszxcvXkzms+0/0Q/f/0j00wH64cc4erRD+MGe/eOPP7Y22kpq3SC9q4Ik5LikcLyJGZZEeWFEkufaBd2G9TlMYF9/dBJNmf8Z0ZZimvL4M/T6yp0UT9zXrVun7AGuuuqq5EhG6GxQFnKxEEuUl1hEEv+vnYxbtGiRWp+b7DKqzgS2Z2JUDp2BPTsCTiD169dPWWdhjR716XvQhBw3HEd5gRYAbRWVPQdc/3Ry7SM69g537NgxnTIYey3W5n379qVt27bV+zGDMwQsTaKcTCDkGH92xPnCCy/wIcnjIKBV0LGexWaPMDhHhHuokpIStQX15JNPVo4lKioq1GwFnnJuueWWyLo5MoGQ476MvoYdhA899FBk24OxSJZrFXQIApIV+vMkqMCe/ZNPPlFkjzPaCnS2cGuEDRarVq1ShhvPPfdcJL2VmkDIcRNiByFmXhLlhRFpmGsVdHh7RTr++OMbvsmiI3APBXVaog0UEH5YybHhxtChQyNnf20KIYduBc5kwoQJEuUliYxpFXRM22GeaLx9exJAnO6hQPYkSuhcMMHEFB+jGwghRBmBKi4KyUnImVDfnJwcNQMbN26ccixpQplMKoNWQf/ggw8I61mbE6bsSEzyNFYXrBHhGBLaBqSePXtSXl6eIvAau9fm8yYRcowjR3mZMWMGH5L8EALaBB2GMtA5t23b1lpwoaLBhhWQO+kSithZ9cYbb9Q7SID7LASVxAwhjMkkQo7xBYcC68VRo0ZFkjdhHOLl2gR9586d6vk2E3HgGEDqZOoeCtN5rN/xwcBWVnQ4qOPC6q7YJEKOOzc76pQoL4xIXa5N0NesWaOeaCsRh7U2NuOA1HFrEIPR7sEHHyRWxw0ZMiSU6jiTCDnu1sCe3U5JlBdGhUiboGPaDv/tNhJxUKfBCAa20yB1dCVWx2E5wOq4iRMnhmZaaRohx+3GobnBlUiqQ0CboCNaia0WcVCn4evvVbQVfDzYNTQMO1gdZ/v63URCDt0aMzKJ8nL4J06boCMsso3rc6c6zUsnBuh8N9xww2HquJEjR1ptTmsiIcfdW6K8MBJ1uRZBB+OO1KxZrJe1w19m4n8Id4zEJI7XZWR1HJxMIsGc1mZ1nImEHLchGHgEESkoKOBDkc21CLqtjDvYcdizg7zB6ORngjEOq+NAAkIdB3Na26bzJhJy3I6YoUH7AbdT2JgU5aRF0Ddv3qwwbNGihVVYfvjhh0qdFpR7KFbHvfvuu/XmtLap40wl5LgjIsoLVKZRj/KiRdB3796tcHWrluLG8SNn91AmRFvBbALmtFDHISAl1HEwp7VBPWQqIcd9CNiy26momCdz3Z25FkHH1BM27rYk3p1mWrQVTDWnTJmizGmrqqoUpli/MwdiIr4mE3KMF2ZsUJ0m2qTE14U51yLoBw8eJJs8vsICDiM6yBoTExhjVsdh6yWwRXQSU9fvJhNyaF/MNNHWUY7yokXQX3zxRWtUa4i2ggbHFlMv1WluPyCsjgNhiE6K3XGmercxmZDjdnBGeTF5hsTl1Z1rEXQUyhbVGruHgm9wGxKmxjDkgToOloesjjNp/W46IcftjKUa1G3Tpk3jQ5HJXQs6qy2OPfZY40FDtBUw3CBnIEA2Jajj2LsNcyKmqONMJ+S4nTGDY2chJn0ouXxe5q4FHTG4kExXrYGAgytq26OtxHq3wXQ+6OiiNhByLES5ublK3Ra1KC+uBf2rr75iDI3Ot27dquKaY71rkxowHqgoP6vjTPFuYzohxzgCO1a3Rcmbr2tB37Fjh8LQZNYdbDWHVAIpE5aEqSh7t4E6jr3bBEE22UDIcbtjVoSZXZTcTrkWdAbP5Pzzzz9X3m9SdQ9lcl3ilY3VcbwPGx9dv73b2ELIMX7QqYOYQ7z7KCTXgg4hAhtsaoJ6aubMmYqESdc9lKl1ilcuTEnjebfxa3pqCyHH2DmjvAQxA+Jy+JW7FvSNGzcavQ99y5YtinwBCROFBGLM6d3Gr2ATNhFy3A94x2IUory4FnQGzcQc7qEQOw0NajsBly6+WL9DHeendxtbCDnGEh+nqER5cS3ovKGFwTMlhzpt6dKlinTBlDaqib3bQH/stXcbmwg57g9RifLiWtCxp9pEzzJQp8EKLsobGbgzszrOGWwC3m107+ayjZADPsCGo7yE1Vsv6ula0PEQ08xfoU577bXXlI04SBdJdQiwdxuvgk3YRshxv+AoL/n5+cZuHOKyZpprEfRMX+7VfXD2iMRki1fvsfW5UMfFerfRoY6zkZDjNgx7lJfQCTrUabNnz1YkCzqepPgIsDoOtv86g03YRsgxOiAvGYcwqttcCToDctRRRzFegefw0uIm2krgFfC5APgYsjpOh3ebxgm5nVR6Xx9q0qRJ3d+QQqqsSVLpmvVUOKT9oevb05DC9ZTs8iRPavQU3E4hhTHKiytBZ6eQrVu3bhREPy7Yvn07zZ8/X0u0FT/Ka9I7MKLBuw3UcezdJpNgE40Tcm1p0BPldHDb2zQ6i4iKZ9G8lXWuyBriUUN7yqZSfnE1UU4BVRysoqLRXfQQSw1fpnY0snVh2Ha3uRL0OFgFdgjqtCVLlih1ms5oK4FVKKAXszqO9cvperdJlZBrelxranv2tXTDwBX01P+V0vZ4w3TNFzT/re108bXdfEMjrFFeQiPoUKchLJRX0VZ862kGvAjrdwSbcHq3STXYRFqE3FH96b/GDCMqfJOKNuyPqXkN7V05l+Z2GkaXn+JfvADUnaO8BL39NwYQV/+GQtBZnQYyBVNQSXoQgNCydxs8MVXvNqkTcsdQh4uG0fVZMyi/YAntcRa75gsqnrKdLh/Zh/xeGEIrAW80WKub6qfPCVUqv0Mh6BjJkZhMSaXick3qCKTr3aZxQs7x7lb9KXfC1VQ96QWaWfnzqF6zoYz+3mk45Zzs32juKJWywQhTlBfrBR3uoWDPHkS0FWfHiMJvmBLHBpuIN71tnJBzotWcOl91G+VlLaJX531Ce9WpnVRWsJTO/013aum81MffmBmy2yl2l+bj67W/ynpBh+tmqNOCiraivUUMfyCm8+zdBuo4eKeNDTaRKiFXX9VWvWjEPb2o7Kk59NGeGqqpnEdP0Qi6qnPz+kuC+AEHomGJ8mK1oCNUM/ZbmxBtJYiOGOQ7WR0Hc1pWx3GwCSbkcC61NW5r6jlyNI2mV+iJmQuppOAjGpbbn1oFWUEipW5jt1O69wX4XTVrBR3qNLhAhg+4MLmH8rsDuH1fPO820L9DC4IPMJyGxpvex7636cn/Ttdd356Kcy+kUQaM5lw+Z5SX1D5afKdZubWCjsCOmLaDHZUULAJsTgt1HCKXYjss+xJEyTC9jzVAqfl2N+08rNhtaWDu7ZRDl9I9I3oFPppz0VA3jvJis9spKwUd0VbghN/0aCvcWaKSY8oO1Vq8tGbNmkOH60xgjzhlOBX+PZeyj+hD95XWiXzTToPp9j/eTiN7QqFWQ3tKH6T2TdrR4EnLiYpxrbcmsPHKjWOYtUDYoW5js+9E15p63EpBx2gOksSWaCumNr4X5TrxxBPjPhajYZ2Q1JnA1tbWUt1fOT0xqG3dPU070OV3/gedrHplU2o1aCJV1V+H6701gY1b8EMH4VjU5igv1gk6/MjDRRJIEowgksxCoFevXuoj7CwVWHgETBg6dGhK63Xnvab8xl5+VrfFLkNMKWOyclgl6CDgsC63PdpKsgax/RzWtCDfYKWI6S5yaEac3m2gjrORxeYoL4hwa1tqUov5U4YJXzbERR8zZowvYZOxO62goECx7eI5JsNGC/g2CD0EBRFt8SHAlNgmN9z4iIFchOrQJm2PNSM6VBvLli1TnUOEPGBpdfF6CAd7t4HAn3rqqb4Hm3BRfOU738YoL9YI+pdffklgbsMabcVN57PtXqc6jr26IFikLc4Z2e2UTeo2KwQd+lnsjwYZYtM0zzYB9Lu8IFPZuw2i/QwZMoRuueWWBjp3v8vV2PtgFYhlB6bwtqjbtAj6jz/+2Bg2rs4jGgzUaVGJtuIKLAtvhuD4HWzCLUzseNSWKC+uBB0NhORlEAeo07BjCsBiyicpvAjEerdhdZyJpqeYjbAXHhvUba4E3esuB3XawoULIx9txWucTXo+PubwbuNUx6Xq3cbvenCUF2zmMT0ZLeiwl4beXNxDmd6N9JePg01g4xISe7cxaW84PkqI8gJVoelEorGCjukadOYgPXiJoL87yRNNRwCqVFbHlZWVKXXcc889l0OLotUAAAiISURBVOL2V+9rh+UGNlaZHuVFi6AfOHBAO6KfffaZeiaTHtpfIA+0BgFWx8V6tzFlFMVgBDv4GTNmGIupa0G/6667CISZzgR1GlhYkB1iz64TWbufhb7g9G4DdVysd5sgaogZJ9sDmKpucy3oRx99tHZsV69erdRpIDskCQKxCECwEGwinneb2Gv9+p8dk5oa5cW1oOsGEtFfwLSLeyjdyIbveezdBjM/mNOmG2xCJyKYbXCUFxM37LgWdIRj0qVHZ3UayA2bNgzo7DDyrPQQYHWcM9gEzGmxecbvxFFeHnnkEb9f3ej7XAt6ly5daO3atY2+KJULsL6BPTvIDUmCQDoIYETlYBMwp2V1nJ/GLPjomBrlxbWgp9MYya6FeyiYE4LUEHVaMqTkXDIE0g02kexZmZzDTBQDlWlRXlwLesuWdS723ZopsiEEkxqZgCz3CAKMAIJNLFiwoN4rDKbzqXij5fvd5Fh6mhblxbWgd+zYUWGyb9++jLHBGn/mzJkSbSVjBOXGeAhgKs3qODit5GATXpNlmJGy2ykewOKVz89jrgVdR2EBvLiH0oGkPCMeAhA8rN9ZHdezZ0/iYBPxrtdxzLQoL64FvW3bOg+emTLvcA8FdRrWNfgCSxIEvEKA1XGsBoM6burUqZ6Y04IchFUn1H5BaABiMXQt6KgQUiZ70qFOKy8vV0Iu6rTYppH/vUAAgwnW71DHsTWbV+o4jvICYXfLYbnFwrWgcwEysXfHaL5q1SqJtsIgSu4bAhignN5toI7T7d0GHxXo1LG7LWi3U1oEHdPudO3dne6hRJ3mW/+WF8UggL7npXcbqPtY3RakHbwWQY/BLqV/v/nmG4m2khJScpEfCLB3G7DliB2n07sNR3kJ0u2UFkFHJI4lS5ak3B4Y/V977TWJtpIyYnKhHwiwOi7Wu41bdRxHecEHxE9LPSdmWgQdJoepJhBwFRUV4h4qVcDkOt8RYO82UMchsTrOjU486CgvWgSdA+vBjLWxBMOa0tJSRVI0dq2cFwSCRACaoFjvNpmq4zBbgNspxKALQt2mRdDbtWun2qMxFRtUDO+//74iJ0BSSBIETEeA1XHwbuNUx2Xi3QY8QFBRXrQIeqpGM4i2AnWaRFsxvXtL+WIRcKrjevTooYJNZOLdJqgoL1oEPRWjGUzZJdpKbPeR/21DAOo4eLcpKiqiqqoqFWR04sSJKUdswf1Qt/kd5cVVNFVnI+HrBjb9oosuch6u/w3msrq6Wu0ownRIkiBgOwJYisIh5KhRo1RVYFoLa7jG+jf06TC/hcD75cpcy4iOWuJLlaiCIOnmzZunyIhE19je6FL+6CGAvoxgE07vNqkEm8AMmN1f+aVu0ybo3bp1o5KSkgatDXUaCDiQECAjJAkCYUMAgsvebVC3VLzb+B3lRZugswMKCLYzYZoCAs6vKYrz3fJbEPATgXS822A2wG6nMmHw062XNkHHiI4E01ZOWMPA7A9rEUztJQkCUUAAu+Nig03E824DPb1fUV60CXqLFi1UGzo9zYB8Q5JoK1Ho3lJHJwKYzrN3G6jj2LtN7Jocg6AfUV60CTrMBpG+++47lUPgYc8OJpLVb04g5LcgEAUEWB3H3m2ys7MP826D82yI48bEtjEstanX8CLs54XrZ6jYULFvv/1W1GmNtYCcjwwCWMpiXzpGdyRWx+G41+o2bSM6Cn7CCScQHFBAwGHPLtFWItOHpaIpIAACjr3bsNEMvNusW7fO8ygvWkd0EA74WmFN0rdvX2VBlEL95RJBIJIIYL0OJ5XwQHPjjTcStsdCewVHGLqTVkHHrhzoEJGwFVWYdt3NJc8LIwIYIBHwAaQcEoxpYIijM2kVdHyh+vXrpy0Wm86KyrMEAVsQwHS+rKxMa3G1CjpKBpv2TZs2aS2kPEwQiAoCMKfFYNm9e3etVdYu6FpLJw8TBAQBLQhoZd21lEgeIggIAtoRcAh6De0pfZDaN2lCTeL+nU2jJk+n0srG3UVpL6U8UBAQBH5GoKaals+dRpNHnX1IVtvThfdNobnLq+mnyiKaW7n/52sP/XIIelNqNWgiVdV+RSV5vYlyCqjiYC3V1uLvAFWV59FJfx9HgweOo8L1IuwNkJQDgoAPCNRUL6anb8qhy2ZV0el3FtNBJZ9VVHJ3Lzr4/v30y+wC2hWnHA5Bj3O2/tDRlNX7enrsLxMop7qQ8l8uJxH1enDkhyDgDwI1W2hO/m10z9bR9M7z99Cw3lnEAtw0qzcNu/eP9M6Tx1DFtr0NynNkgyNyQBAQBAxEoIb2lP2FxhY2o7yS66l3SxZxZ1FbU+///h/assJ5rO53yoJeU/0RvVownYqzRlPBTX2oVcNnyRFBQBDwCoGaSpr5xCtUTQMo+5SWid/S6jwaNrDh6cSCXpxL2UfkxtzRm/JKXqfRXUTMY4CRfwUBbxHYW0UVq6uJsjpRx5OOTvtd8cb/uoc0IOPepidvOECTBl9IowrXUMNVQNrvlhsEAUHAJwQSC/phBQAZN4zufXkGFeTsoqm5j9JbcSj8w26RfwQBQUAfAi3bU/bZWRk/L0VBP/T8pm2p4zmIs7YpLrOXcSnkRkFAEEiOQNOONOCaAUTVxVRU/nXya+OcTU/Qa76j3TsPENHpyQmBOC+SQ4KAIOAGgebU+arbKC9rOU3631dp+d6auA+rqV5JZXGM2lIW9Jrq5TTrqfE0tnAXDXzgFrqkU/O4L5KDgoAg4BECrQbSw6Wv0A0V91Cfyx6gwtJKB1e2hypLp9ErS5tRn85xyPLa+nSw9puSB2qziGop0d/AvNqCOeW1VQfrb5IfgoAg4DMCB6vKa+cU5NUOrJfTrNqBeS/Wzimvqk0kmrJ7zaOPrzxWEDAJgZSn7iYVWsoiCAgC6SEggp4eXnK1IGAlAv8PPbGxMHRxc+wAAAAASUVORK5CYII="></p>
<p>The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find AM.</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>Find \({\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}\) in radians.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>PC \( = \frac{{\sqrt 3 }}{2}\) or 0.8660 <em><strong>(M1)</strong></em></p>
<p>PM \( = \frac{1}{2}\)PC \( = \frac{{\sqrt 3 }}{4}\) or 0.4330 <strong>(A1)</strong></p>
<p>AM \( = \sqrt {\frac{1}{4} + \frac{3}{{16}}} \)</p>
<p>\( = \frac{{\sqrt 7 }}{4}\) or 0.661 (m) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using the cosine rule</p>
<p>AM<sup>2</sup> \( = {1^2} + {\left( {\frac{{\sqrt 3 }}{4}} \right)^2} - 2 \times \frac{{\sqrt 3 }}{4} \times {\text{cos}}\left( {30^\circ } \right)\) <em><strong>M1A1</strong></em></p>
<p>AM \( = \frac{{\sqrt 7 }}{4}\) or 0.661 (m) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>tan (\({\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}\)) \( = \frac{2}{{\sqrt 3 }}\) or equivalent <em><strong>(M1)</strong></em></p>
<p>= 0.857 <em><strong>A1</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><strong>EITHER</strong></p>
<p>\(\frac{1}{2}{\text{A}}{{\text{M}}^2}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - {\text{sin}}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}} \right)} \right)\) <em><strong>(M1)A1</strong></em></p>
<p><strong>OR </strong></p>
<p>\(\frac{1}{2}{\text{A}}{{\text{M}}^2} \times 2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - = \frac{{\sqrt 3 }}{8}\) <em><strong>(M1)A1</strong></em></p>
<p>= 0.158(m<sup>2</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to calculate area of a sector minus area of a triangle.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p 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 following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the smaller circle lies on the circumference of the bigger circle. O is the centre of the bigger circle and the two circles intersect at points A and B.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-15_om_11.00.51.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;"> </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:</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) \({\rm{B\hat OC}}\);</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) the area of the shaded region.</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) <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;">\(2{\text{arcsin}}\left( {\frac{{1.5}}{4}} \right)\) <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;">\(\alpha = {0.769^c}{\text{ (44.0}}^\circ {\text{)}}\) <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 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;">using the cosine rule:</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;">\({3^2} = {4^2} + {4^2} - 2(4)(4)\cos \alpha \) <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;">\(\alpha = {0.769^c}{\text{ (44.0}}^\circ {\text{)}}\) <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>
<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) one segment</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;">\({{\text{A}}_1} = \frac{1}{2} \times {4^2} \times 0.76879 - \frac{1}{2} \times {4^2}\sin (0.76879)\) <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;">\( = 0.58819{\text{K}}\) <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;">\(2{{\text{A}}_1} = 1.18{\text{ }}({\text{c}}{{\text{m}}^2})\) <strong><em>A1</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>only if both sector and triangle are considered.</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>[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;"><strong><em>Total [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 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;">Triangle ABC has AB = 5 cm, BC = 6 cm and area 10 \({\text{c}}{{\text{m}}^2}\).</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;">(a) Find \(\sin \hat B\).</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) <strong>Hence</strong>, find the two possible values of AC, giving your answers correct to two decimal places.</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;">(a) area \( = \frac{1}{2} \times {\text{BC}} \times {\text{AB}} \times \sin B\) <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;">\(\left( {10 = \frac{1}{2} \times 5 \times 6 \times \sin B} \right)\)</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;">\(\sin \hat B = \frac{2}{3}\) <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> </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;">(b) \(\cos B = \pm \frac{{\sqrt 5 }}{3}{\text{ }}( = \pm 0.7453 \ldots ){\text{ or }}B = 41.8 \ldots {\text{ and }}138.1 \ldots \) <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;">\({\text{A}}{{\text{C}}^2} = {\text{B}}{{\text{C}}^2} + {\text{A}}{{\text{B}}^2} - 2 \times {\text{BC}} \times {\text{AB}} \times \cos B\) <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;">\({\text{AC}} = \sqrt {{5^2} + {6^2} - 2 \times 5 \times 6 \times 0.7453 \ldots } {\text{ or }}\sqrt {{5^2} + {6^2} + 2 \times 5 \times 6 \times 0.7453 \ldots } \)</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;">\({\text{AC}} = 4.03{\text{ or }}10.28\) <strong><em>A1A1</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>[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 attempted this question and part (a) was answered correctly by most candidates but in (b), despite the wording of the question, the obtuse angle was often omitted leading to only one solution.</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;">In many cases early rounding led to inaccuracy in the final answers and many candidates failed to round their answers to two decimal places as required.</span></p>
</div>
<br><hr><br><div class="question">
<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;">In a triangle ABC, \(\hat A = 35^\circ \), BC = 4 cm and AC = 6.5 cm. Find the possible values of \(\hat B\) and the corresponding values of AB.</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 Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sin B}}{{6.5}} = \frac{{\sin 35^\circ }}{4}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\hat B = 68.8^\circ {\text{ or }}111^\circ \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\hat C = 76.2^\circ \) or \(33.8^\circ \) (accept \(34^\circ \)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{AB}}}}{{\sin C}} = \frac{{{\text{BC}}}}{{\sin A}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{AB}}}}{{\sin 76.2^\circ }} = \frac{4}{{\sin 35^\circ }}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">AB = 6.77 cm <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{AB}}}}{{\sin 33.8^\circ }} = \frac{4}{{\sin 35^\circ }}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">AB = 3.88 cm\(\,\,\,\,\,\)(accept 3.90) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 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;">Most candidates realised that the sine rule was the best option although some used the more difficult cosine rule which was an alternative method. Many candidates failed to realise that there were two solutions even though the question suggested this. Many candidates were given an arithmetic penalty for giving one of the possible of values \({\hat B}\) as 112.2° instead of 111°.</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\arctan \frac{1}{2} - \arctan \frac{1}{3} = \arctan a,{\text{ }}a \in {\mathbb{Q}^ + }\), find the value of <em>a</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, solve the equation \(\arcsin x = \arctan a\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \left( {\arctan \frac{1}{2} - \arctan \frac{1}{3}} \right) = \tan (\arctan a)\) <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;">\(a = 0.14285 \ldots = \frac{1}{7}\) <strong><em>(A1)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">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;">\(\arctan \left( {\frac{1}{7}} \right) = \arcsin (x) \Rightarrow x = \sin \left( {\arctan \frac{1}{7}} \right) \approx 0.141\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px 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;"> Accept exact value of \(\left( {\frac{1}{{\sqrt {50} }}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px 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: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<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 candidates failed to give the answer for (a) in rational form. The GDC can render the answer in this form as well as the decimal approximation, but this was obviously missed by many 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: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) was generally answered successfully.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the triangle \({\text{PQR}}\) where \({\rm{Q\hat PR = 30^\circ }}\), \({\text{PQ}} = (x + 2){\text{ cm}}\) and \({\text{PR}} = {(5 - x)^2}{\text{ cm}}\), where \( - 2 < x < 5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the area, \(A\;{\text{c}}{{\text{m}}^2}\), of the triangle is given by \(A = \frac{1}{4}({x^3} - 8{x^2} + 5x + 50)\).</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">(i) <span class="Apple-converted-space"> </span>State \(\frac{{{\text{d}}A}}{{{\text{d}}x}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Verify that \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\).</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">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}\) and hence justify that \(x = \frac{1}{3}\) gives the maximum area of triangle \(PQR\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State the maximum area of triangle \(PQR\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find \(QR\) when the area of triangle \(PQR\) is a maximum.</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 class="p1">use of \(A = \frac{1}{2}qr\sin \theta \) to obtain \(A = \frac{1}{2}(x + 2){(5 - x)^2}\sin 30^\circ \) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\( = \frac{1}{4}(x + 2)(25 - 10x + {x^2})\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(A = \frac{1}{4}({x^3} - 8{x^2} + 5x + 50)\) <span class="Apple-converted-space"> </span><strong><em>AG</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">(i) <span class="Apple-converted-space"> </span>\(\frac{{{\text{d}}A}}{{{\text{d}}x}} = \frac{1}{4}(3{x^2} - 16x + 5) = \frac{1}{4}(3x - 1)(x - 5)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">\(\frac{{{\text{d}}A}}{{{\text{d}}x}} = \frac{1}{4}\left( {3{{\left( {\frac{1}{3}} \right)}^2} - 16\left( {\frac{1}{3}} \right) + 5} \right) = 0\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">\(\frac{{{\text{d}}A}}{{{\text{d}}x}} = \frac{1}{4}\left( {3\left( {\frac{1}{3}} \right) - 1} \right)\left( {\left( {\frac{1}{3}} \right) - 5} \right) = 0\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1">so \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">solving \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) for \(x\)<em> <span class="Apple-converted-space"> </span></em><strong><em>M1</em></strong></p>
<p class="p1">\( - 2 < x < 5 \Rightarrow x = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1">a correct graph of \(\frac{{{\text{d}}A}}{{{\text{d}}x}}\) versus \(x\)<em> <span class="Apple-converted-space"> </span></em><strong><em>M1</em></strong></p>
<p class="p1">the graph clearly showing that \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\) <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}} = \frac{1}{2}(3x - 8)\) <strong><em>A1</em></strong></p>
<p>for \(x = \frac{1}{3},{\text{ }}\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}} = - 3.5{\text{ }}( < 0)\) <strong><em>R1</em></strong></p>
<p>so \(x = \frac{1}{3}\) gives the maximum area of triangle \(PQR\) <strong><em>AG</em></strong></p>
<p>(ii) \({A_{\max }} = \frac{{343}}{{27}}{\text{ }}( = 12.7){\text{ (c}}{{\text{m}}^2}{\text{)}}\) <strong><em>A1</em></strong></p>
<p>(iii) \({\text{PQ}} = \frac{7}{3}{\text{ (cm)}}\) and \({\text{PR}} = {\left( {\frac{{14}}{3}} \right)^2}{\text{ (cm)}}\) <strong><em>(A1)</em></strong></p>
<p>\({\text{Q}}{{\text{R}}^2} = {\left( {\frac{7}{3}} \right)^2} + {\left( {\frac{{14}}{3}} \right)^4} - 2\left( {\frac{7}{3}} \right){\left( {\frac{{14}}{3}} \right)^2}\cos 30^\circ \) <strong><em>(M1)(A1)</em></strong></p>
<p>\( = 391.702 \ldots \)</p>
<p>\({\text{QR = 19.8 (cm)}}\) <strong><em>A1</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
<p><strong><em>Total [12 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">This question was generally well done. Parts (a) and (b) were straightforward and well answered.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally well done. Parts (a) and (b) were straightforward and well answered.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This question was generally well done. Parts (c) (i) and (ii) were also well answered with most candidates correctly applying the second derivative test and displaying sound reasoning skills.</p>
<p>Part (c) (iii) required the use of the cosine rule and was reasonably well done. The most common error committed by candidates in attempting to find the value of \(QR\) was to use \({\text{PR}} = \frac{{14}}{3}{\text{ (cm)}}\) rather than \({\text{PR}} = {\left( {\frac{{14}}{3}} \right)^2}{\text{ (cm)}}\). The occasional candidate used \(\cos 30^\circ = \frac{1}{2}\).</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f(x) = 3\sin x + 4\cos x\) is defined for \(0 < x < 2\pi \) .</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;">Write down the coordinates of the minimum point on the graph 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><span style="font-family: 'times new roman', times; font-size: medium;">The points \({\text{P}}(p,{\text{ }}3)\) and \({\text{Q}}(q,{\text{ }}3){\text{, }}q > p\), lie on the graph of \(y = f(x)\) .</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;">Find <em>p </em>and <em>q </em>.</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 coordinates of the point of intersection of the normals to the graph at the points P and Q.</span></p>
<div class="marks">[7]</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;">\((3.79, - 5)\) <span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>A1 </em></strong></span></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>[1 mark] </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;">\(p = 1.57{\text{ or }}\frac{\pi }{2},{\text{ }}q = 6.00\) <strong><em>A1A1</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;">[2 marks]</span><br></em></strong></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;">\(f'(x) = 3\cos x - 4\sin x\) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(3\cos x - 4\sin x = 3 \Rightarrow x = 4.43...\) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\((y = -4)\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Coordinates are \((4.43, -4)\)</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;">[4 marks]</span><br></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;">\({m_{{\text{normal}}}} = \frac{1}{{{m_{{\text{tangent}}}}}}\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">gradient at P is \( - 4\) so gradient of normal at P is \(\frac{1}{4}\) </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;"><em>(A1)</em></strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">gradient at Q is 4 so gradient of normal at Q is \( - \frac{1}{4}\) </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;"><em>(A1)</em></strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">equation of normal at P is \(y - 3 = \frac{1}{4}(x - 1.570...){\text{ }}({\text{or }}y = 0.25x + 2.60...)\) </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;"><em>(M1)</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;">equation of normal at Q is \(y - 3 = \frac{1}{4}(x - 5.999...){\text{ }}({\text{or }}y = -0.25x + \underbrace {4.499...}_{})\) <strong><em>(M1)</em></strong></span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px;"> </span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Award the previous two </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;"><em>M1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">even if the gradients are incorrect in \(y - b = m(x - a)\) where \((a,b)\) are coordinates of P and Q (or in \(y = mx + c\) with </span><em style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">c </em><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">determined using coordinates of P and Q.</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;">intersect at \((3.79,{\text{ }}3.55)\) <strong><em>A1A1</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>Award <strong><em>N2 </em></strong>for 3.79 without other working.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px;"> </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]<br></em></strong></span></p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<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 answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.</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;">Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.</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;">Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.</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: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates answered parts (a) and (b) of this question well and, although many were also successful in part (c), just a few candidates gave answers to the required level of accuracy. Part d) was rather challenging for many candidates. The most common errors among the candidates who attempted this question were the confusion between tangents and normals and incorrect final answers due to premature rounding.</span></p>
<div class="question_part_label">d.</div>
</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;">A circle of radius 4 cm , centre O , is cut by a chord [AB] of length 6 cm.</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;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\rm{A\hat OB}}\), expressing your answer in radians correct to four significant figures.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the area of the shaded region.</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</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;">\({\rm{A\hat OB}} = 2\arcsin \left( {\frac{3}{4}} \right)\) or equivalent (<em>eg</em> \({\rm{A\hat OB}} = 2\arctan \left( {\frac{3}{{\sqrt 7 }}} \right),{\rm{ A\hat OB}} = 2\arccos \left( {\frac{{\sqrt 7 }}{4}} \right)\)) <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;"><strong>OR</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;">\(\cos {\rm{A\hat OB}} = \frac{{{4^2} + {4^2} - {6^2}}}{{2 \times 4 \times 4}}{\text{ }}\left( { = - \frac{1}{8}} \right)\) <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;"><strong>THEN</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;">\( = 1.696\) (correct to 4sf) <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>[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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">use of area of segment = area of sector – area of triangle <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;">\( = \frac{1}{2} \times {4^2} \times 1.696 - \frac{1}{2} \times {4^2} \times \sin 1.696\) <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;">\( = 5.63{\text{ (c}}{{\text{m}}^2})\) <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>[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;">This was generally well done. In part (a), a number of candidates expressed the required angle either in degrees or in radians stated to an incorrect number of significant figures.</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;">This was generally well done. In part (b), some candidates demonstrated a correct method to calculate the shaded area using an incorrect formula for the area of a sector.</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the triangle ABC where \({\rm{B\hat AC}} = 70^\circ \), AB = 8 cm and AC = 7 cm. The point D on the side BC is such that \(\frac{{{\text{BD}}}}{{{\text{DC}}}} = 2\).</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;">Determine the length of AD.</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">use of cosine rule: \({\text{BC}} = \sqrt {({8^2} + {7^2} - 2 \times 7 \times 8\cos 70)} = 8.6426 \ldots \) <strong><em>(M1)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>Note:</strong> Accept an expression for \({\text{B}}{{\text{C}}^2}\).</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;"> </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;">\({\text{BD}} = 5.7617 \ldots \,\,\,\,\,{\text{(CD}} = 2.88085 \ldots )\) <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;">use of sine rule: \(\hat B = \arcsin \left( {\frac{{7\sin 70}}{{{\text{BC}}}}} \right) = 49.561 \ldots ^\circ \,\,\,\,\,(\hat C = 60.4387 \ldots ^\circ )\) <strong><em>(M1)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;">use of cosine rule: \({\text{AD}} = \sqrt {{8^2} + {\text{B}}{{\text{D}}^2} - 2 \times {\text{BD}} \times 8\cos B} = 6.12{\text{ (cm)}}\) <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>Note:</strong> Scale drawing method not acceptable.</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;"> </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>[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;">Well done.</span></p>
</div>
<br><hr><br><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;">The depth, <em>h</em>(<em>t</em>) metres, of water at the entrance to a harbour at <em>t</em> hours after midnight on a particular day is given by</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;">\[h(t) = 8 + 4\sin \left( {\frac{{\pi t}}{6}} \right),{\text{ }}0 \leqslant t \leqslant 24.\]</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;">(a) Find the maximum depth and the minimum depth of the water.</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;">(b) Find the values of <em>t</em> for which \(h(t) \geqslant 8\).</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Either finding depths graphically, using \(\sin \frac{{\pi t}}{6} = \pm 1\) or solving \(h'(t) = 0\) for <em>t</em> <strong><em>(M1)</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;">\(h{(t)_{\max }} = 12{\text{ (m), }}h{(t)_{\min }} = 4{\text{ (m)}}\) <strong><em>A1A1</em></strong> <strong><em>N3</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> </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;">(b) Attempting to solve \(8 + 4\sin \frac{{\pi t}}{6} = 8\) algebraically or graphically <strong><em>(M1)</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;">\(t \in [{\text{0}},{\text{6}}] \cup [{\text{12}},{\text{18}}] \cup \{ {\text{24}}\} \)</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>N3</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;"><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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Not as well done as expected with most successful candidates using a graphical approach. Some candidates confused <em>t</em> and <em>h</em> and subsequently stated the values of <em>t</em> for which the water depth was either at a maximum and a minimum. Some candidates simply gave the maximum and minimum coordinates without stating the maximum and minimum depths.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"> </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;">In part (b), a large number of candidates left out <em>t</em> = 24 from their final answer. A number of candidates experienced difficulties solving the inequality via algebraic means. A number of candidates specified incorrect intervals or only one correct interval.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"> </p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Farmer Bill owns a rectangular field, 10 m by 4 m. Bill attaches a rope to a wooden post at one corner of his field, and attaches the other end to his goat Gruff.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to graze. Give your answer correct to the nearest integer.</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>Bill replaces Gruff’s rope with another, this time of length \(a,{\text{ }}4 < a < 10\), so that Gruff can now graze exactly one half of Bill’s field.</p>
<p>Show that \(a\) satisfies the equation</p>
<p>\[{a^2}\arcsin \left( {\frac{4}{a}} \right) + 4\sqrt {{a^2} - 16} = 40.\]</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\).</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"><img src="images/Schermafbeelding_2016-01-06_om_16.06.57.png" alt></p>
<p class="p2"><strong>EITHER</strong></p>
<p class="p1">area of triangle \( = \frac{1}{2} \times 3 \times 4\;\;\;( = 6)\) <span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">area of sector \( = \frac{1}{2}\arcsin \left( {\frac{4}{5}} \right) \times {5^2}\;\;\;( = 11.5911 \ldots )\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><strong>OR</strong></p>
<p class="p1">\(\int_0^4 {\sqrt {25 - {x^2}} {\text{d}}x} \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p2"><strong>THEN</strong></p>
<p class="p1">total area \( = 17.5911 \ldots {\text{ }}{{\text{m}}^2}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">percentage \( = \frac{{17.5911 \ldots }}{{40}} \times 100 = 44\% \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2"><img src="images/Schermafbeelding_2016-01-06_om_16.38.28.png" alt></p>
<p>area of triangle \( = \frac{1}{2} \times 4 \times \sqrt {{a^2} - 16} \) <strong><em>A1</em></strong></p>
<p>\(\theta = \arcsin \left( {\frac{4}{a}} \right)\) <strong><em>(A1)</em></strong></p>
<p>area of sector \( = \frac{1}{2}{r^2}\theta = \frac{1}{2}{a^2}\arcsin \left( {\frac{4}{a}} \right)\) <strong><em>A1</em></strong></p>
<p>therefore total area \( = 2\sqrt {{a^2} - 16} + \frac{1}{2}{a^2}\arcsin \left( {\frac{4}{a}} \right) = 20\) <strong><em>A1</em></strong></p>
<p>rearrange to give: \({a^2}\arcsin \left( {\frac{4}{a}} \right) + 4\sqrt {{a^2} - 16} = 40\) <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\(\int_0^4 {\sqrt {{a^2} - {x^2}} {\text{d}}x = 20} \) <strong><em>M1</em></strong></p>
<p>use substitution \(x = a\sin \theta ,{\text{ }}\frac{{{\text{d}}x}}{{{\text{d}}\theta }} = a\cos \theta \)</p>
<p>\(\int_0^{\arcsin \left( {\frac{4}{a}} \right)} {{a^2}{{\cos }^2}\theta {\text{d}}\theta = 20} \)</p>
<p>\(\frac{{{a^2}}}{2}\int_0^{\arcsin \left( {\frac{4}{a}} \right)} {(\cos 2\theta + 1){\text{d}}\theta = 20} \) <strong><em>M1</em></strong></p>
<p>\({a^2}\left[ {\left( {\frac{{\sin 2\theta }}{2} + \theta } \right)} \right]_0^{\arcsin \left( {\frac{4}{a}} \right)} = 40\) <strong><em>A1</em></strong></p>
<p>\({a^2}\left[ {(\sin \theta \cos \theta + \theta } \right]_0^{\arcsin \left( {\frac{4}{a}} \right)} = 40\)</p>
<p>\({a^2}\arcsin \left( {\frac{4}{a}} \right) + {a^2}\left( {\frac{4}{a}} \right)\sqrt {\left( {1 - {{\left( {\frac{4}{a}} \right)}^2}} \right)} = 40\) <strong><em>A1</em></strong></p>
<p>\({a^2}\arcsin \left( {\frac{4}{a}} \right) + 4\sqrt {{a^2} - 16} = 40\) <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">solving using \({\text{GDC}} \Rightarrow a = 5.53{\text{ cm}}\) <em><strong>A2</strong></em></p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<p class="p1"><em><strong>Total [10 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In triangle \({\text{ABC}}\), \({\text{AB}} = 5{\text{ cm}}\), \({\text{BC}} = 12{\text{ cm}}\) and \({\rm{A\hat BC}} = 100^\circ \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the area of the triangle.</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 \(AC\).</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 class="p1">\(A = \frac{1}{2} \times 5 \times 12 \times \sin 100^\circ \) <span class="Apple-converted-space"> </span><span class="s1"><strong>(M1)</strong></span></p>
<p class="p1">\( = 29.5{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <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">\({\text{A}}{{\text{C}}^2} = {5^2} + {12^2} - 2 \times 5 \times 12 \times \cos 100^\circ \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">therefore \({\text{AC}} = 13.8{\text{ (cm)}}\) <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>
<p class="p1"><span class="s1"><strong><em>Total [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">
<p>Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.<br>“<em>Seaview</em>” \((S)\) is at an angle of depression of 25°.<br>“<em>Nauti Buoy</em>” \((N)\) is at an angle of depression of 35°.<br>The following three dimensional diagram shows Barry and the two yachts at S and N.<br>X lies at the foot of the cliff and angle \({\text{SXN}} = \) 70°.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-08_om_11.45.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/05"></p>
<p>Find, to 3 significant figures, the distance between the two yachts.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to use tan, or sine rule, in triangle BXN or BXS <strong><em>(M1)</em></strong></p>
<p>\({\text{NX}} = 80\tan 55{\rm{^\circ }}\left( { = \frac{{80}}{{\tan 35{\rm{^\circ }}}} = 114.25} \right)\) <strong><em>(A1)</em></strong></p>
<p>\({\text{SX}} = 80\tan 65{\rm{^\circ }}\left( { = \frac{{80}}{{\tan 25{\rm{^\circ }}}} = 171.56} \right)\) <strong><em>(A1)</em></strong></p>
<p>Attempt to use cosine rule <strong><em>M1</em></strong></p>
<p>\({\text{S}}{{\text{N}}^2} = {171.56^2} + {114.25^2} - 2 \times 171.56 \times 114.25\cos 70\)° <strong><em>(A1)</em></strong></p>
<p>\({\text{SN}} = 171{\text{ }}({\text{m}})\) <strong><em>A1</em></strong></p>
<p> </p>
<p>Note: Award final <strong><em>A1 </em></strong>only if the correct answer has been given to 3 significant figures.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p 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;">An electricity station is on the edge of a straight coastline. A lighthouse is located in the sea 200 m from the electricity station. The angle between the coastline and the line joining the lighthouse with the electricity station is 60°. A cable needs to be laid connecting the lighthouse to the electricity station. It is decided to lay the cable in a straight line to the coast and then along the coast to the electricity station. The length of cable laid along the coastline is <em>x</em> metres. This information is illustrated in the diagram below.</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="font: normal normal normal 24px/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: 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;">The cost of laying the cable along the sea bed is US$80 per metre, and the cost of laying it on land is US$20 per metre.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in terms of <em>x</em>, an expression for the cost of laying the cable.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>x</em>, to the nearest metre, such that this cost is minimized.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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 distance the cable is laid along the seabed be <em>y</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;">\({y^2} = {x^2} + {200^2} - 2 \times x \times 200\cos 60^\circ \) <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;">(or equivalent method)</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;">\({y^2} = {x^2} - 200x + 40000\) <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;">cost</span><span style="font-family: 'times new roman', times; font-size: medium;"> = <em>C</em> = 80<em>y</em> + 20<em>x</em> </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;">\(C = 80{({x^2} - 200x + 40000)^{\frac{1}{2}}} + 20x\) <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>[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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 55.2786 \ldots = 55\) (m to the nearest metre) <strong><em>(A1)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;">\(\left( {x = 100 - \sqrt {2000} } \right)\)</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;"><strong><em>[2 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><span style="font-family: times new roman,times; font-size: medium;">Some surprising misconceptions were evident here, using right angled trigonometry in non right angled triangles etc. Those that used the cosine rule, usually managed to obtain the correct answer to part (a).</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;">Some surprising misconceptions were evident here, using right angled trigonometry in non right angled triangles etc. Many students attempted to find the value of the minimum algebraically instead of the simple calculator solution.</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The vertices of an equilateral triangle, with perimeter <em>P</em> and area <em>A</em> , lie on a circle with radius <em>r</em> . Find an expression for \(\frac{P}{A}\) in the form \(\frac{k}{r}\), where \(k \in {\mathbb{Z}^ + }\).</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: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">let the length of one side of the triangle be <em>x</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">consider the triangle consisting of a side of the triangle and two radii</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({x^2} = {r^2} + {r^2} - 2{r^2}\cos 120^\circ \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 3{r^2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><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: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 2r\cos 30^\circ \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><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: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = r\sqrt 3 \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">so perimeter \( = 3\sqrt 3 r\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">now consider the area of the triangle</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">area \( = 3 \times \frac{1}{2}{r^2}\sin 120^\circ \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 3 \times \frac{{\sqrt 3 }}{4}{r^2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{P}{A} = \frac{{3\sqrt 3 r}}{{\frac{{3\sqrt 3 }}{4}{r^2}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{4}{r}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times; color: #3f3f3f;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept alternative methods</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 44.0px Helvetica; color: #3f3f3f;"><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: 32.0px Times; color: #3f3f3f;"><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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It was pleasing to see some very slick solutions to this question. There were various reasons for the less successful attempts: not drawing a diagram; drawing a diagram, but putting one vertex of the triangle at the centre of the circle; drawing the circle inside the triangle; the side of the triangle being denoted by <em>r</em> the symbol used in the question for the radius of the circle.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}\).</p>
</div>
<div class="specification">
<p>Let \(u = \tan x\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for \(f’(x)\) in terms of \(x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of \(y = f’(x)\) for \(0 \leqslant x < \frac{\pi }{2}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(x\)-coordinate(s) of the point(s) of inflexion of the graph of \(y = f(x)\), labelling these clearly on the graph of \(y = f’(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \(\sin x\) in terms of \(\mu \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \(\sin 2x\) in terms of \(u\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \(f(x) = 0\) can be expressed as \({u^3} - 7{u^2} + 15u - 9 = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation \(f(x) = 0\), giving your answers in the form \(\arctan k\) where \(k \in \mathbb{Z}\).</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>\(f’(x) = 4\sin x\cos x + 14\cos 2x + {\sec ^2}x\) (or equivalent) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.47.49.png" alt="N17/5/MATHL/HP2/ENG/TZ0/11.a.ii/M"> <strong><em>A1A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correct behaviour at \(x = 0\), <strong><em>A1 </em></strong>for correct domain and correct behaviour for \(x \to \frac{\pi }{2}\), <strong><em>A1 </em></strong>for two clear intersections with \(x\)-axis and minimum point, <strong><em>A1 </em></strong>for clear maximum point.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(x = 0.0736\) <strong><em>A1</em></strong></p>
<p>\(x = 1.13\) <strong><em>A1</em></strong></p>
<p><em style="font-size: 14px;"><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to write \(\sin x\) in terms of \(u\) only <strong><em>(M1)</em></strong></p>
<p>\(\sin x = \frac{u}{{\sqrt {1 + {u^2}} }}\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\cos x = \frac{1}{{\sqrt {1 + {u^2}} }}\) <strong><em>(A1)</em></strong></p>
<p>attempt to use \(\sin 2x = 2\sin x\cos x{\text{ }}\left( { = 2\frac{u}{{\sqrt {1 + {u^2}} }}\frac{1}{{\sqrt {1 + {u^2}} }}} \right)\) <strong><em>(M1)</em></strong></p>
<p>\(\sin 2x = \frac{{2u}}{{1 + {u^2}}}\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(2{\sin ^2}x + 7\sin 2x + \tan x - 9 = 0\)</p>
<p>\(\frac{{2{u^2}}}{{1 + {u^2}}} + \frac{{14u}}{{1 + {u^2}}} + u - 9{\text{ }}( = 0)\) <strong><em>M1</em></strong></p>
<p>\(\frac{{2{u^2} + 14u + u(1 + {u^2}) - 9(1 + {u^2})}}{{1 + {u^2}}} = 0\) (or equivalent) <strong><em>A1</em></strong></p>
<p>\({u^3} - 7{u^2} + 15u - 9 = 0\) <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(u = 1\) or \(u = 3\) <strong><em>(M1)</em></strong></p>
<p>\(x = \arctan (1)\) <strong><em>A1</em></strong></p>
<p>\(x = \arctan (3)\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only accept answers given the required form.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<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">The diagram shows two circles with centres at the points <span class="s1">A </span>and <span class="s1">B </span>and radii \(2r\) and \(r\), respectively. The point <span class="s1">B </span>lies on the circle with centre <span class="s1">A</span>. The circles intersect at the points <span class="s1">C </span>and <span class="s1">D</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.29.37.png" alt="N16/5/MATHL/HP2/ENG/TZ0/09"></p>
<p class="p1">Let \(\alpha \) be the measure of the angle <span class="s1">CAD </span>and \(\theta \) be the measure of the angle <span class="s1">CBD </span>in radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for the shaded area in terms of \(\alpha \), \(\theta \) and \(r\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\alpha = 4\arcsin \frac{1}{4}\).</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">Hence find the value of \(r\) given that the shaded area is equal to 4.</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 class="p1"><span class="Apple-converted-space">\(A = 2(\alpha - \sin \alpha ){r^2} + \frac{1}{2}(\theta - \sin \theta ){r^2}\) </span><strong><em>M1A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>M1A1A1 </em></strong>for alternative correct expressions <em>eg</em>. \(A = 4\left( {\frac{\alpha }{2} - \sin \frac{\alpha }{2}} \right){r^2} + \frac{1}{2}\theta {r^2}\).</p>
<p class="p2"> </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"><strong>METHOD 1</strong></p>
<p class="p1">consider for example triangle <span class="s1">ADM </span>where <span class="s1">M </span>is the midpoint of <span class="s1">BD <span class="Apple-converted-space"> </span></span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\sin \frac{\alpha }{4} = \frac{1}{4}\) </span><strong><em>A1</em></strong></p>
<p class="p1">\(\frac{\alpha }{4} = \arcsin \frac{1}{4}\)</p>
<p class="p1"><span class="Apple-converted-space">\(\alpha = 4\arcsin \frac{1}{4}\) </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">attempting to use the cosine rule (to obtain \(1 - \cos \frac{\alpha }{2} = \frac{1}{8}\)) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">\(\sin \frac{\alpha }{4} = \frac{1}{4}\) (obtained from \(\sin \frac{\alpha }{4} = \sqrt {\frac{{1 - \cos \frac{\alpha }{2}}}{2}} \)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(\frac{\alpha }{4} = \arcsin \frac{1}{4}\)</p>
<p class="p1"><span class="Apple-converted-space">\(\alpha = 4\arcsin \frac{1}{4}\) </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1">\(\sin \left( {\frac{\pi }{2} - \frac{\alpha }{4}} \right) = 2\sin \frac{\alpha }{2}\) where \(\frac{\theta }{2} = \frac{\pi }{2} - \frac{\alpha }{4}\)</p>
<p class="p1"><span class="Apple-converted-space">\(\cos \frac{\alpha }{4} = 4\sin \frac{\alpha }{4}\cos \frac{\alpha }{4}\) </span><strong><em>M1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>M1 </em></strong>either for use of the double angle formula or the conversion from sine to cosine.</p>
<p class="p2"> </p>
<p class="p1"><span class="Apple-converted-space">\(\frac{1}{4} = \sin \frac{\alpha }{4}\) </span><strong><em>A1</em></strong></p>
<p class="p1">\(\frac{\alpha }{4} = \arcsin \frac{1}{4}\)</p>
<p class="p1"><span class="Apple-converted-space">\(\alpha = 4\arcsin \frac{1}{4}\) </span><strong><em>AG</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">(from triangle <span class="s1">ADM), \(\theta = \pi - \frac{\alpha }{2}{\text{ }}\left( { = \pi - 2\arcsin \frac{1}{4} = 2\arcsin \frac{1}{4} = 2.6362 \ldots } \right)\)</span> <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2">attempting to solve \(2(\alpha - \sin \alpha ){r^2} + \frac{1}{2}(\theta - \sin \theta ){r^2} = 4\)</p>
<p class="p3">with \(\alpha = 4\arcsin \frac{1}{4}\)<span class="s2"> </span>and \(\theta = \pi - \frac{\alpha }{2}{\text{ }}\left( { = 2\arccos \frac{1}{4}} \right)\)<span class="s2"> </span>for \(r\) <span class="Apple-converted-space"> </span><span class="s3"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\(r = 1.69\) </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p 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 shaded region <em>S </em>is enclosed between the curve \(y = x + 2\cos x\), for \(0 \leqslant x \leqslant 2\pi \), and the line \(y = x\), as shown in the diagram below.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-12_om_06.15.17.png" alt></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;">Find the coordinates of the points where the line meets the curve.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region \(S\) is rotated by \(2\pi \) about the \(x\)-axis to generate a solid.</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) Write down an integral that represents the volume \(V\) of the solid.</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) Find the volume \(V\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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) \(\frac{\pi }{2}(1.57),{\text{ }}\frac{{3\pi }}{2}(4.71)\) <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;">hence the coordinates are \(\left( {\frac{\pi }{2},{\text{ }}\frac{\pi }{2}} \right),{\text{ }}\left( {\frac{{3\pi }}{2},{\text{ }}\frac{{3\pi }}{2}} \right)\) <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;"><strong><em>[3 marks]<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: 21.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">(i) \(\pi \int_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} {\left( {{x^2} - {{(x + 2\cos x)}^2}} \right){\text{d}}x} \) <strong style="font-weight: bold;"><em style="font-style: italic;">A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"><span style="font-size: medium; font-family: 'times new roman', times;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;"><strong style="font-weight: bold;">Note:</strong> Award <strong style="font-weight: bold;"><em style="font-style: italic;">A1 </em></strong>for \({x^2} - {(x + 2\cos x)^2}\), <strong style="font-weight: bold;"><em style="font-style: italic;">A1 </em></strong>for correct limits and <strong style="font-weight: bold;"><em style="font-style: italic;">A1 </em></strong>for \(\pi \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"><span style="font-size: medium; font-family: 'times new roman', times;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">(ii) \(6{\pi ^2}{\text{ }}( = 59.2)\) <strong style="font-weight: bold;"><em style="font-style: italic;">A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"><span style="font-size: medium; font-family: 'times new roman', times;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;"><strong style="font-weight: bold;">Notes:</strong> Do not award <strong style="font-weight: bold;">ft </strong>from (b)(i).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"><span style="font-size: medium; font-family: 'times new roman', times;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;"><strong style="font-weight: bold;"><em style="font-style: italic;">[5 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 class="p1">In a triangle \({\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}\) and \({\rm{B\hat AC}} = \frac{\pi }{9}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the cosine rule to find the two possible values for <span class="s1">AC</span><span class="s2">.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the difference between the areas of the two possible triangles <span class="s1">ABC</span><span class="s2">.</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 class="p1"><strong>METHOD 1</strong></p>
<p class="p1">let \({\text{AC}} = x\)</p>
<p class="p1"><span class="Apple-converted-space">\({3^2} = {x^2} + {4^2} - 8x\cos \frac{\pi }{9}\) </span><strong><em>M1A1</em></strong></p>
<p class="p1">attempting to solve for \(x\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(x = 1.09,{\text{ }}6.43\) </span><strong><em>A1A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">let \({\text{AC}} = x\)</p>
<p class="p1">using the sine rule to find a value of \(C\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({4^2} = {x^2} + {3^2} - 6x\cos (152.869 \ldots ^\circ ) \Rightarrow x = 1.09\) </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({4^2} = {x^2} + {3^2} - 6x\cos (27.131 \ldots ^\circ ) \Rightarrow x = 6.43\) </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1">let \({\text{AC}} = x\)</p>
<p class="p1">using the sine rule to find a value of \(B\) and a value of \(C\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1">obtaining \(B = 132.869 \ldots ^\circ ,{\text{ }}7.131 \ldots ^\circ \) and \(C = 27.131 \ldots ^\circ ,{\text{ }}152.869 \ldots ^\circ \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\((B = 2.319 \ldots ,{\text{ }}0.124 \ldots \) and \(C = 0.473 \ldots ,{\text{ }}2.668 \ldots )\)</p>
<p class="p1">attempting to find a value of \(x\) using the cosine rule <strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(x = 1.09,{\text{ }}6.43\) </span><strong><em>A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>M1A0(M1)A1A0 </em></strong>for one correct value of \(x\)</p>
<p class="p2"> </p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\frac{1}{2} \times 4 \times 6.428 \ldots \times \sin \frac{\pi }{9}\) and \(\frac{1}{2} \times 4 \times 1.088 \ldots \times \sin \frac{\pi }{9}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">(\(4.39747 \ldots \) and \(0.744833 \ldots \))</p>
<p class="p1">let \(D\) be the difference between the two areas</p>
<p class="p1"><span class="Apple-converted-space">\(D = \frac{1}{2} \times 4 \times 6.428 \ldots \times \sin \frac{\pi }{9} - \frac{1}{2} \times 4 \times 1.088 \ldots \times \sin \frac{\pi }{9}\) </span><strong><em>(M1)</em></strong></p>
<p class="p1">\((D = 4.39747 \ldots - 0.744833 \ldots )\)</p>
<p class="p1"><span class="Apple-converted-space">\( = 3.65{\text{ (c}}{{\text{m}}^2})\) </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(z = r(\cos \alpha + {\text{i}}\sin \alpha )\), where \(\alpha \) is measured in degrees, be the solution of \({z^5} - 1 = 0\) which has the smallest positive argument.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Use the binomial theorem to expand \({(\cos \theta + {\text{i}}\sin \theta )^5}\).</p>
<p>(ii) Hence use De Moivre’s theorem to prove</p>
<p>\[\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta .\]</p>
<p>(iii) State a similar expression for \(\cos 5\theta \) in terms of \(\cos \theta \) and \(\sin \theta \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(r\) and the value of \(\alpha \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using (a) (ii) and your answer from (b) show that \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Hence express \(\sin 72^\circ \) </span>in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) \({(\cos \theta + {\text{i}}\sin \theta )^5}\)</p>
<p>\( = {\cos ^5}\theta + 5{\text{i}}{\cos ^4}\theta \sin \theta + 10{{\text{i}}^2}{\cos ^3}\theta {\sin ^2}\theta + \)</p>
<p>\(10{{\text{i}}^3}{\cos ^2}\theta {\sin ^3}\theta + 5{{\text{i}}^4}\cos \theta {\sin ^4}\theta + {{\text{i}}^5}{\sin ^5}\theta \) <strong><em>A1A1</em></strong></p>
<p>\(( = {\cos ^5}\theta + 5{\text{i}}{\cos ^4}\theta \sin \theta - 10{\cos ^3}\theta {\sin ^2}\theta - \)</p>
<p>\(10{\text{i}}{\cos ^2}\theta {\sin ^3}\theta + 5\cos \theta {\sin ^4}\theta + {\text{i}}{\sin ^5}\theta )\)</p>
<p> </p>
<p><strong>Note:</strong> Award first <strong><em>A1</em></strong> for correct binomial coefficients.</p>
<p> </p>
<p>(ii) \({({\text{cis}}\theta )^5} = {\text{cis}}5\theta = \cos 5\theta + {\text{i}}\sin 5\theta \) <strong><em>M1</em></strong></p>
<p>\( = {\cos ^5}\theta + 5{\text{i}}{\cos ^4}\theta \sin \theta - 10{\cos ^3}\theta {\sin ^2}\theta - 10{\text{i}}{\cos ^2}\theta {\sin ^3}\theta + \)</p>
<p>\(5\cos \theta {\sin ^4}\theta + {\text{i}}{\sin ^5}\theta \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Previous line may be seen in (i)</p>
<p> </p>
<p>equating imaginary terms <strong><em>M1</em></strong></p>
<p>\(\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta \) <strong><em>AG</em></strong></p>
<p>(iii) equating real terms</p>
<p>\(\cos 5\theta = {\cos ^5}\theta - 10{\cos ^3}\theta {\sin ^2}\theta + 5\cos \theta {\sin ^4}\theta \) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({(r{\text{cis}}\alpha )^5} = 1 \Rightarrow {r^5}{\text{cis}}5\alpha = 1{\text{cis}}0\) <strong><em>M1</em></strong></p>
<p>\({r^5} = 1 \Rightarrow r = 1\) <strong><em>A1</em></strong></p>
<p>\(5\alpha = 0 \pm 360k,{\text{ }}k \in \mathbb{Z} \Rightarrow a = 72k\) <strong><em>(M1)</em></strong></p>
<p>\(\alpha = 72^\circ \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1A0</em></strong> if final answer is given in radians.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of \(\sin (5 \times 72) = 0\) <strong>OR</strong> the imaginary part of \(1\) is \(0\) <strong><em>(M1)</em></strong></p>
<p>\(0 = 5{\cos ^4}\alpha \sin \alpha - 10{\cos ^2}\alpha {\sin ^3}\alpha + {\sin ^5}\alpha \) <strong><em>A1</em></strong></p>
<p>\(\sin \alpha \ne 0 \Rightarrow 0 = 5{(1 - {\sin ^2}\alpha )^2} - 10(1 - {\sin ^2}\alpha ){\sin ^2}\alpha + {\sin ^4}\alpha \) <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1</em></strong> for replacing \({\cos ^2}\alpha \).</p>
<p> </p>
<p>\(0 = 5(1 - 2{\sin ^2}\alpha + {\sin ^4}\alpha ) - 10{\sin ^2}\alpha + 10{\sin ^4}\alpha + {\sin ^4}\alpha \) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em></strong> for any correct simplification.</p>
<p> </p>
<p>so \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\) <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\sin ^2}\alpha = \frac{{20 \pm \sqrt {400 - 320} }}{{32}}\) <strong><em>M1A1</em></strong></p>
<p>\(\sin \alpha = \pm \sqrt {\frac{{20 \pm \sqrt {80} }}{{32}}} \)</p>
<p>\(\sin \alpha = \frac{{ \pm \sqrt {10 \pm 2\sqrt 5 } }}{4}\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1</em></strong> regardless of signs. Accept equivalent forms with integral denominator, simplification may be seen later.</p>
<p> </p>
<p>as \(72 > 60\), \(\sin 72 > \frac{{\sqrt 3 }}{2} = 0.866 \ldots \) we have to take both positive signs (or equivalent argument) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow verification of correct signs with calculator if clearly stated</p>
<p> </p>
<p>\(\sin 72 = \frac{{\sqrt {10 + 2\sqrt 5 } }}{4}\) <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<p><strong><em>Total [19 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;">
<p class="p1">In part (i) many candidates tried to multiply it out the binomials rather than using the binomial theorem. In parts (ii) and (iii) many candidates showed poor understanding of complex numbers and made no attempt to equate real and imaginary parts. In a some cases the correct answer to part (iii) was seen although it was unclear how it was obtained.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was poorly done. Very few candidates made a good attempt to apply De Moivre’s theorem and most of them could not even equate the moduli to obtain \(r\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was poorly done. From the few candidates that attempted it, many candidates started by writing down what they were trying to prove and made no progress.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Very few made a serious attempt to answer this question. Also very few realised that they could use the answers given in part (c) to attempt this part.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="text-align: center;"><img src="images/12b.png" 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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In triangle ABC, BC = <em>a</em> , AC = <em>b</em> , AB = <em>c</em> and [BD] is perpendicular to [AC].</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="font: normal normal normal 26px/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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \({\text{CD}} = b - c\cos A\).</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) <strong>Hence</strong>, by using Pythagoras’ Theorem in the triangle BCD, prove the cosine rule for the triangle ABC.</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) If \({\rm{A\hat BC}} = 60^\circ \) , use the cosine rule to show that \(c = \frac{1}{2}a \pm \sqrt {{b^2} - \frac{3}{4}{a^2}} \) .</span></p>
<div class="marks">[12]</div>
<div class="question_part_label">Part 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: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The above three dimensional diagram shows the points P and Q which are respectively west and south-west of the base R of a vertical flagpole RS on horizontal ground. The angles of elevation of the top S of the flagpole from P and Q are respectively 25° and 40° , and PQ = 20 m .</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;">Determine the height of the flagpole.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">Part 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: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{CD}} = {\text{AC}} - {\text{AD}} = b - c\cos A\) <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;"><strong><em>[1 mark]</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> </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;">(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;">\({\text{B}}{{\text{C}}^2} = {\text{B}}{{\text{D}}^2} + {\text{C}}{{\text{D}}^2}\) <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;">\({a^2} = {(c\sin A)^2} + {(b - c\cos A)^2}\) <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;">\( = {c^2}{\sin ^2}A + {b^2} - 2bc\cos A + {c^2}{\cos ^2}A\) <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;">\( = {b^2} + {c^2} - 2bc\cos A\) <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><em>[4 marks]</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;">\({\text{B}}{{\text{D}}^2} = {\text{A}}{{\text{B}}^2} - {\text{A}}{{\text{D}}^2} = {\text{B}}{{\text{C}}^2} - {\text{C}}{{\text{D}}^2}\) <strong><em>(M1)(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;">\( \Rightarrow {c^2} - {c^2}{\cos ^2}A = {a^2} - {b^2} + 2bc\cos A - {c^2}{\cos ^2}A\) <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;">\( \Rightarrow {a^2} = {b^2} + {c^2} - 2bc\cos A\) <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><em>[4 marks]</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> </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;">(c) <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;">\({b^2} = {a^2} + {c^2} - 2ac\cos 60^\circ \Rightarrow {b^2} = {a^2} + {c^2} - ac\) <strong><em>(M1)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;">\( \Rightarrow {c^2} - ac + {a^2} - {b^2} = 0\) <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;">\( \Rightarrow c = \frac{{a \pm \sqrt {{{( - a)}^2} - 4({a^2} - {b^2})} }}{2}\) <strong><em>(M1)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;">\( = \frac{{a \pm \sqrt {4{b^2} - 3{a^2}} }}{2} = \frac{a}{2} \pm \sqrt {\frac{{4{b^2} - 3{a^2}}}{4}} \) <strong><em>(M1)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;">\( = \frac{1}{2}a \pm \sqrt {{b^2} - \frac{3}{4}{a^2}} \) <strong><em>AG</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> Candidates can only obtain a maximum of the first three marks if they <strong>verify</strong> that the answer given in the question satisfies the equation.</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;"><em><span style="font-family: 'times new roman',times; font-size: medium;"><strong>[7 marks]</strong></span></em></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;">\({b^2} = {a^2} + {c^2} - 2ac\cos 60^\circ \Rightarrow {b^2} = {a^2} + {c^2} - ac\) <strong><em>(M1)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;">\({c^2} - ac = {b^2} - {a^2}\) <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;">\({c^2} - ac + {\left( {\frac{a}{2}} \right)^2} = {b^2} - {a^2} + {\left( {\frac{a}{2}} \right)^2}\) <strong><em>M1A1</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;">\({\left( {c - \frac{a}{2}} \right)^2} = {b^2} - \frac{3}{4}{a^2}\) <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;">\(c - \frac{a}{2} = \pm \sqrt {{b^2} - \frac{3}{4}{a^2}} \) <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;">\( \Rightarrow c = \frac{1}{2}a \pm \sqrt {{b^2} - \frac{3}{4}{a^2}} \) <strong><em>AG</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>[7 marks]</em></strong></span></p>
<div class="question_part_label">Part A.</div>
</div>
<div class="question" style="padding-left: 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;">\({\text{PR}} = h\tan 55^\circ {\text{ , QR}} = h\tan 50^\circ {\text{ where RS}} = h\) <strong><em>M1A1A1</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;">Use the cosine rule in triangle PQR. <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;">\({20^2} = {h^2}{\tan ^2}55^\circ + {h^2}{\tan ^2}50^\circ - 2h\tan 55^\circ h\tan 50^\circ \cos 45^\circ \) <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;">\({h^2} = \frac{{400}}{{{{\tan }^2}55^\circ + {{\tan }^2}50^\circ - 2\tan 55^\circ \tan 50^\circ \cos 45^\circ }}\) <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;">\( = 379.9…\) <strong>(A1)</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;">\(h = 19.5{\text{ (m)}}\) <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><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">Part 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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The majority of the candidates attempted part A of this question. Parts (a) and (b) were answered reasonably well. In part (c), many candidates scored the first two marks, but failed to recognize that the result was a quadratic equation, and hence did not progress further.</span></p>
<div class="question_part_label">Part 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;">Correct answers to part B were rarely seen. Although many candidates expressed RS correctly in two different ways, they failed to go on to use the cosine rule.</span></p>
<div class="question_part_label">Part B.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)\) where \(0 < x < \frac{\pi }{2}\).</p>
<p>Express \(f\left( x \right)\) in terms of sin \(x\) and cos \(x\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\({\text{tan}}\left( {x + \pi } \right) = \tan x\left( { = \frac{{{\text{sin}}\,x}}{{{\text{cos}}\,x}}} \right)\)<em><strong> (M1)A1</strong></em></p>
<p>\({\text{cos}}\left( {x - \frac{\pi }{2}} \right) = {\text{sin}}\,x\)<em><strong> (M1)A1</strong></em></p>
<p><strong>Note:</strong> The two <em><strong>M1</strong></em>s can be awarded for observation or for expanding.</p>
<p>\({\text{tan}}\left( {x + \pi } \right) = {\text{cos}}\left( {x - \frac{\pi }{2}} \right) = \frac{{{\text{si}}{{\text{n}}^2}\,x}}{{{\text{cos}}\,x}}\) <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">
<p class="p1">ABCD is a quadrilateral where \({\text{AB}} = 6.5,{\text{ BC}} = 9.1,{\text{ CD}} = 10.4,{\text{ DA}} = 7.8\) and \({\rm{C\hat DA}} = 90^\circ \). Find \({\rm{A\hat BC}}\)<span class="s1">, giving your answer correct to the nearest degree.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><span class="Apple-converted-space">\({\text{A}}{{\text{C}}^2} = {7.8^2} + {10.4^2}\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\({\text{AC}} = 13\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2">use of cosine rule <em>eg</em>, \(\cos ({\rm{A\hat BC}}) = \frac{{{{6.5}^2} + {{9.1}^2} - {{13}^2}}}{{2(6.5)(9.1)}}\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\rm{A\hat BC}} = 111.804 \ldots ^\circ {\text{ }}( = 1.95134 \ldots )\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = 112^\circ \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Well done by most candidates. A small number of candidates did not express the required angle correct to the nearest degree.</p>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is \(\theta \) radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of \(0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water \(V{\text{ }}({{\text{m}}^3})\) in the trough in terms of \(\theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\) when \(\theta = \frac{\pi }{3}\).</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>area of segment \( = \frac{1}{2} \times {0.5^2} \times (\theta - \sin \theta )\) <strong><em>M1A1</em></strong></p>
<p>\(V = {\text{area of segment}} \times 10\)</p>
<p>\(V = \frac{5}{4}(\theta - \sin \theta )\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{5}{4}(1 - \cos \theta )\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\) <strong><em>M1A1</em></strong></p>
<p>\(0.0008 = \frac{5}{4}\left( {1 - \cos \frac{\pi }{3}} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\) <strong><em>(M1)</em></strong></p>
<p>\(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128{\text{ }}({\text{rad}}\,{s^{ - 1}})\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>\(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}\theta }}{{{\text{d}}V}} \times \frac{{{\text{d}}V}}{{{\text{d}}t}}\) <strong><em>(M1)</em></strong></p>
<p>\(\frac{{{\text{d}}V}}{{{\text{d}}\theta }} = \frac{5}{4}(1 - \cos \theta )\) <strong><em>A1</em></strong></p>
<p>\(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{4 \times 0.0008}}{{5\left( {1 - \cos \frac{\pi }{3}} \right)}}\) <strong><em>(M1)</em></strong></p>
<p>\(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128\left( {\frac{4}{{3125}}} \right)({\text{rad }}{s^{ - 1}})\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Triangle \(ABC\) has area \({\text{21 c}}{{\text{m}}^{\text{2}}}\). The sides \(AB\) and \(AC\) have lengths \(6\) cm and \(11\) cm respectively. Find the two possible lengths of the side \(BC\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>\(21 = \frac{1}{2} \bullet 6 \bullet 11 \bullet \sin A\) <strong><em>(M1)</em></strong></p>
<p>\(\sin A = \frac{7}{{11}}\) <strong><em>(A1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p>\(\hat A = 0.6897 \ldots ,{\text{ }}2.452 \ldots \left( {\hat A = \arcsin \frac{7}{{11}},{\text{ }}\pi - \arcsin \frac{7}{{11}} = 39.521 \ldots ^\circ ,{\text{ }}140.478 \ldots ^\circ } \right)\) <strong><em>(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p>\(\cos A = \pm \frac{{6\sqrt 2 }}{{11}}\;\;\;( = \pm 0.771 \ldots )\) <strong><em>(A1)</em></strong></p>
<p><strong>THEN</strong></p>
<p>\({\text{B}}{{\text{C}}^2} = {6^2} + {11^2} - 2 \bullet 6 \bullet 11\cos A\) <strong><em>(M1)</em></strong></p>
<p>\({\text{BC}} = 16.1\) or \(7.43\) <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A1A0M1A1A0 </em></strong>if only one correct solution is given.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p 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;">Consider the planes \({\pi _1}:x - 2y - 3z = 2{\text{ and }}{\pi _2}:2x - y - z = k\) .</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;">Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .</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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector equation of</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;">\({L_1}\) is \(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\)</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and \({L_2}\) intersect at a point X. Find the coordinates of X.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and \({L_2}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ and the area of the triangle XYZ is 3. Find the perimeter of the triangle XYZ.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</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><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{n} = \left( {\begin{array}{*{20}{c}}<br> 1 \\ <br> { - 2} \\ <br> { - 3} <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> and \(\boldsymbol{m} = \left( {\begin{array}{*{20}{c}}<br> 2 \\ <br> { - 1} \\ <br> { - 1} <br>\end{array}} \right)\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <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;">\(\cos \theta = \frac{{\boldsymbol{n} \cdot \boldsymbol{m}}}{{\left| \boldsymbol{n} \right|\left| \boldsymbol{m} \right|}}\) <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;">\(\cos \theta = \frac{{2 + 2 + 3}}{{\sqrt {1 + 4 + 9} \sqrt {4 + 1 + 1} }} = \frac{7}{{\sqrt {14} \sqrt 6 }}\) <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;">\(\theta = 40.2^\circ \,\,\,\,\,(0.702{\text{ rad}})\) <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;"><strong><em>[4 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;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</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;"><strong>METHOD 1</strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">eliminate <em>z </em>from <em>x</em> – 2<em>y</em> – 3<em>z</em> = 2 and 2<em>x</em> – <em>y</em> – <em>z</em> = <em>k</em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(5x - y = 3k - 2 \Rightarrow x = \frac{{y - (2 - 3k)}}{5}\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">eliminate <em>y </em>from <em>x</em> – 2<em>y</em> – 3<em>z</em> = 2 and 2<em>x</em> – <em>y</em> – <em>z</em> = <em>k</em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(3x + z = 2k - 2 \Rightarrow x = \frac{{z - (2k - 2)}}{{ - 3}}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em>x</em> =<em> t</em>,<em> y </em>= (2 − 3<em>k</em>) + 5t and <em>z</em> = (2<em>k </em>− 2) − 3<em>t <strong>A1A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <strong><em>AG</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">METHOD 2</strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br>1\\<br>{ - 2}\\<br>{ - 3}<br>\end{array}} \right) \times \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 1}\\<br>{ - 1}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>{ - 5}\\<br>3<br>\end{array}} \right) \Rightarrow {\text{direction is }}\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <em><strong> M1A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>x</em> = 0</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(0 - 2y - 3z = 2{\text{ and }}2 \times 0 - y - z = k\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">solve simultaneously <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(y = 2 - 3k{\text{ and }}z = 2k - 2\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">therefore <strong>r</strong> \( = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <em><strong>AG</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[5 marks]</strong></em></span></p>
<p><strong style="font-family: 'times new roman', times; font-size: medium;">METHOD 3</strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">substitute \(x = t,{\text{ }}y = (2 - 3k) + 5t{\text{ and }}z = (2k - 2) - 3t{\text{ into }}{\pi _1}{\text{ and }}{\pi _2}\) <em><strong> M1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">for \({\pi _1}:t - 2(2 - 3k + 5t) - 3(2k - 2 - 3t) = 2\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">for \({\pi _2}:2t - (2 - 3k + 5t) - (2k - 2 - 3t) = k\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">the planes have a unique line of intersection <em><strong>R2</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">therefore the line is \(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) <em><strong>AG</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[5 marks]</strong></em></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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"> </p>
<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;">\(5 - t = (2 - 3k + 5t) + 3 = 2 - 2(2k - 2 - 3t)\) <em><strong>M1A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 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>M1A1 </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">if candidates use vector or parametric equations of \({L_2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><em><span style="font-family: 'times new roman',times; font-size: medium;">eg </span></em><span style="font-family: 'times new roman',times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>5\\<br>{ - 3}\\<br>1<br>\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}<br>{ - 2}\\<br>2\\<br>{ - 1}<br>\end{array}} \right)\) or \( \Rightarrow \left\{ {\begin{array}{*{20}{l}}<br>{t = 5 - 2s}\\<br>{2 - 3k + 5t = - 3 + 2s}\\<br>{2k - 2 - 3t = 1 + s}<br>\end{array}} \right.\)<br></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;">solve simultaneously <strong><em>M1</em></strong></span></p>
<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;">\(k = 2,{\text{ }}t = 1{\text{ }}(s = 2)\) <em><strong>A1</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;">intersection point (\(1\), \(1\), \( - 1\)) <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;"><strong><em> </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;"><strong><em>[5 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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\overrightarrow l _2} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 2}\\<br>1<br>\end{array}} \right)\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\overrightarrow l _1} \times {\overrightarrow l _2} = \left| {\begin{array}{*{20}{c}}<br>\boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\<br>1&5&{ - 3}\\<br>2&{ - 2}&1<br>\end{array}} \right| = \left( {\begin{array}{*{20}{c}}<br>{ - 1}\\<br>{ - 7}\\<br>{ - 12}<br>\end{array}} \right)\) <em><strong>(M1)A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\boldsymbol{r} \cdot \left( {\begin{array}{*{20}{c}}<br>1\\<br>7\\<br>{12}<br>\end{array}} \right) = \left( {\begin{array}{*{20}{c}}<br>1\\<br>1\\<br>{ - 1}<br>\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}<br>1\\<br>7\\<br>{12}<br>\end{array}} \right)\) <em><strong>(M1)</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x + 7y + 12z = - 4\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[5 marks]</strong></em></span></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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept alternative notation for vectors (<em>eg</em> \(\langle a{\text{, }}b{\text{, }}c\rangle {\text{ or }}\left( {a{\text{, }}b{\text{, }}c} \right)\)).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"> </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;">Let \(\theta \) be the angle between the lines \({\overrightarrow l _1} = \left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\) and \({\overrightarrow l _2} = \left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 2}\\<br>1<br>\end{array}} \right)\)<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;">\(\cos \theta = \frac{{\left| {2 - 10 - 3} \right|}}{{\sqrt {35} \sqrt 9 }} \Rightarrow \theta = 0.902334...{\text{ }}51.699...^\circ )\) <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;">as the triangle XYZ has a right angle at Y,</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;">\({\text{XZ}} = a \Rightarrow {\text{YZ}} = a\sin \theta {\text{ and XY}} = a\cos \theta \) <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;">\({\text{area = 3}} \Rightarrow \frac{{{a^2}\sin \theta \cos \theta }}{2} = 3\) <strong><em>(M1)</em></strong></span></p>
<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;">\(a = 3.5122...\) <em><strong>(A1)</strong></em><br></span></p>
<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;">perimeter \( = a + a\sin \theta + a\cos \theta = 8.44537... = 8.45\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 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 candidates attempt to find coordinates of Y and Z award </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;">for expression of vector YZ in terms of two parameters, </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;">for attempt to use perpendicular condition to determine relation between parameters, </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;">for attempt to use the area to find the parameters and </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 final answer.</span></p>
<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;"> </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>[5 marks]</em></strong></span></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 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;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></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;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</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; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</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 Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</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: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></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: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Although this was the last question in part B, it was answered surprisingly well by many candidates, except for part (e). Even those who had not done so well elsewhere often gained a number of marks in some parts of the question. Nevertheless the presence of parameters seemed to have blocked the abilities of weaker candidates to solve situations in which vectors were involved. Mathematical skills for this particular question were sometimes remarkable, however, calculations proved incomplete due to the way that planes were presented. Most candidates found a correct angle in part (a). Occasional arithmetic errors in calculating the magnitude of a vector and dot product occurred. In part (b) the vector product approach was popular. In some case candidates simply verified the result by substitution. There was a lot of simultaneous equation solving, much of which was not very pretty. In part (c), a number of candidates made errors when attempting to solve a system of equations involving parameters. Many of the results for the point were found in terms of <em>k</em>. It was notorious that candidates did not use their GDC to try to find the coordinates of the intersection point between lines. In part (d), a number of candidates used an incorrect point but this part was often done well. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few excellent answers to part (e) were seen using an efficient method. Most candidates attempted methods involving heavy algebraic manipulation and had little success in this part of the question.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In triangle \(ABC\),</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(3\sin B + 4\cos C = 6\) and</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(4\sin C + 3\cos B = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\sin (B + C) = \frac{1}{2}\).</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 class="p1">Robert conjectures that \({\rm{C\hat AB}}\) can have two possible values.</p>
<p class="p1">Show that Robert’s conjecture is incorrect by proving that \({\rm{C\hat AB}}\) has only one possible value.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>squaring both equations <strong><em>M1</em></strong></p>
<p>\(9{\sin ^2}B + 24\sin B\cos C + 16{\cos ^2}C = 36\) <strong><em>(A1)</em></strong></p>
<p>\(9{\cos ^2}B + 24\cos B\sin C + 16{\sin ^2}C = 1\) <strong><em>(A1)</em></strong></p>
<p>adding the equations and using \({\cos ^2}\theta + {\sin ^2}\theta = 1\) to obtain \(9 + 24\sin (B + C) + 16 = 37\) <strong><em>M1</em></strong></p>
<p>\(24(\sin B\cos C + \cos B\sin C) = 12\) <strong><em>A1</em></strong></p>
<p>\(24\sin (B + C) = 12\) <strong><em>(A1)</em></strong></p>
<p>\(\sin (B + C) = \frac{1}{2}\) <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>substituting for \(\sin B\) and \(\cos B\) to obtain</p>
<p>\(\sin (B + C) = \left( {\frac{{6 - 4\cos C}}{3}} \right)\cos C + \left( {\frac{{1 - 4\sin C}}{3}} \right)\sin C\) <strong><em>M1</em></strong></p>
<p>\( = \frac{{6\cos C + \sin C - 4}}{3}\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p>substituting for \(\sin C\) and \(\cos C\) to obtain</p>
<p>\(\sin (B + C) = \sin B\left( {\frac{{6 - 3\sin B}}{4}} \right) + \cos B\left( {\frac{{1 - 3\cos B}}{4}} \right)\) <strong><em>M1</em></strong></p>
<p>\( = \frac{{\cos B + 6\sin B - 3}}{4}\;\;\;\)(or equivalent) <strong><em>A1</em></strong></p>
<p>Adding the two equations for \(\sin (B + C)\):</p>
<p>\(2\sin (B + C) = \frac{{(18\sin B + 24\cos C) + (4\sin C + 3\cos B) - 25}}{{12}}\) <strong><em>A1</em></strong></p>
<p>\(\sin (B + C) = \frac{{36 + 1 - 25}}{{24}}\) <strong><em>(A1)</em></strong></p>
<p>\(\sin (B + C) = \frac{1}{2}\) <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>substituting \(\sin B\) and \(\sin C\) to obtain</p>
<p>\(\sin (B + C) = \left( {\frac{{6 - 4\cos C}}{3}} \right)\cos C + \cos B\left( {\frac{{1 - 3\cos B}}{4}} \right)\) <strong><em>M1</em></strong></p>
<p>substituting for \(\cos B\) and \(\cos B\) to obtain</p>
<p>\(\sin (B + C) = \sin B\left( {\frac{{6 - 3\sin B}}{4}} \right) + \left( {\frac{{1 - 4\sin C}}{3}} \right)\sin C\) <strong><em>M1</em></strong></p>
<p>Adding the two equations for \(\sin (B + C)\):</p>
<p>\(2\sin (B + C) = \frac{{6\cos C + \sin C - 4}}{3} + \frac{{6\sin B + \cos B - 3}}{4}\;\;\;\)(or equivalent) <strong><em>A1A1</em></strong></p>
<p>\(2\sin (B + C) = \frac{{(18\sin B + 24\cos C) + (4\sin C + 3\cos B) - 25}}{{12}}\) <strong><em>A1</em></strong></p>
<p>\(\sin (B + C) = \frac{{36 + 1 - 25}}{{24}}\) <strong><em>(A1)</em></strong></p>
<p>\(\sin (B + C) = \frac{1}{2}\) <strong><em>AG</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\sin A = \sin \left( {180^\circ - (B + C)} \right)\) so \(\sin A = \sin (B + C)\) <strong><em>R1</em></strong></p>
<p>\(\sin (B + C) = \frac{1}{2} \Rightarrow \sin A = \frac{1}{2}\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow A = 30^\circ \) or \(A = 150^\circ \) <strong><em>A1</em></strong></p>
<p>if \(A = 150^\circ \), then \(B < 30^\circ \) <strong><em>R1</em></strong></p>
<p>for example, \(3\sin B + 4\cos C < \frac{3}{2} + 4 < 6\), <em>ie </em>a contradiction <strong><em>R1</em></strong></p>
<p>only one possible value \((A = 30^\circ )\) <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<p><strong><em>Total [11 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>Most candidates found this a difficult question with a large number of candidates either not attempting it or making little to no progress. In part (a), most successful candidates squared both equations, added them together, used \({\cos ^2}\theta + {\sin ^2}\theta = 1\) and then simplified their result to show that \(\sin (B + C) = \frac{1}{2}\). A number of candidates started with a correct alternative method (see the markscheme for alternative approaches) but were unable to follow them through fully.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), a small percentage of candidates were able to obtain \(B + C = 30^\circ {\text{ }}(A = 150^\circ )\) or \(B + C = 150^\circ {\text{ }}(A = 30^\circ )\) but were then unable to demonstrate or explain why \(A = 30^\circ \) is the only possible value for triangle ABC.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of \(k\) that satisfy the inequality \({k^2} - k - 12 < 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that \(\cos B < \frac{1}{4}\), find the range of possible values for AB.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({k^2} - k - 12 < 0\)</p>
<p>\((k - 4)(k + 3) < 0\) <strong><em>(M1)</em></strong></p>
<p>\( - 3 < k < 4\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\cos B = \frac{{{2^2} + {c^2} - {4^2}}}{{4c}}{\text{ }}({\text{or }}16 = {2^2} + {c^2} - 4c\cos B)\) <strong><em>M1</em></strong></p>
<p>\( \Rightarrow \frac{{{c^2} - 12}}{{4c}} < \frac{1}{4}\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow {c^2} - c - 12 < 0\)</p>
<p>from result in (a)</p>
<p>\(0 < {\text{AB}} < 4\) or \( - 3 < {\text{AB}} < 4\) <strong><em>(A1)</em></strong></p>
<p>but AB must be at least 2</p>
<p>\( \Rightarrow 2 < {\text{AB}} < 4\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow \( \leqslant {\text{AB}}\) for either of the final two <strong><em>A </em></strong>marks.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A triangle <span class="s1">\(ABC\) </span>has \(\hat A = 50^\circ \), <span class="s1">\({\text{AB}} = 7{\text{ cm}}\) </span>and <span class="s1">\({\text{BC}} = 6{\text{ cm}}\)</span>. Find the area of the triangle given that it is smaller than \(10{\text{ c}}{{\text{m}}^2}\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><img src="images/Schermafbeelding_2016-01-05_om_17.12.42.png" alt></p>
<p class="p2"><strong>METHOD 1</strong></p>
<p class="p1">\(\frac{6}{{\sin 50}} = \frac{7}{{\sin C}} \Rightarrow \sin C = \frac{{7\sin 50}}{6}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(C = 63.344 \ldots \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">or\(\;\;\;C = 116.655 \ldots \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">\(B = 13.344 \ldots \;\;\;({\text{or }}B = 66.656 \ldots )\) <span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">\({\text{area}} = \frac{1}{2} \times 6 \times 7 \times \sin 13.344 \ldots \;\;\;\left( {{\text{or }}\frac{1}{2} \times 6 \times 7 \times \sin 66.656 \ldots } \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(4.846 \ldots \;\;\;({\text{or }} = 19.281 \ldots )\)</p>
<p class="p1">so answer is \(4.85{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong>METHOD 2</strong></p>
<p class="p1">\({6^2} = {7^2} + {b^2} - 2 \times 7b\cos 50\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)(A1)</em></strong></span></p>
<p class="p2"><span class="s2">\({b^2} - 14b\cos 50 + 13 = 0\;\;\;\)</span>or equivalent method to solve the above equation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(b = 7.1912821 \ldots \;\;\;{\text{or}}\;\;\;b = 1.807744 \ldots \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">\({\text{area}} = \frac{1}{2} \times 7 \times 1.8077 \ldots \sin 50 = 4.846 \ldots \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(\left( {{\text{or }}\frac{1}{2} \times 7 \times 7.1912821 \ldots \sin 50 = 19.281 \ldots } \right)\)</p>
<p class="p1">so answer is \(4.85{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong>METHOD 3</strong></p>
<p class="p1"><img src="images/Schermafbeelding_2016-01-05_om_17.46.05.png" alt></p>
<p>Diagram showing triangles \(ACB\) and \(ADB\) <strong><em>(M1)</em></strong></p>
<p>\(h = 7\sin (50) = 5.3623 \ldots {\text{ (cm)}}\) <strong><em>(M1)</em></strong></p>
<p>\(\alpha = \arcsin \frac{h}{6} = 63.3442 \ldots \) <strong><em>(M1)</em></strong></p>
<p>\({\text{AC}} = {\text{AD}} - {\text{CD}} = 7\cos 50 - 6\cos \alpha = 1.8077 \ldots {\text{ (cm)}}\) <strong><em>(M1)</em></strong></p>
<p>\({\text{area}} = \frac{1}{2} \times 1.8077 \ldots \times 5.3623 \ldots \) <strong><em>(M1)</em></strong></p>
<p>\( = 4.85{\text{ (c}}{{\text{m}}^{\text{2}}}{\text{)}}\) <strong><em>A1</em></strong></p>
<p><strong><em>Total [6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Most candidates scored 4/6 showing that candidates do not have enough experience with the ambiguous case. Very few candidates drew a suitable diagram that would have illustrated this fact which could have helped them to understand the requirement that the answer should be less than 10. In fact many candidates ignored this requirement or used it incorrectly to solve an inequality.</p>
</div>
<br><hr><br><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;">A rectangle is drawn around a sector of a circle as shown. If the angle of the sector is 1 radian and the area of the sector is \(7{\text{ c}}{{\text{m}}^2}\), find the dimensions of the rectangle, giving your answers to the nearest millimetre.</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;"> </span></p>
<p style="font: normal normal normal 22px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<h2 style="margin-top: 1em">Markscheme</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;">\(\frac{1}{2}{r^2} \times 1 = 7\) <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;">\(r = 3.7 \ldots \left( { = \sqrt {14} } \right)\) (or 37… mm) <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;">\({\text{height}} = 2r\cos \left( {\frac{{\pi - 1}}{2}} \right){\text{ }}\left( {{\text{or }}2r\sin \frac{1}{2}} \right)\) <strong><em>(M1)(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;">3.59 or anything that rounds to 3.6 <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;">so the dimensions are 3.7 by 3.6 (cm or 37 by 36 mm) <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>[6 marks]</em></strong></span></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Most students found the value of \(r\) , but a surprising number had difficulties finding the height of the rectangle by any one of the many methods possible. Those that did, frequently failed to round their final answer to the required accuracy leading to few students obtaining full marks on this question. A surprising number of students found the area – clearly misinterpreting the meaning of “dimensions”.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P and Q lie on the larger circle and \({\rm{P}}\hat {\text{O}}{\text{Q}} = x\) , where \(0 < x < \frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Show that the area of the shaded region is \(8\sin x - 2x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the maximum area of the shaded region.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) shaded area area of triangle area of sector, <em>i.e.</em> <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\frac{1}{2} \times {4^2}\sin x} \right) - \left( {\frac{1}{2}{2^2}x} \right) = 8\sin x - 2x\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1AG</span></strong></em></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;">(b) <strong>EITHER</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">any method from GDC gaining \(x \approx 1.32\) <em><strong>(M1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum value for given domain is \(5.11\) <em><strong>A2</strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 8\cos x - 2\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">set \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\)</span><span style="font-family: times new roman,times; font-size: medium;">, hence \(8\cos x - 2 = 0\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\cos x = \frac{1}{4} \Rightarrow x \approx 1.32\) </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;">hence \({A_{\max }} = 5.11\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Generally a well answered question.</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;"><span style="font-family: 'times new roman', times; font-size: medium;">In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO of the cone is 8 cm and the angle \({\rm{B\hat OC}}\) is 60°.</span> </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;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></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;">Calculate the size of the angle \({\rm{B\hat AC}}\).</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: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">AC = AB = 10 (cm) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">triangle OBC is equilateral <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">BC = 6 (cm) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{B\hat AC}} = 2\arcsin \frac{3}{{10}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{B\hat AC}} = 34.9^\circ \,\,\,\,\,\)(accept 0.609 radians) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><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: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos {\rm{B\hat AC = }}\frac{{{{10}^2} + {{10}^2} - {6^2}}}{{2 \times 10 \times 10}} = \frac{{164}}{{200}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{B\hat AC}} = 34.9^\circ \,\,\,\,\,\)(accept 0.609 radians) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Other valid methods may be seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times; color: #3f3f3f;"><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 Times; color: #3f3f3f;"><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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The question was generally well answered, but some students attempted to find the length of arc BC.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Points A , B and T lie on a line on an indoor soccer field. The goal, [AB] , is 2 metres wide. A player situated at point P kicks a ball at the goal. [PT] is perpendicular to (AB) and is 6 metres from a parallel line through the centre of [AB] . Let PT <span class="s1">be \(x\) metros and let \(\alpha = {\rm{A\hat PB}}\) measured in degrees. Assume that the ball travels along the floor.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-03_om_11.38.31.png" alt="M16/5/MATHL/HP2/ENG/TZ2/11"></span></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The maximum for \(\tan \alpha \) </span>gives the maximum for \(\alpha \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(\alpha \) when \(x = 10\).</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">Show that \(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise find the value of \(\alpha \) <span class="s1">such that \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = 0\).</span></p>
<p class="p2"><span class="s1">(iii) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) </span>and hence show that the value of \(\alpha \) <span class="s1">never exceeds 10°.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of values of \(x\) for which \(\alpha \geqslant 7^\circ \).</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 class="p1"><strong>EITHER</strong></p>
<p class="p1"><span class="s1">\(\alpha = \arctan \frac{7}{{10}} - \arctan \frac{5}{{10}}{\text{ }}( = 34.992 \ldots ^\circ - 26.5651 \ldots ^\circ )\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)(A1)</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M1) </em></strong>for \(\alpha = {\rm{A\hat PT}} - {\rm{B\hat PT}}\), <strong><em>(A1) </em></strong><span class="s1">for a correct \({\rm{A\hat PT}}\) </span>and <strong><em>(A1) </em></strong><span class="s1">for a correct \({\rm{B\hat PT}}\)</span>.</p>
<p class="p1"><strong>OR</strong></p>
<p class="p1"><span class="s1">\(\alpha = \arctan {\text{ }}2 - \arctan \frac{{10}}{7}{\text{ }}( = 63.434 \ldots ^\circ - 55.008 \ldots ^\circ )\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)(A1)</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M1) </em></strong>for \(\alpha = {\rm{P\hat BT}} - {\rm{P\hat AT}}\), <strong><em>(A1) </em></strong><span class="s1">for a correct \({\rm{P\hat BT}}\) </span>and <strong><em>(A1) </em></strong><span class="s1">for a correct \({\rm{P\hat AT}}\).</span></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1"><span class="s1">\(\alpha = \arccos \left( {\frac{{125 + 149 - 4}}{{2 \times \sqrt {125} \times \sqrt {149} }}} \right)\) <span class="Apple-converted-space"> </span></span><strong><em>(M1)(A1)(A1)</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M1) </em></strong>for use of cosine rule, <strong><em>(A1) </em></strong>for a correct numerator and <strong><em>(A1) </em></strong>for a correct denominator.</p>
<p class="p1"><strong>THEN</strong></p>
<p class="p3"><span class="Apple-converted-space">\( = 8.43^\circ \) </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\tan \alpha = \frac{{\frac{7}{x} - \frac{5}{x}}}{{1 + \left( {\frac{7}{x}} \right)\left( {\frac{5}{x}} \right)}}\) </span><span class="s1"><strong><em>M1A1A1</em></strong></span></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong><span class="s2">for use of \(\tan (A - B)\)</span>, <strong><em>A1 </em></strong>for a correct numerator and <strong><em>A1 </em></strong>for a correct denominator.</p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{\frac{2}{x}}}{{1 + \frac{{35}}{{{x^2}}}}}\) </span><strong><em>M1</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\tan \alpha = \frac{{\frac{x}{5} - \frac{x}{7}}}{{1 + \left( {\frac{x}{5}} \right)\left( {\frac{x}{7}} \right)}}\) </span><span class="s1"><strong><em>M1A1A1</em></strong></span></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong><span class="s2">for use of xxx</span>, <strong><em>A1 </em></strong>for a correct numerator and <strong><em>A1 </em></strong>for a correct denominator.</p>
<p class="p1">\( = \frac{{\frac{{2x}}{{35}}}}{{1 + \frac{{{x^2}}}{{35}}}}\) <strong><em>M1</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\cos \alpha = \frac{{{x^2} + 35}}{{\sqrt {({x^2} + 25)({x^2} + 49)} }}\) </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p2"><span class="s1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong></span>for either use of the cosine rule or use of \(\cos (A - B)\)<span class="s1">.</span></p>
<p class="p1"><span class="Apple-converted-space">\(\sin \alpha \frac{{2x}}{{\sqrt {({x^2} + 25)({x^2} + 49)} }}\) </span><strong><em>A1</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(\tan \alpha = \frac{{\frac{{2x}}{{\sqrt {({x^2} + 25)({x^2} + 49)} }}}}{{\frac{{{x^2} + 35}}{{\sqrt {({x^2} + 25)({x^2} + 49)} }}}}\) </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p2"><span class="Apple-converted-space">\(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\) </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = \frac{{2({x^2} + 35) - (2x)(2x)}}{{{{({x^2} + 35)}^2}}}{\text{ }}\left( { = \frac{{70 - 2{x^2}}}{{{{({x^2} + 35)}^2}}}} \right)\)</span> <strong><em>M1A1A1</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong>for attempting product or quotient rule differentiation, <strong><em>A1 </em></strong>for a correct numerator and <strong><em>A1 </em></strong>for a correct denominator.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p3"><span class="Apple-converted-space">\(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = 0 \Rightarrow 70 - 2{x^2} = 0\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\(x = \sqrt {35} {\text{ (m) }}\left( { = 5.9161 \ldots {\text{ (m)}}} \right)\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><span class="Apple-converted-space">\(\tan \alpha = \frac{1}{{\sqrt {35} }}{\text{ }}( = 0.16903 \ldots )\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">attempting to locate the stationary point on the graph of</p>
<p class="p1"><span class="Apple-converted-space">\(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\) </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(x = 5.9161 \ldots {\text{ (m) }}\left( { = \sqrt {35} {\text{ (m)}}} \right)\) </span><strong><em>A1</em></strong></p>
<p class="p4"><span class="Apple-converted-space">\(\tan \alpha = 0.16903 \ldots {\text{ }}\left( { = \frac{1}{{\sqrt {35} }}} \right)\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p4"><span class="Apple-converted-space">\(\alpha = 9.59^\circ \) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p5"><strong>METHOD 2</strong></p>
<p class="p5"><strong>EITHER</strong></p>
<p class="p6"><span class="Apple-converted-space">\(\alpha = \arctan \left( {\frac{{2x}}{{{x^2} + 35}}} \right) \Rightarrow \frac{{{\text{d}}\alpha }}{{{\text{d}}x}} = \frac{{70 - 2{x^2}}}{{{{({x^2} + 35)}^2} + 4{x^2}}}\) </span><span class="s2"><strong><em>M1</em></strong></span></p>
<p class="p6"><span class="Apple-converted-space">\(\frac{{{\text{d}}\alpha }}{{{\text{d}}x}} = 0 \Rightarrow x = \sqrt {35} {\text{ (m) }}\left( { = 5.9161{\text{ (m)}}} \right)\) </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p5"><strong>OR</strong></p>
<p class="p5">attempting to locate the stationary point on the graph of</p>
<p class="p5"><span class="Apple-converted-space">\(\alpha = \arctan \left( {\frac{{2x}}{{{x^2} + 35}}} \right)\) </span><strong><em>(M1)</em></strong></p>
<p class="p5"><span class="Apple-converted-space">\(x = 5.9161 \ldots {\text{ (m) }}\left( { = \sqrt {35} {\text{ (m)}}} \right)\) </span><strong><em>A1</em></strong></p>
<p class="p5"><strong>THEN</strong></p>
<p class="p7"><span class="Apple-converted-space">\(\alpha = 0.1674 \ldots {\text{ }}\left( { = \arctan \frac{1}{{\sqrt {35} }}} \right)\) </span><span class="s2"><strong><em>(A1)</em></strong></span></p>
<p class="p6"><span class="Apple-converted-space">\( = 9.59^\circ \) </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p5">(iii) <span class="Apple-converted-space"> \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha ) = \frac{{{{({x^2} + 25)}^2}( - 4x) - (2)(2x)({x^2} + 35)(70 - 2{x^2})}}{{{{({x^2} + 35)}^4}}}{\text{ }}\left( { = \frac{{4x({x^2} - 105)}}{{{{({x^2} + 35)}^3}}}} \right)\)</span> <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p5">substituting \(x = \sqrt {35} {\text{ }}( = 5.9161 \ldots )\) into \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p5"><span class="s3">\(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha ) < 0{\text{ }}( =- 0.004829 \ldots )\) </span>and so \(\alpha = 9.59^\circ \) is the maximum value of \(\alpha \) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p6">\(\alpha \) never exceeds 10° <span class="Apple-converted-space"> </span><span class="s2"><strong><em>AG</em></strong></span></p>
<p class="p5"><strong><em>[11 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempting to solve \(\frac{{2x}}{{{x^2} + 35}} \geqslant \tan 7^\circ \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M1) </em></strong>for attempting to solve \(\frac{{2x}}{{{x^2} + 35}} = \tan 7^\circ \).</p>
<p class="p1">\(x = 2.55\) and \(x = 13.7\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p3"><span class="Apple-converted-space">\(2.55 \leqslant x \leqslant 13.7{\text{ (m)}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[3 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;">
<p class="p1">This question was generally accessible to a large majority of candidates. It was pleasing to see a number of different (and quite clever) trigonometric methods successfully employed to answer part (a) and part (b).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to a large majority of candidates. It was pleasing to see a number of different (and quite clever) trigonometric methods successfully employed to answer part (a) and part (b).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The early parts of part (c) were generally well done. In part (c) (i), a few candidates correctly found \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )\) in unsimplified form but then committed an algebraic error when endeavouring to simplify further. A few candidates merely stated that \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = {\sec ^2}\alpha \).</p>
<p class="p1">Part (c) (ii) was reasonably well done with a large number of candidates understanding what was required to find the correct value of \(\alpha \) in degrees. In part (c)(iii), a reasonable number of candidates were able to successfully find \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) in unsimplified form. Some however attempted to solve \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha ) = 0\) for \(\chi \) rather than examine the value of \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) at \(x = \sqrt {35} \).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (d), which required use of a GCD to determine an inequality, was surprisingly often omitted by candidates. Of the candidates who attempted this part, a number stated that \(x \geqslant 2.55\). Quite a sizeable proportion of candidates who obtained the correct inequality did not express their answer to 3 significant figures.</p>
<div class="question_part_label">d.</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the equation \(3{\cos ^2}x - 8\cos x + 4 = 0\), where \(0 \leqslant x \leqslant 180^\circ \), expressing your answer(s) to the nearest degree.</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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the exact values of \(\sec x\) satisfying the equation \(3{\sec ^4}x - 8{\sec ^2}x + 4 = 0\).</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve for \(\cos x\) or for <em>u</em> where \(u = \cos x\) or for <em>x</em> graphically. <strong><em>(M1)</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>EITHER</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;">\(\cos x = \frac{2}{3}{\text{ (and 2)}}\) <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>OR</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;">\(x = 48.1897 \ldots ^\circ \) <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>THEN</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;">\(x = 48^\circ \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px 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>(M1)(A1)A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \(x = 48^\circ ,{\text{ }}132^\circ \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><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: 24.0px 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>(M1)(A1)A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for 0.841 radians.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px 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: 24.0px 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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve for \(\sec x\) or for \(v\) where \(v = \sec x\). <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;">\(\sec x = \pm \sqrt 2 {\text{ }}\left( {{\text{and }} \pm \sqrt {\frac{2}{3}} } \right)\) <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;">\(\sec x = \pm \sqrt 2 \) <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>[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;">Part (a) was generally well done. Some candidates did not follow instructions and express their final answer correct to the nearest degree. A large number of candidates successfully employed a graphical approach.</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;">Part (b) was not well done. Common errors included attempting to solve for <em>x </em>rather than for \(\sec x\), either omitting or not considering \(\sec x = - \sqrt 2 \), not rejecting \(\sec x = \pm \sqrt {\frac{2}{3}} \) and not working with exact values.</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: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The points P and Q lie on a circle, with centre O and radius 8 cm, such that \({\rm{P\hat OQ}} = 59^\circ \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 20px/normal Times; text-align: center; margin: 0px;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px 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: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of the shaded segment of the circle contained between the arc PQ and the chord [PQ].</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;">area of triangle \({\text{POQ}} = \frac{1}{2}{8^2}\sin 59^\circ \) <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;">= 27.43 <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;">area of sector \( = \pi {8^2}\frac{{59}}{{360}}\) <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;">= 32.95 <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;">area between arc and chord \( = 32.95 - 27.43\)</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;">\( = 5.52{\text{ (c}}{{\text{m}}^2})\) <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;"><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 an easy starter question, with most candidates gaining full marks. Others lost marks through premature rounding or the incorrect use of radian measure.</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 graph below shows \(y = a\cos (bx) + c\).</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;"><img 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myVMLnuhQzArXmWOMxpKQ+3iHbNHSzuG9zEYR5LeWyVgFzv7wxwVnDrl8zZKmFyvVMuALfmGZ9ZmsdGD9wi2jV38EYIg5s4RHkbrRvsmg/f+5atzwXZKgFu3rcsyDXPGId57KHHLaJFczeL+wY3cSh7twFY/AfZKgYH7ws0OCu49dsA2CoB1n8Ne4Fb84xxmMe0FLeIFs0d7BKMfC8H38j6hhq2isHB3TgGlwW3u/zLVjE4uBsnALfm2V/wZP2iBW4RLZo72CUY4CwOxfZFC7aKuLobx+Cy4HY3B7FVDGYhwc08G7fmfeLQ/GBdx/VPflMplS9fuRrzNRypxq3bdyql8vzHl9x8u3PnL1RK5fDI73udOXO2Uirff7DovchDM4a4pNXx6tEj73rXGKZRHa9Wx6veNXIdPuNwmV1re0xPn6qUyo+WHrv5diYOz52/EIz8vtdC7Z75Xt6LPDSjOl6dmDjuXSOPMf/xpUqpvFC7591kOIb57XM2z/Y1co/DeMxqm627WGz9SO6xYu7gWYIRXC6WitULXQVvFcn/md0RHBfc4oYytoq4eqxHAG7NB4pDu9sgcYtoxdzZ7YYBjuPQ4jbIgreK7L2LO+/bDQPcF9zWNki2irh6rEcAbs0HikOx+gQ83CJaMXd2u2GA4zi0uBu54K3i7HbDAPcFtzU7LHiriMPbDQNwaz5oHFo8p8Et4uDmLm83DHAch2LvPURFbhXZ+6VzcLthgJeCWzmnKXiriMPbDQNwaz5oHHLKLzbMXd5uGPmmLr+jrSteRW4VcXi7YYCXglu5lavgreLydsMA3JoPGodi7zIGbhEHN3d5u2GA+zgUTvkxFxK8FNzK5KnIrSJubzcMwK25hTjklH9AczcvJejESxxa2eRW2FYRVy8liOCr4INPE4vcKiIyNTnpeCFBkGtuIQ6FU/7BzN28lKATL3FonqUyYPYXtlXE00KCr4IPnv1FbhUrv2sZwK25nTi0MmPFLeKA5o63QQd4iUPhlH8Ac8e3Gwb4KvjgD6IrbKuIp4UEQa65nTjklD+zuftt0AG+4nDwpzcUs1XE30KCx4IPuG+osK0iIlOTkw7ebtgJbs3txKHYWKTGLeIg5u63QQf4isPBN4MUs1XE30KCx4IPuBmksK3iayFBkGtuLQ4H38KEW8TM5u7vpw7jKw6FU/5MFPMAN+DvSDFbRfwtJAhyza3FIaf8Gf5D9w9mC+MxDge8kbyArSJeFxL8FnyQH7yYrSL+FhIEuebW4lAGfmAbbhEzm7u/nzqMxzjklD8tfhcS/BZ8kOvrBWwV8T3Pxq25zTgc8DPALWI2cy/3U4fxGIcy2APbitYqUvgDXOZzHe/mmQFdSBDkmtuMQ075U+HlgRFh/MbhIFfCitYq4nshwXvBM18J826eGdCFBEGuuc04lMEe2IZbxGzmXu6nDuM3DoVT/sR4X0jwXnDz5KYMkyfv5pnJbG7xTWrZwK255TjklD8hvh4YEcZ7HHLKnxDvCwkaCp5t+qjBPBugCwmCXHPLcSic8ifD1wMjwniPw8xzgkK1iihYSNBQ8Gy/MhrMswG6kCDINbcfh5zyJ8HXAyPCeI9D4ZQ/ARoWEjQU3Bzo0z6wTYN5NrKZa5hn49bcfhxmXi/FLWJacyf3U7/aflY7OVKulMqV0rH53z7bbke/QkMccsrfFx7gAjIsAyoxzwDoQoIg19x+HErWUx/cIqY1z/+BEe3dzTsn5+qN3ba0t/94+ycHSm8cvfFsd/8XaYjDbKc+xWkV8Xo/dYCSgme4hqrEPAOgCwmCXPNc4pBT/hharVb+B7iv12882NzdOx9sP1889kZl9Mbm631fpCEOJdN8tiCtIl4fzBZGScEzXBhTYp6BDOYeH8wWBrfmucQhp/wx+Lif+qvVE2MH5tZf7v9TJXHIKX8MPMBFSLteqsc8LaALCYJc81ziUDjl742HB0a8XJ8fGZ9f/zryx0rikFP+GHiAi5D2pjo95mkBXUgQ5JrnFYec8nfFxwMjXr2on33zVP2Fyq00Bk75u8IDXCdpf4P0mKcFdCFBkGveJw7ND5Zh3H+wWCmVz5w5m/lfGMpx+crVSql8/ZPfOPuOj+9eOPoPJ3/96Pedf1UplSPDb1lu3b7j/QNSNd5//8cHR8e8a2gbLEvneLT0+ODo2PT0Ke8mHofeOFxeeToxcbw6XvVeI1XD9W/yo1v/OvHRpc+edP1bPXH4aOlxpVQu+C8za5JwuJ9T6h+sybKDOByEtHtGbP1I7klo7vqBEe0X67/81eLzl73+Xs9iqaS8pDr0rSK+X2ERQVvBk19S1WaenFTmfl9hEQG35jnGIVf5Izh98mQ0C19tr396c/9uGlVxmOqddkPfKqLgyZNhtBU8+XUybebJSW7u/RUWEXBrnmMcCqf8+3F3gGu/WL94LLoWOnJx/eW+7TSq4lA45Q+h4cmTYbQVPPkmI23myUm7kODxFRYRcGuebxym2hqHW8Qk5koeGBFGWxxyyh/g/RUWERQWPOGjrxSaJwR0IUGQa55vHAqn/HtoePJkBG1xaN5pl2TGMNytIiJTk5M8wMWT8BdKoXlCQBcSBLnmucchp/wGDY/WjaAtDiVxlYa7VRQuJCgseMIqKTRPCOhCgiDXPPc45Cq/aLqfOozCOOSUX1QuJOgseJJFQp3mSUi+kKBtno1b89zjUBKvl+IWsa+5ngdGhFEYh5zyi453YUbQWfAkJ0Y6zZMAupAgyDV3EYec8it58mQEhXEoyWa7Q9wqOhcSdBY8yWUzneZJAF1IEOSau4jDgk/5Vd1PHUZnHBqr+N/wYW0V0bqQoLbgfddL1Zr3JYm5znk2bs1dxKEUe8qv6oERYXTGYcGn/DzApaLvCy7UmvcFdCFBkGvuKA4Lu8qv7YERYXTGoRR4ys8DXFr6/n6pNe8L6EKCINfcURwWdsqvdqVUFMdh36dsDGWryN5CAg9wqYhffdFsHk+8eavVOlSt6pxn49bcURyKyNzsbAGn/H1/ao+ojcNiTvk1LyRoLnj80241m8cDuiNBkGvuLg77fn64RexlrvCBEWHUxqH0O08avlYRfU+eDKO84DEXXJWbxwC6I0GQa+4uDgs45Vf4wIgwmuMw/ira8LWK6HvyZBjlBY+5iqbcPAbQhQRBrrm7OBSR2kKtUFN+bU+ejKA5DqVgU37lCwnKCx5zK5dy8xhizPvup/ULbs2dxmGhpvw6HxgRRnkcmil/1xsQh6xVROv91AH6C97rVi795r0AXUgQ5Jo7jUMROVSt9noAFW4Ru5rHHM2VoDwOCzXlV/jkyTD6C97r6Q36zXvRy9z8XqzW6+6VEoJbc9dxGDMLxi1iV3Od91OHUR6HUpgpf/I3W/lCf8F7rTbrN+8F6EKCINfcdRzGTG1wi9hpHr/5Wwn649AYdu5FGqZWEcX3UwdAFLzrEiKEeVd6mccssCkBt+au41B6L3zjFrHTXPM26AD9cViEKb+5n1r5QgJEwbveqQJh3pWu5mqfWxQGt+Ye4rDX7Qe4RYyYm4O42m3QAfrjUAow5dd8P3UARMG73n4AYd6VXgsJOp9bFAa35h7icOin/MpvNwyAiMOuaTE0rSIic7Oz+hcSUAre+fQGFPNOui4kQMyzcWvuIQ6lx1oibhEj5sq3QQdAxOFwT/mV324YgFLwzrVEFPNOQBcSBLnmfeLQ/GDWx+UrVyul8uUrV3P69z2Ohdq9Sql87vwF7yZ9x7nzF4yqGWo/junpU5VS+dHSY+8mOX0E9x8sejcZmnFwdGxi4rh3jTzG0SPvVser3jXUDtQ4XF55enB07P33f+y9gtbHmTNnUQ5w5lgcDLUHkVu37wzr5Kk6Xj165F3vGsM0gH4BU437DxYrpfKZM2e9m6gducdhfnTepW7rR3JP2Fz/NugAiMVSQ+QGxOFoFf23GwYAFdysPwe3cgGZR4iY93rOgEJwa+4tDjufOYJbxMAcYht0AFAcRm5AHIJWEYTbDQOwCh6+eI9lHiZirvy5RWFwa+4tDqVjywluEQNzta9v7QpQHEa2nAxBq0DcbhiAVfDwlhMs8zBhc6x5Nm7NfcZh5LnsuEU05ii3GwYAxaHs342M3iqCs0vQgFXw8JvisczDhM2x5tm4NfcZh5E99LhFNOa9niimFqw4DD/3Dr1VBOduHANcwYOFaDjzgPBCAtY8G7fmPuNQ9l8+wS2iMQda3DdgxaGEtimht0pku4d+4AoebFOCMw8IzFEe6xGAW3PPcRjeXIdbxOWVp9c/+Q3Q4r4BLg6DZ/lDt8ryylP9LyWIgFhwMz1FNDcE5srfIt4Jbs09x6GEdkzhFnF55en09CmgxX0DXBwGG2qgW2V55SnQ3TgGxIKbk6qF2j04c4OpOdDdOAGI3WLwH4fBUgBuER8tPcZa3DfAxaHsPeETt1WW9x7JhLWQgFhwM3manj4FZ24wNQe6GycAsVsM/uMw2JCJW0TzeBegxX0DYhyaDTXXP/kNaKssrzydmDiu/5ndEUB/N69fu3ZwdOzR0mPfIllYXnkKOs8G7RbREIeyt43YPJTSt0sWzKO2fFukBjEOReRQtWoe7+dbJAvmUVv6n9kdAfQAZ27Xu3zlqm+RLCyvPJ3/+FLnGxz1A9otoiQOg65FLCLurxxoHJrlo/sPFn2LZME8URNoE40B9wBXHa9Wx6u+LbKwvPK0Ol7F2kRjwO0WFXEoIlOTk+Zh7b5FUoO7IAMah2Z1/cyZs75FUtNqtcyT632LpAb3AAd6giUi5sn1WJtoDLjdoiUOzYaaW7fv+BZJB/TletA4FJGJieMHR8d8W6QmuPDpWyQ1uAc40MtvImLea4a1icaA2y1a4tA8eWF6+pRvkXSYRAHdzI0bh4ibM0Xk9MwM6BII7gFuee99mVi5Ano8NOB2i5Y4lL3ZENZlFehbfXHj0FxWwXoGkHmFi3kvtG+X1IB2uIgs770vE+gZQLL3uwm3WmbA7RZFcWjeIw/UtcFTNEE/fug4hLu5JdgBhNgqoB0ue+ZYT4gVkanJSfNeaN8iWcDtFkVxCHdLlrklXGA/fug4NHcsoFwTCp7CDNoqoNqyZx5+/rt+jC3oTntB7hZdcQh0TcisfZlzWdCPHzoOgwd2QKyum1I3Gg3QVgHVlpA50IPxhuDRS6DmuuJwGee9EOFjMejHjx6H5nGOEKvrh6pV6AfzgmpLyDyYkfg26kPQ1UNQczjUxSFE10be9Av68aPHoYTWqzUzBG9qBNUWwLcGDtObGuFQF4cQXRt50y/oxz8EcQhxTSj8gh7QVgHVlv3m+lfXw/Ps4ag5Fori0GxDF/Vd22q1wtchms3mxMTxhdo9v1YZwI3DM2fOBk+lUf42uMgLeoImx2Khdm9i4jjQPt4AswPZ/O/gBWF+lWIIH/omJo5PTBz3bZSFSqms4cCys7OTNkQ0xmFkKVIb5gE6wXKuOd4h3iGEG4fhI4Xy1XWznBvcAw4ah+bWPbVFjiEch9LxcagiMs9mHA6ImVvMzc4m71uNcSi6uzZyOsI4dE/4SBE5iKjCbD8On44wDh0TiUPN268iEzvG4YDs7OzUFmqHqlUTikkuqSiNQ7Vd23mxinHonsiRQu0JYueyP+PQMZE4FK2r652zOsahFVqt1mq9bkJxanJyY2Mj5izrWyIyNzurYVRK5ep4Nfi///Wf//nvRkZ+fv68d7Hw+M7Ywe+MHQz/yemZmUqpfPTIu97d+pY3+fBuGz8Ojo4dHB0L/u/Pz5+vlMqH337bu1h4/PvPflYplY9973uRTyHc5Cjj6JF3K6Xy6ZkZ7yZpR3W8Gunnjz74sFIqf/TBh97d4q0iTQ40dB5DPvrgw++MHayUyoeq1dV6vWsoKopDMxsK/u+/nj5dKZWnfvhD72J9lSLmOscwxWFnwY9973uVUvnff/Yz727xShCtkqTgKKOreeek1vvoVBqymvsdPz9/fuqHP/y7kZH4c0RFi6Wd23PnlF1B7OUDurEYd7G0s2AV11gAABG2SURBVODatl/18gFtFVBt6WGu7f6crj5DVnNfpLqCqDoOVV1BjJFR9fEnZ5jiUJTdn9NLBrRVQLWlt7mqK4hdZYav5o5pNBrm17BSKl+/di105fv1dv2nlVK5MnJ29cUraW//8fZPDpTKb97YVB2HoukE8fTMTC8TJR9/WoYsDvWcIMaYgLYKqLb0NtdzgtjLZPhq7hiThbWFWvcp8u6zm4fH3nu48cfbVxafvzR/pj0OlZwgmpbtpaHk44+yN+uplI7N//bZdjv690MWh6LmBDFGQ2mr9ANUW2LNlZwg9tIYypq7pNVqxR4KzDniWzP1L4NDo/Y4FB0niKZlezko+fj3883mjR9UDl9Z337V3v7i0uHx8KduGL44NKdl169dc68Uceh1kqqyVfoDqi2x5hpOEM3vYFeHoay5KtqNh++Vfrq6/Tr4EwVx+NeTmHLl7yeXNqOnMd5PEPv+2ij8+NuNh++VxufXvxYRkT83Hv6oMnJx/eW+0g5fHMremZnHO+TmZme7nBr2a3LlKOzwhCyvPLn7i/9xtHTk5uZu59/GT3Pzxtxr2OsFPrA1by/evvwfH75zoFSu7A8bbew/SIooiMNvNm+cna8/3xV5/NnVE28fqJR+cHPzm8gXmRNEL4tgpmXjF1X0Ne6fGw9/FO7Fzg9ehjQO/T6kxkzdOp6KmajJNaOvw5Py+O6Fo39TrvSIw62tLY9T7fjHR2DW/GWjfvG7pf/y3Yl/+7/K53ztL1cvzs2feOvA3PrLvT/zHYcv//Pm7WemT5dXnj6+fea9UvnNG5uRGYV52JWXXRKmZbe2tmK+Rl/jfr0+N14ZDdVxu36y9MZ7D5+HG3Qo41CSfWQ50X2/VbIm14y+Dk9G+8tzR77z3b8f7RWH4u9ajFlUj5m3Adb81Yv62QOlt975t1uL2s1fPn945dPNnZfrFw8ce9h4/eXqL5802t7jMMTyytPlzy6c7DiJMXhZBNvZ2UlyqqGvcb9aPTHWEYfRQ/CwxmGr1fKyCBa/38oQ3+Rq0dfhSXj5/OFP/+nDq7fOfz8mDs0JvfuptjmgxbwkBK7m7Rf1mZHygVP12hPN5mbl7JhZrZGX6/Mj5QMnan/cfiX+zw5DLK/8/n//YupAxyUuQ/w6e04kzGB9jds1DotydijJkskuQQbHf1l8k6tFX4f3pb27+el7J5ZuPnpyNzYOxcdUO0kGo9XcxMxbJ69feedvypXSXzMGCE1x+OTuuSPf7dwAGWAO38Gr4/Im+bRRX+NG98683rzxZjGuHQZ039KSGwlXaPs2uU70dXg/dp/dPPHr9e1XyysrfePQ/VQ75ibmALCat58vHnujcnhuaXP78cqTu//r1MmRcuXYwwZUozuIw93NG0d6PBtz7GT9q70ve1U7/4N/+vBaz54VkWRtZIupycmE30th4xZ2Z2lA32szFjHXthN8r0RNrhCFHR5H+8X6xbNms1KSOJS9l5i6mWonnNaD1fz15s3RstmWsrzydHnlt5s3flAp/Wix8WffZilQcnbY3n3+u+P//Rd3n/T5+M0Zm4Mby2LuB+pEY+O2v1w99daBE0vPd1/vNurzxbjvMIKz5QQzS+t3Jpq0yRWiscNj2K6f7Dr/PlHf7v0fJfsQByXhdgSBq3k0Dle26z/tOwvRhoo4bG9/8au539189PskH//nS0t57xtMPNn/C0obd/f56tyxinkqjbluvJ+hj0NxcoxLGLqpmlwbSjs8AQnPDmXvtz7vqbZZw4/ZQROAVvOv1+fGzRLUX+MQ7Rq5/zg0h4nnu+2/fPztF+u/vBNTRLNnIb8l0+DfT34MRWvcv1CEOEw7s0lLwn8/bZNrA7TDJU0cyt5UO7/lBLMkm3CHF1zN2y/qMyNvHL34xd0nTx9/9otLh8dPPvwT0rmh9zhsb39x6fAbkTWN8H2RXTFLpjkd4zKcfcI1rqEIcSh5Lpm2Wq0kZ5/ZmlwVoB0uKeNQ9pYTkpy9pcXMnJJv2AGseXu3UZ//S6v/Y9fnJCvH/9lhQIZjnPWd9OYpFWkXTAAbV6QwcSgip2dmEq5QpcJs0E/10EvQVgHVlvTmzWbT7DK1u/iUcOYUpjg11wNqHMreKrzFi4jmNyHD7dugH39x4tDsX7C7wG4WvtLevg3aKqDaksnc3LRq9yLi9WvXCjJzEmRz4Dg0dwvZWtkILhlm+NdAP/7ixKHsLbDbmvWbVYQMd6qBtgqotmQ1N1dMbD2qxqwidDzJtg9Fq7kGgONQ9s7nBt86aJYyMp9rgn78hYpDsTfrD7ouQ7KCtgqotgxgnuF8rium6zIkawFr7h3sOJTQBerMs/4gCzO3PujHX7Q4FBuz/iALs61JgLYKqLYMYD74YUH2sjDb0amANfcOfBzKXs9NTU5mOEJZaXrQj7+AcSh7K1e1hVqGI9SAWSiwrQKqLYOZD3hwMFeXM8/Ui1lzvwxDHIrI1taWOU6lWu1sNptTk5ODL4mAfvzFjEPZS8TTMzOp1tjNrGvAa9WgrQKqLQObB4mYakWh1WoFPZZ51aqwNffIkMShhLLt86WlJC1owiBtgnYF9OMvbBxKKNuS3I+4s7NjriSlTdBOQFsFVFtsmIezLclMqNFoZEjQTopcc18MTxyKSKvVMoetQ9Xqar3eKxQ3NjZMcM7Nzlp5fBfox1/kOBSRZrNpDltzs7O9Xu7TarVW6/VD1WryaVY8oK0Cqi32zNfW1kwb1BZqvQ4azWbTBOehanXwPTisuXuGKg4NW1tbJu3MNsLVer3RaGxtba3W67WFmunpqcnJwfs1APTjN3HYCOHbKCkWCx6k3dTkZG2htra2FnSLubE1Pi/TAtoqoNpi1XxnZ8eknTlT/HxpyXTL2tra50tLZmqV+bJ0J6y5e+Li0PxUoOPW7TvT06cOjo6Fn4x1cHTs/fd/fPnKVe96Gsa58xciTw7zruRlPFp6fPnK1fff/3GkGtXx6vT0qVu373g35NAz7j9YPHf+wtEj70a65eiRd+c/vnT/waJ3w8IOxmH/8Wjp8a3bd3hQ6xwmDk1xWCIz7j9YZCk4Eo6F2r1bt+8s1O55N+FYzjsOHWPrR3IPqHnBrx16AdQcVFto7gNcc8ahBUDNGYfuATUH1Raa+wDXnHFoAVBzxqF7QM1BtYXmPsA1ZxxaANSccegeUHNQbaG5D3DNGYcWADVnHLoH1BxUW2juA1xzxqEFQM0Zh+4BNQfVFpr7ANeccWgBUHPGoXtAzUG1heY+wDVnHFoA1Jxx6B5Qc1BtobkPcM0ZhxYANWccugfUHFRbaO4DXHPGoQVAzRmH7gE1B9UWmvsA15xxaAFQc8ahe0DNQbWF5j7ANWccWgDUnHHoHlBzUG2huQ9wzRmHFgA1Zxy6B9QcVFto7gNcc8ahBUDNGYfuATUH1Raa+wDXnHFoAVBzxqF7QM1BtYXmPsA1ZxxaANSccegeUHNQbaG5D3DNGYcWADVnHLoH1BxUW2juA1xzxqEFQM0Zh+4BNQfVFpr7ANeccWgBUHPGoXtAzUG1heY+wDVnHFoA1Jxx6B5Qc1BtobkPcM0ZhxZQaf5q+1nt5Ei5UipXSsfmf/tsux39Csahe0DNQbWF5j7ANWccWkCfeXt3887JuXpjty3t7T/e/smB0htHbzzb3f9FjEP3gJqDagvNfYBrzji0gD7zr9dvPNjc3TsfbD9fPPZGZfTG5ut9X8Q4dA+oOai20NwHuOaMQwuoN/9q9cTYgbn1l/v/lHHoHlBzUG2huQ9wzRmHFtBu/nJ9fmR8fv3ryB8zDt0Dag6qLTT3Aa55XByan4oDfDypnf9v3z5yofYk+leVUjkyfKtycHBwZByMwwKP5U8+/Ntonu2N75/7bMV82eO7F47+w8lfP/p957/AOOTg4BiakW8cOsbWj+Qevea7f1qcu7G+/arrX3Kx1D2g5qDaQnMf4JozDi2g1Lz9Yv2Xv1p8/rLX3zMO3QNqDqotNPcBrjnj0AIazaNZ+Gp7/dOb+3fTMA7dA2oOqi009wGuOePQAurM2y/WLx6LXhocubj+ct+TaRiH7gE1B9UWmvsA15xxaAFQc8ahe0DNQbWF5j7ANWccWgDUnHHoHlBzUG2huQ9wzRmHFgA1Zxy6B9QcVFto7gNcc8ahBUDNGYfuATUH1Raa+wDXnHFoAVBzxqF7QM1BtYXmPsA1ZxxaANSccegeUHNQbaG5D3DNGYcWADVnHLoH1BxUW2juA1xzxqEFQM0Zh+4BNQfVFpr7ANeccWgBUHPGoXtAzUG1heY+wDVnHFoA1Jxx6B5Qc1BtobkPcM0ZhxYANWccugfUHFRbaO4DXHPGoQVAzRmH7gE1B9UWmvsA15xxaAFQc8ahe0DNQbWF5j7ANWccWgDUnHHoHlBzUG2huQ9wzRmHFgA1Zxy6B9QcVFto7gNcc8ahBUDNGYfuATUH1Raa+wDXnHFoAVBzxqF7QM1BtYXmPsA1ZxxaANSccegeUHNQbaG5D3DNGYcWADVnHLoH1BxUW2juA1xzxqEFQM0Zh+4BNQfVFpr7ANeccWgBUHPGoXtAzUG1heY+wDVnHFoA1Jxx6B5Qc1BtobkPcM3j4tD8VBzDOs6dv1AplScmjgfDuxIHBwdHtsE45Mg+GIccHBxDM/KNQ8fY+pHcA2rOxVL3gJqDagvNfYBrzji0AKg549A9oOag2kJzH+CaMw4tAGrOOHQPqDmottDcB7jmjEMLgJozDt0Dag6qLTT3Aa4549ACoOaMQ/eAmoNqC819gGvOOLQAqDnj0D2g5qDaQnMf4JozDi0Aas44dA+oOai20NwHuOaMQwuAmjMO3QNqDqotNPcBrjnj0AKg5oxD94Cag2oLzX2Aa844tACoOePQPaDmoNpCcx/gmjMOLQBqzjh0D6g5qLbQ3Ae45oxDC4CaMw7dA2oOqi009wGuOePQAqDmjEP3gJqDagvNfYBrzji0AKg549A9oOag2kJzH+CaMw4tAGrOOHQPqDmottDcB7jmjEMLgJozDt0Dag6qLTT3Aa4549ACoOaMQ/eAmoNqC819gGvOOLQAqDnj0D2g5qDaQnMf4JozDi0Aas44dA+oOai20NwHuOaMQwuAmjMO3QNqDqotNPcBrjnj0AKqzdsv1i8eq4ze2Hwd/RvGoXtAzUG1heY+wDVnHFpAsfmrF/WzB0plxqESQM1BtYXmPsA1ZxxaQK15+0V9ZvRfjv7LG4xDJYCag2oLzX2Aa844tIBS890/LZ744Obm/1s9McY4VAKoOai20NwHuOaMQwuoNP9m88YHJx/+aVe+YhzqAdQcVFto7gNcc8ahBfSZt3c375y8+MV2W4RxqAlQc1BtobkPcM3j4tD8VBw6x++uffTtUrnSfbz5j29P//rR75dXni6vfHbu7ZHK33706+Xov9D5H3r/oTg4ODiyDcYhR+dYuXv++90ycuSd85+Fv5JxyMHBMTQj3zh0jK0fyT26zblYqghQc1BtobkPcM0ZhxbQbc44VASoOai20NwHuOaMQwvoNmccKgLUHFRbaO4DXHPGoQVAzRmH7gE1B9UWmvsA15xxaAFQc8ahe0DNQbWF5j7ANWccWgDUnHHoHlBzUG2huQ9wzRmHFgA1Zxy6B9QcVFto7gNcc8ahBUDNGYfuATUH1Raa+wDXnHFoAVBzxqF7QM1BtYXmPsA1ZxxaANSccegeUHNQbaG5D3DNGYcWADVnHLoH1BxUW2juA1xzxqEFQM0Zh+4BNQfVFpr7ANeccWgBUHPGoXtAzUG1heY+wDVnHFoA1Jxx6B5Qc1BtobkPcM0ZhxYANWccugfUHFRbaO4DXHPGoQVAzRmH7gE1B9UWmvsA15xxaAFQc8ahe0DNQbWF5j7ANWccWgDUnHHoHlBzUG2huQ9wzRmHFgA1Zxy6B9QcVFto7gNcc0VxSByDG4eEEGIdxmFxYRwSQkgA47C4MA4JISSAcVhcGIeEEBLAOCwujENCCAlgHBYXxiEhhAQwDosL45AQQgIYh8WFcUgIIQGMw+LCOCSEkADGYXFpNBqr9bpvC0IIUcH/B4nTItvmI1bJAAAAAElFTkSuQmCC" alt></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;">Find the value of <em>a</em>, the value of <em>b</em> and the value of <em>c</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 3\) <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;">\(c = 2\) <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;">period \( = \frac{{2\pi }}{b} = 3\) <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;">\(b = \frac{{2\pi }}{3}{\text{ }}( = 2.09)\) <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>
<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 find <em>a</em> and <em>c</em>, but many had difficulties with finding <em>b</em>.</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;">A system of equations is given by</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;">\[\cos x + \cos y = 1.2\]</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;">\[\sin x + \sin y = 1.4{\text{ .}}\]</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;">(a) For each equation express <em>y</em> in terms of <em>x</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;">(b) <strong>Hence</strong> solve the system for \(0 < x < \pi ,{\text{ }}0 < y < \pi \) .</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(y = \arccos (1.2 - \cos x)\) <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;">\(y = \arcsin (1.4 - \sin x)\) <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;"><strong><em> </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;">(b) The solutions are</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;"><em>x</em> = 1.26, <em>y</em> = 0.464 <strong><em>A1A1</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;"><em>x</em> = 0.464, <em>y</em> = 1.26 <strong><em>A1A1</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>[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 first two marks. Candidates who used their GDC to solve this question did so successfully, although few candidates provided a sketch as the rubric requires. Attempts to use “solver” only gave one solution.</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;">Some candidates did not give the solutions as coordinate pairs, but simply stated the <em>x</em> and <em>y</em> values.</span></p>
</div>
<br><hr><br><div class="question">
<p>This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius \({\text{OB}} = 9{\text{ cm}}\) and four equal sectors of a smaller circle of radius \({\text{OA}} = 3{\text{ cm}}\).<br>The angle \({\text{BOC}} = \) 20°.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-08_om_11.16.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/03"></p>
<p>Find the area of the pendant.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>area = (four sector areas radius 9) + (four sector areas radius 3) <strong><em>(M1)</em></strong></p>
<p>\( = 4\left( {\frac{1}{2}{9^2}\frac{\pi }{9}} \right) + 4\left( {\frac{1}{2}{3^2}\frac{{7\pi }}{{18}}} \right)\) <strong><em>(A1)(A1)</em></strong></p>
<p>\( = 18\pi + 7\pi \)</p>
<p>\( = 25\pi {\text{ }}( = 78.5{\text{ c}}{{\text{m}}^2})\) <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>area =</p>
<p>(area of circle radius 3) + (four sector areas radius 9) – (four sector areas radius 3) <strong><em>(M1)</em></strong></p>
<p>\(\pi {3^2} + 4\left( {\frac{1}{2}{9^2}\frac{\pi }{9}} \right) - 4\left( {\frac{1}{2}{3^2}\frac{\pi }{9}} \right)\) <strong><em>(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for the second term and <strong><em>A1 </em></strong>for the third term.</p>
<p> </p>
<p>\( = 9\pi + 18\pi - 2\pi \)</p>
<p>\( = 25\pi {\text{ }}( = {\text{ }}78.5{\text{ c}}{{\text{m}}^2})\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept working in degrees.</p>
<p> </p>
<p><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="specification">
<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;">A ship, S, is 10 km north of a motorboat, M, at 12.00pm. The ship is travelling northeast with a constant velocity of \(20{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\). The motorboat wishes to intercept the ship and it moves with a constant velocity of \(30{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\) in a direction \(\theta \) degrees east of north. In order for the interception to take place, determine</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;">the value of \(\theta \).<br></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: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the time at which the interception occurs, correct to the nearest minute.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img 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" alt></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;">let the interception occur at the point P, <em>t</em> hrs after 12:00</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;">then, SP = 20<em>t</em> and MP = 30<em>t</em> <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;">using the sine rule,</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;">\(\frac{{{\text{SP}}}}{{{\text{MP}}}} = \frac{2}{3} = \frac{{\sin \theta }}{{\sin 135}}\) <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;">whence \(\theta = 28.1\) <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">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;">using the sine rule again,</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;">\(\frac{{{\text{MP}}}}{{{\text{MS}}}} = \frac{{\sin 135}}{{\sin (45 - 28.1255 \ldots )}}\) <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;">\(30t = 10 \times \frac{{\sin 135}}{{\sin 16.8745 \ldots }}\) <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;">\(t = 0.81199 \ldots \) <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;">the interception occurs at 12:49 <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>[5 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 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 straight street of width 20 metres is bounded on its parallel sides by two vertical walls, one of height 13 metres, the other of height 8 metres. The intensity of light at point P at ground level on the street is proportional to the angle \(\theta \) where \(\theta = {\rm{A\hat PB}}\), as shown in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \(\theta \) in terms of <em>x</em>, where <em>x</em> is the distance of P from the base of the wall of height 8 m.</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the value of \(\theta \) when <em>x</em> = 0.</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;">(ii) Calculate the value of \(\theta \) when <em>x</em> = 20.</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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(\theta \), for \(0 \leqslant x \leqslant 20\).</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{{5(744 - 64x - {x^2})}}{{({x^2} + 64)({x^2} - 40x + 569)}}\).</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 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;">Using the result in part (d), or otherwise, determine the value of <em>x</em> corresponding to the maximum light intensity at P. Give your answer to four significant figures.</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point P moves across the street with speed \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\). Determine the rate of change of \(\theta \) with respect to time when P is at the midpoint of the street.</span></p>
<div class="marks">[4]</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</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;">\(\theta = \pi - \arctan \left( {\frac{8}{x}} \right) - \arctan \left( {\frac{{13}}{{20 - x}}} \right)\) (or equivalent) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px 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;"> Accept \(\theta = 180^\circ - \arctan \left( {\frac{8}{x}} \right) - \arctan \left( {\frac{{13}}{{20 - x}}} \right)\) (or equivalent).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><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: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">OR</strong></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;">\(\theta = \arctan \left( {\frac{x}{8}} \right) + \arctan \left( {\frac{{20 - x}}{{13}}} \right)\) (or equivalent) <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;"><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;">(i) \(\theta = 0.994{\text{ }}\left( { = \arctan \frac{{20}}{{13}}} \right)\) <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;"><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;">(ii) \(\theta = 1.19{\text{ }}\left( { = \arctan \frac{5}{2}} \right)\) <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;"><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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correct shape. <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;">correct domain indicated. <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> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img src="data:image/png;base64,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" alt></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>[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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to differentiate one \(\arctan \left( {f(x)} \right)\) term <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;"><strong>EITHER</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;">\(\theta = \pi - \arctan \left( {\frac{8}{x}} \right) - \arctan \left( {\frac{{13}}{{20 - x}}} \right)\)</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;">\(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{8}{{{x^2}}} \times \frac{1}{{1 + {{\left( {\frac{8}{x}} \right)}^2}}} - \frac{{13}}{{{{(20 - x)}^2}}} \times \frac{1}{{1 + {{\left( {\frac{{13}}{{20 - x}}} \right)}^2}}}\) <strong><em>A1A1</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>OR</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;">\(\theta = \arctan \left( {\frac{x}{8}} \right) + \arctan \left( {\frac{{20 - x}}{{13}}} \right)\)</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;">\(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{{\frac{1}{8}}}{{1 + {{\left( {\frac{x}{8}} \right)}^2}}} + \frac{{ - \frac{1}{{13}}}}{{1 + {{\left( {\frac{{20 - x}}{{13}}} \right)}^2}}}\) <strong><em>A1A1</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>THEN</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;">\( = \frac{8}{{{x^2} + 64}} - \frac{{13}}{{569 - 40x + {x^2}}}\) <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;">\( = \frac{{8(569 - 40x + {x^2}) - 13({x^{2}} + 64)}}{{({x^2} + 64)({x^2} - 40x + 569)}}\) <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;">\( = \frac{{5(744 - 64x - {x^2})}}{{({x^2} + 64)({x^2} - 40x + 569)}}\) <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>[6 marks]</em></strong></span></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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Maximum light intensity at P occurs when \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = 0\). <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;">either attempting to solve \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = 0\) for <em>x</em> or using the graph of either \(\theta \) or \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}}\) <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;"><em>x</em> = 10.05 (m) <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>[3 marks]</em></strong></span></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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.5\) <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;">At <em>x</em> = 10, \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = 0.000453{\text{ }}\left( { = \frac{5}{{11029}}} \right)\). <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;">use of \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}\theta }}{{{\text{d}}x}} \times \frac{{{\text{d}}x}}{{{\text{d}}t}}\) <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;">\(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.000227{\text{ }}\left( { = \frac{5}{{22058}}} \right){\text{ (rad }}{{\text{s}}^{ - 1}})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px 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 \(\frac{{{\text{d}}x}}{{{\text{d}}t}} = - 0.5\) and </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 \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = - 0.000227{\text{ }}\left( { = - \frac{5}{{22058}}} \right){\text{ }}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><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: 26.0px 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;"> Implicit differentiation can be used to find \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\). Award as above.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px 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: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[4 marks]</em></strong></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 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) was reasonably well done. While many candidates exhibited sound trigonometric knowledge to correctly express <em>θ </em>in terms of <em>x</em>, many other candidates were not able to use elementary trigonometry to formulate the required expression for <em>θ</em>.</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;">In part (b), a large number of candidates did not realize that <em>θ </em>could only be acute and gave obtuse angle values for <em>θ</em>. Many candidates also demonstrated a lack of insight when substituting endpoint <em>x</em>-values into <em>θ</em>.</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 Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (c), many candidates sketched either inaccurate or implausible graphs.</span></p>
<div class="question_part_label">c.</div>
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<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;">In part (d), a large number of candidates started their differentiation incorrectly by failing to use the chain rule correctly.</span></p>
<div class="question_part_label">d.</div>
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<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;">For a question part situated at the end of the paper, part (e) was reasonably well done. A large number of candidates demonstrated a sound knowledge of finding where the maximum value of <em>θ </em>occurred and rejected solutions that were not physically feasible.</span></p>
<div class="question_part_label">e.</div>
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<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;">In part (f), many candidates were able to link the required rates, however only a few candidates were able to successfully apply the chain rule in a related rates context.</span></p>
<div class="question_part_label">f.</div>
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<br><hr><br><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;">The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas of these sectors form a geometric sequence with common ratio <em>k</em>. The angle of the first sector is \(\theta \) radians.</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;">(a) Show that \(\theta = 2\pi (1 - k)\).</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;">(b) The perimeter of the third sector is half the perimeter of the first sector.</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;">Find the value of <em>k</em> and of \(\theta \).</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) the area of the first sector is \(\frac{1}{2}{2^2}\theta \) <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;">the sequence of areas is \(2\theta ,{\text{ }}2k\theta ,{\text{ }}2{k^2}\theta \ldots \) <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;">the sum of these areas is \(2\theta (1 + k + {k^2} + \ldots )\) <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;">\( = \frac{{2\theta }}{{1 - k}} = 4\pi \) <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;">hence \(\theta = 2\pi (1 - k)\) <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>Note:</strong> Accept solutions where candidates deal with angles instead of area.</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;"><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) the perimeter of the first sector is \(4 + 2\theta \) <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;">the perimeter of the third sector is \(4 + 2{k^2}\theta \) <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;">the given condition is \(4 + 2{k^2}\theta = 2 + \theta \) <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;">which simplifies to \(2 = \theta (1 - 2{k^2})\) <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;">eliminating \(\theta \), obtain cubic in <em>k</em>: \(\pi (1 - k)(1 - 2{k^2}) - 1 = 0\) <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;">or equivalent</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;">solve for <em>k</em> = 0.456 and then \(\theta = 3.42\) <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>[7 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>Total [12 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 a disappointingly answered question.</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;">Part(a) - Many candidates correctly assumed that the areas of the sectors were proportional to their angles, but did not actually state that fact.</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;">Part(b) - Few candidates seem to know what the term ‘perimeter’ means.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The points A and C are 3 m <span class="s1">apart. A goat \(G\) </span>is tied by a 5 m length of rope at point A on the outside edge of the enclosure.</p>
<p class="p1">Given that the corner of the enclosure at C <span class="s1">forms an angle of \(\theta \) </span>radians and the area of field that can be reached by the goat is \({\text{44 }}{{\text{m}}^{\text{2}}}\)<span class="s1">, find the value of \(\theta \).</span></p>
<p class="p1"><span class="s1"><img src="images/Schermafbeelding_2017-01-26_om_08.36.38.png" alt="M16/5/MATHL/HP2/ENG/TZ1/04"></span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempting to use the area of sector formula (including for a semicircle) <strong><em>M1</em></strong></p>
<p>semi-circle \(\frac{1}{2}\pi \times {5^2} = \frac{{25\pi }}{2} = 39.26990817 \ldots \) <strong><em>(A1)</em></strong></p>
<p>angle in smaller sector is \(\pi - \theta \) <strong><em>(A1)</em></strong></p>
<p>area of sector \( = \frac{1}{2} \times {2^2} \times (\pi - \theta )\) <strong><em>(A1)</em></strong></p>
<p>attempt to total a sum of areas of regions to 44 <strong><em>(M1)</em></strong></p>
<p>\(2(\pi - \theta ) = 44 - 39.26990817 \ldots \)</p>
<p class="p1">\(\theta = 0.777{\text{ }}\left( { = \frac{{29\pi }}{4} - 22} \right)\) <strong><em>A1</em></strong></p>
<p><strong>Note: </strong>Award all marks except the final <strong><em>A1 </em></strong>for correct working in degrees.</p>
<p><strong>Note: </strong>Attempt to solve with goat inside triangle should lead to nonsense answer and so should only receive a maximum of the two <strong><em>M </em></strong>marks.</p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Many students experienced difficulties with this question, mostly it seems through failing to understand the question. Some students left their answers in degrees, thereby losing the final mark.</p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm. If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two arcs AB.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></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;">\(\alpha = 2\arcsin \left( {\frac{{4.5}}{7}} \right)\) (\( \Rightarrow \alpha = 1.396... = 80.010^\circ ...\))</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>M1(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\beta = 2\arcsin \left( {\frac{{4.5}}{5}} \right)\) (\( \Rightarrow \beta = 2.239... = 128.31^\circ ...\)) <em><strong>(A1)</strong></em><br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Allow use of cosine rule.</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;">area \(P = \frac{1}{2} \times {7^2} \times \left( {\alpha - \sin \alpha } \right) = 10.08...\) </span><strong><span style="font-family: times new roman,times; font-size: medium;"> M1(A1)</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \(Q = \frac{1}{2} \times {5^2} \times \left( {\beta - \sin \beta } \right) = 18.18...\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> The <em><strong>M1</strong></em> is for an attempt at area of sector minus area of triangle.</span></p>
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> The use of degrees correctly converted is acceptable.</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;">area = 28.3 (cm<sup>2</sup>) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Whilst most candidates were able to make the correct construction to solve the problem some candidates seemed unable to find the area of a segment. In a number of cases candidates used degrees in a formula that required radians. There were a number of candidates who followed a completely correct method but due to premature approximation were unable to obtain a correct solution.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Compactness is a measure of how compact an enclosed region is.</p>
<p class="p1">The compactness, <em>\(C\) </em>, of an enclosed region can be defined by \(C = \frac{{4A}}{{\pi {d^2}}}\), where <em>\(A\) </em>is the area of the region and <em>\(d\) </em>is the maximum distance between any two points in the region.</p>
<p class="p1">For a circular region, \(C = 1\).</p>
<p class="p1">Consider a regular polygon of <em>\(n\) </em>sides constructed such that its vertices lie on the circumference of a circle of diameter <em>\(x\) </em>units.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(n > 2\) and even, show that \(C = \frac{n}{{2\pi }}\sin \frac{{2\pi }}{n}\).</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">If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).</p>
<p class="p1">Find the regular polygon with the least number of sides for which the compactness is more than \(0.99\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).</p>
<p class="p1">Comment briefly on whether <em>C </em>is a good measure of compactness.</p>
<div class="marks">[1]</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">each triangle has area \(\frac{1}{8}{x^2}\sin \frac{{2\pi }}{n}\;\;\;({\text{use of }}\frac{1}{2}ab\sin C)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">there are <em>\(n\) </em>triangles so \(A = \frac{1}{8}n{x^2}\sin \frac{{2\pi }}{n}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(C = \frac{{4\left( {\frac{1}{8}n{x^2}\sin \frac{{2\pi }}{n}} \right)}}{{\pi {n^2}}}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">so \(C = \frac{n}{{2\pi }}\sin \frac{{2\pi }}{n}\) <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">attempting to find the least value of <em>\(n\) </em>such that \(\frac{n}{{2\pi }}\sin \frac{{2\pi }}{n} > 0.99\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(n = 26\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">attempting to find the least value of <em>\(n\) </em>such that \(\frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}} > 0.99\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(n = 21\) (and so a regular polygon with 21 sides) <em><strong>A1</strong></em></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <span class="s1"><strong><em>(M0)A0(M1)A1</em></strong></span> if \(\frac{n}{{2\pi }}\sin \frac{{2\pi }}{n} > 0.99\) is not considered and \(\frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}} > 0.99\) is correctly considered.</p>
<p class="p1">Award <strong><em>(M1)A1(M0)A0 </em></strong>for \(n = 26\).</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">for even and odd values of <em>n</em>, the value of <em>C </em>seems to increase towards the limiting value of the circle \((C = 1)\) <em>ie </em>as <em>n </em>increases, the polygonal regions get closer and closer to the enclosing circular region <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">the differences between the odd and even values of <em>n </em>illustrate that this measure of compactness is not a good one. <span class="Apple-converted-space"> </span><strong><em>R1</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">Most candidates found this a difficult question with a large number of candidates either not attempting it or making little to no progress. In part (a), a number of candidates attempted to show the desired result using specific regular polygons. Some candidates attempted to fudge the result.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), the overwhelming majority of candidates that obtained either \(n = 21\) or \(n = 26\) or both used either a GDC numerical solve feature or a graphical approach rather than a tabular approach which is more appropriate for a discrete variable such as the number of sides of a regular polygon. Some candidates wasted valuable time by showing that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\) (a given result).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (c), the occasional candidate correctly commented that \(C \) was a good measure of compactness either because the value of \(C \) seemed to approach the limiting value of the circle as \(n \) increased or commented that \(C \) was not a good measure because of the disparity in \(C \)-values between even and odd values of \(n \).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Two non-intersecting circles C<sub>1</sub> , containing points M and S , and C<sub>2</sub> , containing points N and R, have centres P and Q where PQ \( = 50\) . The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is \( \alpha\), the size of the obtuse angle NQR is \( \beta\) , and the size of the angle MPQ is \( \theta\) . The arc length MS is \({l_1}\) and the arc length NR is \({l_2}\) . This information is represented in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The radius of C<sub>1</sub> is \(x\) , where \(x \geqslant 10\) and the radius of C<sub>2</sub> is \(10\).</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Explain why \(x < 40\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Show that cosθ = x −10 </span><span style="font-family: times new roman,times; font-size: medium;">50</span><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;">(c) (i) Find an expression for MN in terms of \(x\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the value of \(x\) that maximises MN.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) Find an expression in terms of \(x\) for</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) \( \alpha\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \( \beta\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) The length of the perimeter is given by \({l_1} + {l_2} + {\text{MN}} + {\text{SR}}\).</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Find an expression, \(b (x)\) , for the length of the perimeter in terms of \(x\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the maximum value of the length of the perimeter.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Find the value of \(x\) that gives a perimeter of length \(200\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) PQ \( = 50\) and non-intersecting <em><strong>R1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></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;">(b) a construction QT (where T is on the radius MP), parallel to MN, so that </span><span style="font-family: times new roman,times; font-size: medium;">\({\text{Q}}\hat {\text{T}}{\text{M}} = 90^\circ \) (angle between tangent and radius \( = 90^\circ \) ) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">lengths \(50\), \(x - 10\) and angle \( \theta\) marked on a diagram, or equivalent <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Other construction lines are possible.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></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;">(c) (i) MN \( = \sqrt {{{50}^2} - {{\left( {x - 10} \right)}^2}} \) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) maximum for MN occurs when \(x = 10\) <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>
<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;">(d) (i) \(\alpha = 2\pi - 2\theta \) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 2\pi - 2\arccos \left( {\frac{{x - 10}}{{50}}} \right)\) </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;">(ii) \(\beta = 2\pi - \alpha \) ( \( = 2\theta \) ) <em><strong>A1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 2\left( {{{\cos }^{ - 1}}\left( {\frac{{x - 10}}{{50}}} \right)} \right)\)</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><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></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;">(e) (i) </span><span style="font-family: times new roman,times; font-size: medium;">\(b(x) = x\alpha + 10\beta + 2\sqrt {{{50}^2} - {{\left( {x - 10} \right)}^2}} \) <em><strong>A1A1A1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = x\left( {2\pi - 2\left( {{{\cos }^{ - 1}}\left( {\frac{{x - 10}}{{50}}} \right)} \right)} \right) + 20\left( {\left( {{{\cos }^{ - 1}}\left( {\frac{{x - 10}}{{50}}} \right)} \right)} \right) + 2\sqrt {{{50}^2} - {{\left( {x - 10} \right)}^2}} \)</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;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) maximum value of perimeter \( = 276\) <em><strong>A2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) perimeter of \(200\) cm \(b(x) = 200\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">when \(x = 21.2\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[9 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [18 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This is not an inherently difficult question, but candidates either made heavy weather of it or avoided it almost entirely. The key to answering the question is in obtaining the displayed answer to part (b), for which a construction line parallel to MN through Q is required. Diagrams seen by examiners on some scripts tend to suggest that the perpendicularity property of a tangent to a circle and the associated radius is not as firmly known as they had expected. Some candidates mixed radians and degrees in their expressions.</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 a triangle ABC with \({\rm{B\hat AC}} = 45.7^\circ \) , AB = 9.63 cm and BC = 7.5 cm .<br></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;">By drawing a diagram, show why there are two triangles consistent with this information.</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the possible values of AC .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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><img 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" alt> A2<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>Note: </strong>Accept 2 separate triangles. The diagram(s) should show that one triangle has an acute angle and the other triangle has an obtuse angle. The values 9.63, 7.5 and 45.7 and/or the letters, A, B, C′ and C should be correctly marked on the diagram(s).</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;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></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: 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;">\(\frac{{\sin 45.7}}{{7.5}} = \frac{{\sin C}}{{9.63}}\) <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;">\( \Rightarrow \hat C = 66.77...^\circ \,{\text{, }}113.2...^\circ \) <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;">\( \Rightarrow \hat B = 67.52...^\circ \,{\text{, }}21.07...^\circ \) <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;">\(\frac{b}{{\sin B}} = \frac{{7.5}}{{\sin 45.7}} \Rightarrow b = 9.68({\text{cm}}){\text{, }}b = 3.77({\text{cm}})\) <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>If only the acute value of \({\hat C}\) is found, award <strong><em>M1(A1)(A0)(A0)A1A0</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></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;">\({7.5^2} = {9.63^2} + {b^2} - 2 \times 9.63 \times b\cos 45.7^\circ \) <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;">\({b^2} - 13.45...b + 36.48... = 0\)</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;">\(b = \frac{{13.45... \pm \sqrt {13.45..{.^2} - 4 \times 36.48...} }}{2}\) <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;">\({\text{AC}} = 9.68({\text{cm}})\,{\text{, AC}} = 3.77({\text{cm}})\) <strong> <em>A1A1</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 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;">Surprisingly few candidates were able to demonstrate diagrammatically the situation for the ambiguous case of the sine rule. More were successful in trying to apply it or to use the cosine rule. However, there were still a surprisingly large number of candidates who were only able to find one possible answer for AC.</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;">Surprisingly few candidates were able to demonstrate diagrammatically the situation for the ambiguous case of the sine rule. More were successful in trying to apply it or to use the cosine rule. However, there were still a surprisingly large number of candidates who were only able to find one possible answer for AC.</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: 26.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">Points A, B and C are on the circumference of a circle, centre O and radius \(r\) . </span><span style="font-family: times new roman,times; font-size: medium;">A trapezium OABC is formed such that AB is parallel to OC, and the angle \({\rm{A}}\hat {\text{O}}{\text{C}}\) </span><span style="font-family: times new roman,times; font-size: medium;">is \(\theta\) , \(\frac{\pi }{2} \leqslant \theta \leqslant \pi \)</span><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;"> </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;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></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;">(a) Show that angle \({\rm{B\hat OC}}\) is \(\pi - \theta \).</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) Show that the area, <em>T</em>, of the trapezium can be expressed as</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;">\[T = \frac{1}{2}{r^2}\sin \theta - \frac{1}{2}{r^2}\sin 2\theta .\]</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) (i) Show that when the area is maximum, the value of \(\theta \) satisfies</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;">\[\cos \theta = 2\cos 2\theta .\]</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) <strong>Hence</strong> determine the maximum area of the trapezium when <em>r</em> = 1.</span></p>
<p style="margin: 0px 0px 0px 60px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (Note: It is not required to prove that it is a maximum.)</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) \({\rm{O\hat AB}} = \pi - \theta \,\,\,\,\,\)(allied) <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;">recognizing OAB as an isosceles triangle <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;">so \({\rm{A\hat BO}} = \pi - \theta \)</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;">\({\rm{B\hat OC}} = \pi - \theta \,\,\,\,\,\)(alternate) <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>Note:</strong> This can be done in many ways, including a clear diagram.</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>[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;"><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) area of trapezium is \(T = {\text{are}}{{\text{a}}_{\Delta {\text{BOC}}}} + {\text{are}}{{\text{a}}_{\Delta {\text{AOB}}}}\) <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;">\( = \frac{1}{2}{r^2}\sin (\pi - \theta ) + \frac{1}{2}{r^2}\sin (2\theta - \pi )\) <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;">\( = \frac{1}{2}{r^2}\sin \theta - \frac{1}{2}{r^2}\sin 2\theta \) <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;"><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;">(c) (i) \(\frac{{{\text{d}}T}}{{{\text{d}}\theta }} = \frac{1}{2}{r^2}\cos \theta - {r^2}\cos 2\theta \) <strong><em>M1A1</em></strong></span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for maximum area \(\frac{1}{2}{r^2}\cos \theta - {r^2}\cos 2\theta = 0\) <strong><em>M1</em></strong></span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\cos \theta = 2\cos 2\theta \) <strong><em>AG</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;"><strong><em> </em></strong></span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; font: 25px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\theta _{\max }} = 2.205 \ldots \) <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;">\(\frac{1}{2}\sin {\theta _{\max }} - \frac{1}{2}\sin 2{\theta _{\max }} = 0.880\) <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>[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;"><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;"><strong><em>Total [11 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;">In part (a) students had difficulties supporting their statements and were consequently unable to gain all the marks here. There were some good attempts at parts (b) and (c) although many students failed to recognise r as a constant and hence differentiated it, often incorrectly.</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;">The diagram shows the plan of an art gallery <em>a </em>metres wide. [AB] represents a doorway, leading to an exit corridor <em>b </em>metres wide. In order to remove a painting from the art gallery, CD (denoted by <em>L </em>) is measured for various values of \(\alpha \) , as represented in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">If </span><span style="font: 12.5px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha \)</span> </span><span style="font-family: 'times new roman', times; font-size: medium;">is the angle between [CD] and the wall, show that \(L = \frac{a }{{\sin \alpha }} + \frac{b}{{\cos \alpha }}{\text{, }}0 < \alpha < \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </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: 12.0px Times;"><span style="font-family: arial, helvetica, sans-serif;"><span style="font-family: 'times new roman', times; font-size: medium;">If <em>a </em>= 5 and <em>b </em>= 1, find the maximum length of a painting that can be removed through this doorway.</span><br></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</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;">Find \(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}\).</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 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 <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in terms of <em>k </em>, the maximum length of a painting that can be removed from the gallery through this doorway.</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 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 <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the minimum value of <em>k </em>if a painting 8 metres long is to be removed through this doorway.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(L = {\text{CA}} + {\text{AD}}\) <em style="font-style: italic;"><strong style="font-weight: bold;">M1</strong></em></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;">\({\text{sin}}\alpha {\text{ = }}\frac{a}{{{\text{CA}}}} \Rightarrow {\text{CA}} = \frac{a}{{\sin \alpha }}\) <em style="font-style: italic;"><strong style="font-weight: bold;">A1</strong></em></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;">\(\cos \alpha = \frac{b}{{{\text{AD}}}} \Rightarrow {\text{AD}} = \frac{b}{{\cos \alpha }}\) <em style="font-style: italic;"><strong style="font-weight: bold;">A1</strong></em></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;">\(L = \frac{a}{{\sin \alpha }} + \frac{b}{{\cos \alpha }}\) <em style="font-style: italic;"><strong style="font-weight: bold;">AG</strong></em></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;"><em style="font-style: italic;"><strong style="font-weight: bold;">[2 marks]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 5{\text{ and }}b = 1 \Rightarrow L = \frac{5}{{\sin \alpha }} + \frac{1}{{\cos \alpha }}\)</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong style="font-weight: bold;">METHOD 1</strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><em style="font-style: italic;"><strong style="font-weight: bold;"><img 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" alt> (M1)</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">minimum from graph \( \Rightarrow L = 7.77\) <strong><em>(M1)A1</em></strong></span></p>
<p style="font-family: arial, helvetica, sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">minimum of <em>L </em>gives the max length of the painting <strong><em>R1</em></strong></span></p>
<p style="font-family: arial, helvetica, sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="font-family: arial, helvetica, sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="font-family: arial, helvetica, sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }} = \frac{{ - 5\cos \alpha }}{{{{\sin }^2}\alpha }} + \frac{{\sin \alpha }}{{{{\cos }^2}\alpha }}\) <em style="font-style: italic;"><strong> (M1)</strong></em></span></p>
<p style="font-family: arial, helvetica, sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }} = 0 \Rightarrow \frac{{{{\sin }^3}\alpha }}{{{{\cos }^3}\alpha }} = 5 \Rightarrow \tan \alpha = \sqrt[{3{\text{ }}}]{5}{\text{ }}(\alpha = 1.0416...)\) <strong> <em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">minimum of <em>L </em>gives the max length of the paintin</span></span><span style="font-family: 'times new roman', times; font-size: medium;">g <strong><em>R1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">maximum length = 7.77 </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }} = \frac{{ - 3k\cos \alpha }}{{{{\sin }^2}\alpha }} + \frac{{k\sin \alpha }}{{{{\cos }^2}\alpha }}\,\,\,\,\,{\text{(or equivalent)}}\) <em style="font-style: italic;"><strong style="font-weight: bold;">M1A1A1</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><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;">\(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }} = \frac{{ - 3k{{\cos }^3}\alpha + k{{\sin }^3}\alpha }}{{{{\sin }^2}\alpha {{\cos }^2}\alpha }}\) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }} = 0 \Rightarrow \frac{{{{\sin }^3}\alpha }}{{{{\cos }^3}\alpha }} = \frac{{3k}}{k} \Rightarrow \tan \alpha = \sqrt[3]{3}\,\,\,\,\,(\alpha = 0.96454...)\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\tan \alpha = \sqrt[3]{3} \Rightarrow \frac{1}{{\cos \alpha }} = \sqrt {1 + \sqrt[3]{9}} \,\,\,\,\,(1.755...)\) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{and }}\frac{1}{{\sin \alpha }} = \frac{{\sqrt {1 + \sqrt[3]{9}} }}{{\sqrt[3]{3}}}\,\,\,\,\,(1.216...)\) <strong> <em>(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(L = 3k\left( {\frac{{\sqrt {1 + \sqrt[3]{9}} }}{{\sqrt[3]{3}}}} \right) + k\sqrt {1 + \sqrt[3]{9}} \,\,\,\,\,(L = 5.405598...k)\) <strong><em>A1 N4</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;"><strong><em>[6 marks]</em></strong></span></p>
<p> </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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(L \leqslant 8 \Rightarrow k \geqslant 1.48\) <em><strong>M1A1</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;">the minimum value is 1.48</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>[2 marks]</em></strong></span></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 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 very well done by most candidates. Parts (b), (c) and (d) required a subtle balance between abstraction, differentiation skills and use of GDC. </span></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;">In part (b), although candidates were asked to justify their reasoning, very few candidates offered an explanation for the maximum. Therefore most candidates did not earn the R1 mark in part (b). Also not as many candidates as anticipated used a graphical approach, preferring to use the calculus with varying degrees of success. In part (c), some candidates calculated the derivatives of inverse trigonometric functions. Some candidates had difficulty with parts (d) and (e). In part (d), some candidates erroneously used their alpha value from part (b). In part (d) many candidates used GDC to calculate decimal values for \(\alpha \) and <em>L</em>. The premature rounding of decimals led sometimes to inaccurate results. Nevertheless many candidates got excellent results in this question.</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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was very well done by most candidates. Parts (b), (c) and (d) required a subtle balance between abstraction, differentiation skills and use of GDC. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), although candidates were asked to justify their reasoning, very few candidates offered an explanation for the maximum. Therefore most candidates did not earn the R1 mark in part (b). Also not as many candidates as anticipated used a graphical approach, preferring to use the calculus with varying degrees of success. In part (c), some candidates calculated the derivatives of inverse trigonometric functions. Some candidates had difficulty with parts (d) and (e). In part (d), some candidates erroneously used their alpha value from part (b). In part (d) many candidates used GDC to calculate decimal values for \(\alpha \) and <em>L</em>. The premature rounding of decimals led sometimes to inaccurate results. Nevertheless many candidates got excellent results in this question.</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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was very well done by most candidates. Parts (b), (c) and (d) required a subtle balance between abstraction, differentiation skills and use of GDC. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), although candidates were asked to justify their reasoning, very few candidates offered an explanation for the maximum. Therefore most candidates did not earn the R1 mark in part (b). Also not as many candidates as anticipated used a graphical approach, preferring to use the calculus with varying degrees of success. In part (c), some candidates calculated the derivatives of inverse trigonometric functions. Some candidates had difficulty with parts (d) and (e). In part (d), some candidates erroneously used their alpha value from part (b). In part (d) many candidates used GDC to calculate decimal values for \(\alpha \) and <em>L</em>. The premature rounding of decimals led sometimes to inaccurate results. Nevertheless many candidates got excellent results in this question.</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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was very well done by most candidates. Parts (b), (c) and (d) required a subtle balance between abstraction, differentiation skills and use of GDC. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), although candidates were asked to justify their reasoning, very few candidates offered an explanation for the maximum. Therefore most candidates did not earn the R1 mark in part (b). Also not as many candidates as anticipated used a graphical approach, preferring to use the calculus with varying degrees of success. In part (c), some candidates calculated the derivatives of inverse trigonometric functions. Some candidates had difficulty with parts (d) and (e). In part (d), some candidates erroneously used their alpha value from part (b). In part (d) many candidates used GDC to calculate decimal values for \(\alpha \) and <em>L</em>. The premature rounding of decimals led sometimes to inaccurate results. Nevertheless many candidates got excellent results in this question.</span></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: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was very well done by most candidates. Parts (b), (c) and (d) required a subtle balance between abstraction, differentiation skills and use of GDC. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), although candidates were asked to justify their reasoning, very few candidates offered an explanation for the maximum. Therefore most candidates did not earn the R1 mark in part (b). Also not as many candidates as anticipated used a graphical approach, preferring to use the calculus with varying degrees of success. In part (c), some candidates calculated the derivatives of inverse trigonometric functions. Some candidates had difficulty with parts (d) and (e). In part (d), some candidates erroneously used their alpha value from part (b). In part (d) many candidates used GDC to calculate decimal values for \(\alpha \) and <em>L</em>. The premature rounding of decimals led sometimes to inaccurate results. Nevertheless many candidates got excellent results in this question.</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two discs, one of radius 8 cm and one of radius 5 cm, are placed such that they touch each other. A piece of string is wrapped around the discs. This is shown in the diagram below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<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;">Calculate the length of string needed to go around the discs.</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;"><img 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" alt></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;">\({\text{AC}} = {\text{BD}} = \sqrt {{{13}^2} - {3^2}} = 12.64...\) <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;">\(\cos \alpha = \frac{3}{{13}} \Rightarrow \alpha = 1.337...(76.65...^\circ .)\) <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;">attempt to find either arc length AB or arc length CD <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;">\({\text{arc length AB}} = 5(\pi - 2 \times 0.232...){\text{ }}( = 13.37...)\) <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;">\({\text{arc length CD}} = 8(\pi + 2 \times 0.232...){\text{ }}( = 28.85...)\) <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;">length of string = 13.37... + 28.85... + 2(12.64...) <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;">= 67.5 (cm) <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;">[8 marks]</span><br></em></strong></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: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that this was the last question in section A it was pleasing to see a good number of candidates make a start on the question. As would be expected from a question at this stage of the paper, more limited numbers of candidates gained full marks. A number of candidates made the question very difficult by unnecessarily splitting the angles required to find the final answer into combinations of smaller angles, all of which required a lot of work and time.</span></p>
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