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</div><h2>SL Paper 2</h2><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a pole BT 1.6 m tall on the roof of a vertical building.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The angle of depression from T to a point A on the horizontal ground is \({35^ \circ }\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The angle of elevation of the top of the building from A is \({30^ \circ }\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/building.png" alt></span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the height of the building.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong> M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. completed diagram</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt at set up <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. correct placement of one angle</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\tan 30 = \frac{h}{x}\) , \(\tan 35 = \frac{{h + 1.6}}{x}\) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to set up equation <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. isolate <em>x</em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{h}{{\tan 30}} = \frac{{h + 1.6}}{{\tan 35}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h = 7.52\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\sin 30 = \frac{h}{l}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">in triangle ATB, \(\widehat A = {5^ \circ }\) , \(\widehat T = {55^ \circ }\) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">choosing sine rule <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{h/\sin 30}}{{\sin 55}} = \frac{{1.6}}{{\sin 5}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h = \frac{{1.6 \times \sin 30 \times \sin 55}}{{\sin 5}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h = 7.52\) <em><strong>A1 N3</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">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the following circle with centre O and radius <em>r</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/Jamie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P, R and Q are on the circumference, \({\rm{P}}\widehat {\rm{O}}{\rm{Q}} = 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) . </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the cosine rule to show that \({\rm{PQ}} = 2r\sin \theta \) .</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><span style="font-family: times new roman,times; font-size: medium;">Let <em>l</em> be the length of the arc PRQ .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(1.3{\rm{PQ}} - l = 0\) , find the value of \(\theta \) .</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><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(\theta ) = 2.6\sin \theta - 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down the root of \(f(\theta ) = 0\) .</span></p>
<p> </p>
<div class="marks">[4]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the graph of <em>f</em> to find the values of \(\theta \) for which \(l < 1.3{\rm{PQ}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into cosine rule <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = {r^2} + {r^2} - 2(r)(r)\cos (2\theta )\) , \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 2{r^2} - 2{r^2}(\cos (2\theta ))\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(1 - 2{\sin ^2}\theta \) for \(\cos 2\theta \) (seen anywhere) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 2{r^2} - 2{r^2}(1 - 2{\sin ^2}\theta )\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working towards answer <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 2{r^2} - 2{r^2} + 4{r^2}{\sin ^2}\theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing \(2{r^2} - 2{r^2} = 0\) (including crossing out) (seen anywhere) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 4{r^2}{\sin ^2}\theta \) , \({\rm{PQ}} = \sqrt {4{r^2}{{\sin }^2}\theta } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{PQ = 2}}r{\rm{sin}}\theta \) <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{PRQ}} = r \times 2\theta \) (seen anywhere) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct set up <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1.3 \times 2r\sin \theta - r \times (2\theta ) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to eliminate <em>r</em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation in terms of the one variable \(\theta \) <em><strong>(A1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1.3 \times 2\sin \theta - 2\theta = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">1.221496215 </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 1.22\) (accept \(70.0^\circ \) (69.9)) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/fedup.png" alt> <em><strong>A1A1A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <strong><em>A1</em></strong> for approximately correct shape, <strong><em>A1</em></strong> for <em>x</em>-intercept in approximately </span><span style="font-family: times new roman,times; font-size: medium;">correct position, <strong><em>A1</em></strong> for domain. Do not penalise if sketch starts at origin.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(1.221496215\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 1.22\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[4 marks]</strong> </span></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach (may be seen earlier) <strong><em>M2 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\theta < 2.6\sin \theta \) , \(0 < f(\theta )\) , showing positive part of sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0 < \theta < 1.221496215\)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0 < \theta = 1.22\) (accept \(\theta < 1.22\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This exercise seemed to be challenging for the great majority of the candidates, in particular parts (b), (c) and (d).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally attempted using the cosine rule, but many failed to substitute correctly into the right hand side or skipped important steps. A high percentage could not arrive at the given expression due to a lack of knowledge of trigonometric identities or making algebraic errors, and tried to force their way to the given answer.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The most common errors included taking the square root too soon, and sign errors when distributing the negative after substituting \(\cos 2\theta \) by \(1 - 2{\sin ^2}\theta \) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This exercise seemed to be challenging for the great majority of the candidates, in particular parts (b), (c) and (d). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), most candidates understood what was required but could not find the correct length of the arc PRQ mainly due to substituting the angle by \(\theta \) instead of \(2\theta \) . </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Regarding part (c), many valid approaches were seen for the graph of <em>f</em>, making a good use of their GDC. A common error was finding a second or third solution outside the domain. A considerable amount of sketches were missing a scale. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There were candidates who achieved the correct equation but failed to realize they could use their GDC to solve it. </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was attempted by very few, and of those who achieved the correct answer not many were able to show the method they used. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the triangle ABC, where AB =10 , BC = 7 and \({\rm{C}}\widehat {\rm{A}}{\rm{B}}\) = \({30^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the two possible values of \({\rm{A}}\widehat {\rm{C}}{\rm{B}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Hence, find \({\rm{A}}\widehat {\rm{B}}{\rm{C}}\) , given that it is acute.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: accept answers given in degrees, and minutes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing sine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin A}}{a} = \frac{{\sin B}}{b}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin \theta }}{{10}} = \frac{{\sin {{30}^ \circ }}}{7}\) , \(\sin \theta = \frac{5}{7}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{C}}{\rm{B}} = {45.6^ \circ }\) , \({\rm{A}}\widehat {\rm{C}}{\rm{B}} = {134^ \circ }\) <em><strong>A1A1 N1N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: If candidates only find the acute angle in part (a), award no marks for (b).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute their larger value into angle sum of triangle <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({180^ \circ } - (134.415{ \ldots ^ \circ } + {30^ \circ })\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {15.6^ \circ }\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">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;">Most candidates were comfortable applying the sine rule, although many were then unable to find the obtuse angle, demonstrating a lack of understanding of the ambiguous case. This precluded them from earning marks in part (b). Those who found the obtuse angle generally had no difficulty with part (b). </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;">Most candidates were comfortable applying the sine rule, although many were then unable to find the obtuse angle, demonstrating a lack of understanding of the ambiguous case. This precluded them from earning marks in part (b). Those who found the obtuse angle generally had no difficulty with part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = a\cos (b(x - c))\) . The diagram below shows part of the graph of <em>f</em> , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le x \le 10\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q5.jpg" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph has a local maximum at P(3, 5) , a local minimum at Q(7, − 5) , and crosses the <em>x</em>-axis at R.</span></p>
<p align="LEFT"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(c\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>b</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><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of R.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(a = 5\) (accept \( - 5\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(c = 3\) (accept \(c = 7\) , if \(a = - 5\) ) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept other correct values of <em>c</em>, such as 11, \( - 5\), etc.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find period <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 8 , \(b = \frac{{2\pi }}{{{\rm{period}}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.785398 \ldots \)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{{2\pi }}{8}\) (exact), \(\frac{\pi }{4}\) , 0.785 [\(0.785{\text{, }}0.786\)] (do not accept 45) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = 0\) , symmetry of curve </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 5\) (accept \((5{\text{ ,}}0))\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) (i) was well answered in general. There were more difficulties in finding the correct value of the parameter <em>c</em>. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Finding the correct value of <em>b</em> in part (b) also proved difficult as many did not realize the period was equal to 8. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates could handle part (c) without difficulties using their GDC or working with the symmetry of the curve although follow through from errors in part (b) was often not awarded because candidates failed to show any working by writing down the equations they entered into their GDC. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The height, \(h\) metros, of a seat on a Ferris wheel after \(t\) minutes is given by</p>
<p class="p1">\[h(t) = - 15\cos 1.2t + 17,{\text{ for }}t \geqslant 0{\text{.}}\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the height of the seat when \(t = 0\).</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 class="p1">The seat first reaches a height of 20 m <span class="s1">after \(k\) minutes. Find \(k\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the time needed for the seat to complete a full rotation, giving your answer correct to one decimal place.</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">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(h(0),{\text{ }} - 15\cos (1.2 \times 0) + 17,{\text{ }} - 15(1) + 17\)</p>
<p class="p1"><span class="Apple-converted-space">\(h(0) = 2{\text{ (m)}}\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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">correct substitution into equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(20 = - 15\cos 1.2t + 17,{\text{ }} - 15\cos 1.2k = 3\)</p>
<p class="p1">valid attempt to solve for \(k\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)<img src="images/Schermafbeelding_2017-02-03_om_06.31.48.png" alt="M16/5/MATME/SP2/ENG/TZ2/04.b/M">, \(\cos 1.2k = - \frac{3}{{15}}\)</p>
<p class="p2">1.47679</p>
<p class="p3"><span class="s1">\(k = 1.48\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognize the need to find the period (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)next \(t\) value when \(h = 20\)</p>
<p class="p1">correct value for period <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{period}} = \frac{{2\pi }}{{1.2}},{\text{ }}5.23598,{\text{ }}6.7--1.48\)</p>
<p class="p2">5.2 (min) (must be 1 <span class="s1">dp) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></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;">
<p class="p1">Candidates did quite well at part a). Most substituted correctly but considered \(\cos 0 = 0\), obtaining an incorrect answer of 17.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates understood that they needed to solve \(h(t) = 20\)<span class="s1"><em>, </em></span>but could not do it. A considerable number of students tried to solve the equation algebraically and the most common errors were to obtain \(\cos k = \frac{{ - 0.2}}{{1,2}}\) or \(k = \frac{{ - 3}}{{15\cos 1.2}}\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) proved difficult as many students had difficulties recognizing they needed to find the period of the function and many who could, did not round the final answer to one decimal place.</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;">The diagram below shows a circle centre O, with radius <em>r</em>. The length of arc ABC is </span><span style="font-family: times new roman,times; font-size: medium;">\(3\pi {\text{ cm}}\) and \({\rm{A}}\widehat {\rm{O}}{\rm{C}} = \frac{{2\pi }}{9}\). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/circle.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>r</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the perimeter of sector OABC.</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><span style="font-family: times new roman,times; font-size: medium;">Find the area of sector OABC.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3\pi = r\frac{{2\pi }}{9}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(r = 13.5\) (cm) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">adding two radii plus \(3\pi \) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{perimeter}} = 27 + 3\pi \) (cm) (\(= 36.4\)) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2} \times {13.5^2} \times \frac{{2\pi }}{9}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area \( = 20.25\pi \) (\({\text{cm}}^2\)) (\(= 63.6\)) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Few errors were made in this question. Those that were made were usually arithmetical in </span><span style="font-family: times new roman,times; font-size: medium;">nature.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Few errors were made in this question. Those that were made were usually arithmetical in </span><span style="font-family: times new roman,times; font-size: medium;">nature.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Few errors were made in this question. Those that were made were usually arithmetical in </span><span style="font-family: times new roman,times; font-size: medium;">nature.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Triangle ABC has <em>a</em> = 8.1 cm, <em>b</em> = 12.3 cm and area 15 cm<sup>2</sup>. Find the largest possible perimeter of triangle ABC.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>correct substitution into the formula for area of a triangle <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 15 = \(\frac{1}{2}\) × 8.1 × 12.3 × sin <em>C</em></p>
<p>correct working for angle <em>C</em> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> sin <em>C</em> = 0.301114, 17.5245…, 0.305860</p>
<p>recognizing that obtuse angle needed <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 162.475, 2.83573, cos <em>C</em> < 0</p>
<p>evidence of choosing the cosine rule <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em>a</em><sup>2</sup> = <em>b</em><sup>2</sup> + <em>c</em><sup>2</sup> − 2<em>bc</em> cos(<em>A</em>)</p>
<p>correct substitution into cosine rule to find <em>c</em> <em><strong>(A1)</strong></em></p>
<p><em>eg c</em><sup>2</sup> = (8.1)<sup>2</sup> + (12.3)<sup>2</sup> − 2(8.1)(12.3) cos <em>C</em></p>
<p><em>c</em> = 20.1720 <em><strong>(A1)</strong></em></p>
<p>8.1 + 12.3 + 20.1720 = 40.5720</p>
<p>perimeter = 40.6 <em><strong>A1 N4</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows three towns A, B and C. Town B is 5 km from Town A, on a bearing of 070°. Town C is 8 km from Town B, on a bearing of 115°<span class="s1">.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-01_om_15.15.32.png" alt="M16/5/MATME/SP2/ENG/TZ1/03"></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\rm{A\hat BC}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the distance from Town <span class="s1">A </span>to Town <span class="s1">C</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the sine rule to find \({\rm{A\hat CB}}\).</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">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(70 + (180 - 115),{\text{ }}360 - (110 + 115)\)</p>
<p class="p1"><span class="Apple-converted-space">\({\rm{A\hat BC}} = 135^\circ \) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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">choosing cosine rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({c^2} = {a^2} + {b^2} - 2ab\cos C\)</p>
<p class="p1">correct substitution into RHS <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({5^2} + {8^2} - 2 \times 5 \times 8\cos 135\)</p>
<p class="p2">12.0651</p>
<p class="p2">12.1 (km) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 N2</em></strong></span></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 class="p1">correct substitution (<strong>must </strong>be into sine rule) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\sin {\rm{A\hat CB}}}}{5} = \frac{{\sin 135}}{{{\text{AC}}}}\)</p>
<p class="p2">17.0398</p>
<p class="p1"><span class="s1">\({\rm{A\hat CB}} = 17.0\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some candidates tackled this question very competently, whilst others struggled to obtain a correct answer even for part (a) which would generally be regarded as prior learning.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (b) and (c) were generally answered well, even with follow through from an incorrect angle in part (a). Weaker candidates assumed the triangle to be a right triangle and attempted to use Pythagoras to find AC. One of the most significant errors seen throughout this question was with candidates substituting an angle in degrees into a calculator set in radian mode.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Parts (b) and (c) were generally answered well, even with follow through from an incorrect angle in part (a). Weaker candidates assumed the triangle to be a right triangle and attempted to use Pythagoras to find AC. One of the most significant errors seen throughout this question was with candidates substituting an angle in degrees into a calculator set in radian mode.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>At Grande Anse Beach the height of the water in metres is modelled by the function \(h(t) = p\cos (q \times t) + r\), where \(t\) is the number of hours after 21:00 hours on 10 December 2017. The following diagram shows the graph of \(h\) , for \(0 \leqslant t \leqslant 72\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.10.26.png" alt="M17/5/MATME/SP2/ENG/TZ1/08"></p>
<p>The point \({\text{A}}(6.25,{\text{ }}0.6)\) represents the first low tide and \({\text{B}}(12.5,{\text{ }}1.5)\) represents the next high tide.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>How much time is there between the first low tide and the next high tide?</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the difference in height between low tide and high tide.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(q\);</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>Find the value of \(r\).</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>There are two high tides on 12 December 2017. At what time does the second high tide occur?</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>attempt to find the difference of \(x\)-values of A and B <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(6.25 - 12.5{\text{ }}\)</p>
<p>6.25 (hours), (6 hours 15 minutes) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the difference of \(y\)-values of A and B <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(1.5 - 0.6\)</p>
<p>\(0.9{\text{ (m)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{max}} - {\text{min}}}}{2},{\text{ }}0.9 \div 2\)</p>
<p>\(p = 0.45\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>period \( = 12.5\) (seen anywhere) <strong><em>(A1)</em></strong></p>
<p>valid approach (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{period}} = \frac{{2\pi }}{b},{\text{ }}q = \frac{{2\pi }}{{{\text{period}}}},{\text{ }}\frac{{2\pi }}{{12.5}}\)</p>
<p>0.502654</p>
<p>\(q = \frac{{4\pi }}{{25}},{\text{ 0.503 }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to use a coordinate to make an equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(p\cos (6.25q) + r = 0.6,{\text{ }}p\cos (12.5q) + r = 1.5\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0.45\cos (6.25q) + 1.05 = 0.6,{\text{ }}0.45\cos (12.5q) + 1.05 = 1.5\)</p>
<p>0.502654</p>
<p>\(q = \frac{{4\pi }}{{25}},{\text{ }}0.503{\text{ }}\left( {{\text{or }} - \frac{{4\pi }}{{25}},{\text{ }} - 0.503} \right)\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid method to find \(r\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{max}} + {\text{min}}}}{2},{\text{ }}0.6 + 0.45\)</p>
<p>\(r = 1.05\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to find start or end \(t\)-values for 12 December <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3 + 24,{\text{ }}t = 27,{\text{ }}t = 51\)</p>
<p>finds \(t\)-value for second max <strong><em>(A1)</em></strong></p>
<p>\(t = 50\)</p>
<p>23:00 (or 11 pm) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong>METHOD 2 </strong></p>
<p>valid approach to list either the times of high tides after 21:00 or the \(t\)-values of high tides after 21:00, showing at least two times <strong><em>(M1) </em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{21:00}} + 12.5,{\text{ 21:00}} + 25,{\text{ }}12.5 + 12.5,{\text{ }}25 + 12.5\)</p>
<p>correct time of first high tide on 12 December <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)10:30 (or 10:30 am) </p>
<p>time of second high tide = 23:00 <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>attempt to set <strong>their</strong> \(h\) equal to 1.5 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(h(t) = 1.5,{\text{ }}0.45\cos \left( {\frac{{4\pi }}{{25}}t} \right) + 1.05 = 1.5\)</p>
<p>correct working to find second max <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0.503t = 8\pi ,{\text{ }}t = 50\)</p>
<p>23:00 (or 11 pm) <strong><em>A1</em></strong> <strong><em>N3</em></strong></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">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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The diagram shows a circle of radius \(8\) metres. The points ABCD lie on the </span><span style="font-family: times new roman,times; font-size: medium;">circumference of the circle.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/jack.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">BC = \(14\) m, CD = \(11.5\) m, AD = \(8\) m, \(A\hat DC = {104^ \circ }\) , and \(B\hat CD = {73^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find AC.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(A\hat CD\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find \(A\hat CB\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the area of triangle ADC.</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><span style="font-family: times new roman,times; font-size: medium;">(c) Find the area of triangle ADC.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) </span><span style="font-family: times new roman,times; font-size: medium;">Hence or otherwise, find the total area of the shaded regions.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">cd.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence or otherwise, find the total area of the shaded regions.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({c^2} = {a^2} + {b^2} - 2ab\cos C\) , \({\rm{C}}{{\rm{D}}^2} + {\rm{A}}{{\rm{D}}^2} - 2 \times {\rm{CD}} \times {\rm{AD}}\cos D\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({11.5^2} + {8^2} - 2 \times 11.5 \times 8\cos 104\) , \(196.25 - 184\cos 104\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">AC \( = 15.5\) (m) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) <strong>METHOD 1 </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing sine rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{\sin A}}{a} = \frac{{\sin B}}{b}\) , \(\frac{{\sin {\rm{A}}\widehat {\rm{C}}{\rm{D}}}}{{{\rm{AD}}}} = \frac{{\sin D}}{{{\rm{AC}}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <strong><em> A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{\sin {\rm{A}}\hat {\rm{C}}{\rm{D}}}}{8} = \frac{{\sin 104}}{{15.516 \ldots }}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\hat {\rm{C}}{\rm{D}} = {30.0^ \circ }\) <em><strong>A1 N2 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({c^2} = {a^2} + {b^2} - 2ab\cos C\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({8^2} = {11.5^2} + 15.516{ \ldots ^2} - 2(11.5)(15.516 \ldots )\cos C\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\hat {\rm{C}}{\rm{D}} = {30.0^ \circ }\) <em><strong>A1 N2</strong> </em></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;">(ii) subtracting <strong>their</strong> \({\rm{A}}\hat {\rm{C}}{\rm{D}}\) from \(73\) <strong><em> (M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{7}}3 - {\rm{A}}\hat {\rm{C}}{\rm{D}}\) , \(70 - 30.017 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\hat {\rm{C}}{\rm{B}} = {43.0^ \circ }\) <strong><em>A1 N2 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></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;">correct substitution <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> area \(\Delta {\rm{ADC}} = \frac{1}{2}(8)(11.5)\sin 104\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \( = 44.6\) (m<sup>2</sup>) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(c) correct substitution <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> area \(\Delta {\rm{ADC}} = \frac{1}{2}(8)(11.5)\sin 104\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \( = 44.6\) (m<sup>2</sup>) <em><strong>A1 N2 </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><em><strong><span style="font-family: times new roman,times; font-size: medium;"><br></span></strong></em><span style="font-family: times new roman,times; font-size: medium;">(d) attempt to subtract <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{circle}} - {\rm{ABCD}}\) , \(\pi {r^2} - \Delta {\rm{ADC}} - \Delta {\rm{ACB}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \(\Delta {\rm{ACB = }}\frac{1}{2}(15.516 \ldots )(14)\sin 42.98\) <strong><em> (A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\pi {(8)^2} - 44.6336 \ldots - \frac{1}{2}(15.516 \ldots )(14)\sin 42.98\) , \(64\pi - 44.6 - 74.1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">shaded area is \(82.4\) (m<sup>2</sup>) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 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 [6 marks]<br></span></strong></em></p>
<div class="question_part_label">cd.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to subtract <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{circle}} - {\rm{ABCD}}\) , \(\pi {r^2} - \Delta {\rm{ADC}} - \Delta {\rm{ACB}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \(\Delta {\rm{ACB = }}\frac{1}{2}(15.516 \ldots )(14)\sin 42.98\) <strong><em> (A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\pi {(8)^2} - 44.6336 \ldots - \frac{1}{2}(15.516 \ldots )(14)\sin 42.98\) , \(64\pi - 44.6 - 74.1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">shaded area is \(82.4\) (m<sup>2</sup>) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 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 [6 marks]<br></span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was an error on this question, where the measurements were inconsistent. Whichever method a candidate used to answer the question, the inconsistencies did not cause a problem. The markscheme included a variety of solutions based on possible combinations of solutions, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite this unfortunate error, the question posed few difficulties for candidates and most approached the problem as intended. Although there were other ways to approach the problem (using properties of cyclic quadrilaterals) few considered this, likely due to the fact that cyclic quadrilaterals is not part of the syllabus.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was an error on this question, where the measurements were inconsistent. Whichever method a candidate used to answer the question, the inconsistencies did not cause a problem. The markscheme included a variety of solutions based on possible combinations of solutions, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite this unfortunate error, the question posed few difficulties for candidates and most approached the problem as intended. Although there were other ways to approach the problem (using properties of cyclic quadrilaterals) few considered this, likely due to the fact that cyclic quadrilaterals is not part of the syllabus.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was an error on this question, where the measurements were inconsistent. Whichever method a candidate used to answer the question, the inconsistencies did not cause a problem. The markscheme included a variety of solutions based on possible combinations of solutions, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite this unfortunate error, the question posed few difficulties for candidates and most approached the problem as intended. Although there were other ways to approach the problem (using properties of cyclic quadrilaterals) few considered this, likely due to the fact that cyclic quadrilaterals is not part of the syllabus. Candidates were proficient in their use of sine and cosine rules and most could find the area of the required triangle in part (c). Those who made errors in this question either had their GDC in the wrong mode or were rounding values prematurely while some misinformed candidates treated ADC as a right-angled triangle. <br></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was an error on this question, where the measurements were inconsistent. Whichever method a candidate used to answer the question, the inconsistencies did not cause a problem. The markscheme included a variety of solutions based on possible combinations of solutions, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite this unfortunate error, the question posed few difficulties for candidates and most approached the problem as intended. Although there were other ways to approach the problem (using properties of cyclic quadrilaterals) few considered this, likely due to the fact that cyclic quadrilaterals is not part of the syllabus. Candidates were proficient in their use of sine and cosine rules and most could find the area of the required triangle in part (c). Those who made errors in this question either had their GDC in the wrong mode or were rounding values prematurely while some misinformed candidates treated ADC as a right-angled triangle. In part (d), most candidates recognized what to do and often obtained follow through marks from errors made in previous parts.</span></p>
<div class="question_part_label">cd.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was an error on this question, where the measurements were inconsistent. Whichever method a candidate used to answer the question, the inconsistencies did not cause a problem. The markscheme included a variety of solutions based on possible combinations of solutions, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite this unfortunate error, the question posed few difficulties for candidates and most approached the problem as intended. Although there were other ways to approach the problem (using properties of cyclic quadrilaterals) few considered this, likely due to the fact that cyclic quadrilaterals is not part of the syllabus. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), most candidates recognized what to do and often obtained follow through marks from errors made in previous parts.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the triangle ABC.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sunny.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The angle at C is obtuse, \({\text{AC}} = 5{\text{ cm}}\), \({\text{BC}} =13.6{\text{ cm}}\) and the area is \(20{\text{ c}}{\text{m}}^2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{A}}\widehat {\rm{C}}{\rm{B}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find AB.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into the formula for the area of a triangle <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2} \times 5 \times 13.6 \times \sin C = 20\) , \(\frac{1}{2} \times 5 \times h = 20\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sin C = 0.5882 \ldots \) , \(\sin C = \frac{8}{{13.6}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span lang="EN-US">\(\widehat C = 36.031 \ldots ^\circ \) </span>(\(0.6288 \ldots {\text{ radians}}\)) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{C}}{\rm{B}} = 144^\circ \) \((2.51{\text{ radians}})\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the cosine rule <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({({\rm{AB}})^2} = {5^2} + {13.6^2} - 2(5)(13.6)\cos 143.968 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{AB}} = 17.9\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was well done with the majority of candidates obtaining the acute angle. Unfortunately, the question asked for the obtuse angle which was clearly stated and shown in the diagram. No matter which angle was used, most candidates were able to obtain full marks in part (b) with a simple application of the cosine rule. </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;">Part (a) was well done with the majority of candidates obtaining the acute angle. Unfortunately, the question asked for the obtuse angle which was clearly stated and shown in the diagram. No matter which angle was used, most candidates were able to obtain full marks in part (b) with a simple application of the cosine rule. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows \(\Delta {\rm{PQR}}\) , where RQ = 9 cm, \({\rm{P\hat RQ}} = {70^ \circ }\) and \({\rm{P\hat QR}} = {45^ \circ }\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P2TZ2Q1atri.png" alt></span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{R\hat PQ}}\) .</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;">Find PR .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of \(\Delta {\rm{PQR}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{R\hat PQ = }}{65^ \circ }\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing sine rule <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{PR}}}}{{\sin {{45}^ \circ }}} = \frac{9}{{\sin {{65}^ \circ }}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">7.021854078 </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{PR}} = 7.02\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{area}} = \frac{1}{2} \times 9 \times 7.02 \ldots \times \sin {70^ \circ }\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(29.69273008\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 29.7\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was attempted in a satisfactory manner. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The sine rule was applied satisfactory in part (b) but some obtained an incorrect answer due to having their calculators in radian mode. Some incorrect substitutions were seen, either by choosing an incorrect side or substituting 70 instead of \(\sin {70^ \circ }\) . Approaches using a combination of the cosine rule and/or right-angled triangle trigonometry were seen. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Approaches using a combination of the cosine rule and/or right-angled triangle trigonometry were seen, especially in part (c) to calculate the area of the triangle. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A few candidates set about finding the height, then used the formula for the area of a right-angled triangle. </span></p>
<p> </p>
<p> </p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows triangle ABC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="font: normal normal normal 12px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_1.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: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find AC.</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: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\rm{B\hat CA}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of choosing cosine rule <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><em style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">eg</em><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> \({\text{A}}{{\text{C}}^2} = {\text{A}}{{\text{B}}^2} + {\text{B}}{{\text{C}}^2} - 2({\text{AB}})({\text{BC}})\cos ({\rm{A\hat BC}})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into the right-hand side <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({6^2} + {10^2} - 2(6)(10)\cos {100^ \circ }\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{AC}} = 12.5234\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{AC}} = 12.5{\text{ (cm)}}\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><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: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of choosing a valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>sine rule, cosine rule</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\(\frac{{\sin ({\rm{B\hat CA)}}}}{6} = \frac{{\sin 100^\circ }}{{12.5}},{\text{ }}\cos ({\rm{B\hat CA)}} = \frac{{{{({\text{AC}})}^2} + {{10}^2} - {6^2}}}{{2({\text{AC}})(10)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{B\hat CA}} = 28.1525\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\rm{B\hat CA}} = 28.2^\circ\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two points P and Q have coordinates (3, 2, 5) and (7, 4, 9) respectively.</p>
</div>
<div class="specification">
<p>Let \({\mathop {{\text{PR}}}\limits^ \to }\) = 6<em><strong>i</strong></em> − <em><strong>j</strong></em> + 3<em><strong>k</strong></em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\mathop {{\text{PQ}}}\limits^ \to \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).</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 angle between PQ and PR.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle PQR.</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>Hence or otherwise find the shortest distance from R to the line through P and Q.</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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em>(7, 4, 9) − (3, 2, 5) <em>A − B</em></p>
<p>\(\mathop {{\text{PQ}}}\limits^ \to = \) 4<em><strong>i</strong></em> + 2<em><strong>j</strong></em> + 4<em><strong>k</strong></em> \(\left( { = \left( \begin{gathered}<br> 4 \hfill \\<br> 2 \hfill \\<br> 4 \hfill \\ <br>\end{gathered} \right)} \right)\) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into magnitude formula <em><strong>(A1)</strong></em><br><em>eg</em> \(\sqrt {{4^2} + {2^2} + {4^2}} \)</p>
<p>\(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right| = 6\) <em><strong>A1 N2</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>finding scalar product and magnitudes <em><strong>(A1)(A1)</strong></em></p>
<p>scalar product = (4 × 6) + (2 × (−1) + (4 × 3) (= 34)</p>
<p>magnitude of PR = \(\sqrt {36 + 1 + 9} = \left( {6.782} \right)\)</p>
<p>correct substitution of <strong>their</strong> values to find cos \({\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}\) <em> <strong>M1</strong></em></p>
<p><em>eg </em>cos \({\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}\,\,{\text{ = }}\frac{{24 - 2 + 12}}{{\left( 6 \right) \times \left( {\sqrt {46} } \right)}},\,\,0.8355\)</p>
<p>0.581746</p>
<p>\({\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}\) = 0.582 radians or \({\text{Q}}\mathop {\text{P}}\limits^ \wedge {\text{R}}\) = 33.3° <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong>(A1)</strong></em><br><em>eg</em> \(\frac{1}{2} \times \left| {\mathop {{\text{PQ}}}\limits^ \to } \right| \times \left| {\mathop {{\text{PR}}}\limits^ \to } \right| \times \,\,{\text{sin}}\,P,\,\,\frac{1}{2} \times 6 \times \sqrt {46} \times \,\,{\text{sin}}\,0.582\)</p>
<p>area is 11.2 (sq. units) <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing shortest distance is perpendicular distance from R to line through P and Q <em><strong>(M1)</strong></em></p>
<p><em>eg</em> sketch, height of triangle with base \(\left[ {{\text{PQ}}} \right],\,\,\frac{1}{2} \times 6 \times h,\,\,{\text{sin}}\,33.3^\circ = \frac{h}{{\sqrt {46} }}\)</p>
<p>correct working <em><strong> (A1)</strong></em></p>
<p><em>eg </em>\(\frac{1}{2} \times 6 \times d = 11.2,\,\,\left| {\mathop {{\text{PR}}}\limits^ \to } \right| \times \,\,{\text{sin}}\,P,\,\,\sqrt {46} \times \,\,{\text{sin}}\,33.3^\circ \)</p>
<p>3.72677</p>
<p>distance = 3.73 (units) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a circular play area for children.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q8.jpg" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The circle has centre O and a radius of 20 m, and the points A, B, C and D lie on </span><span style="font-family: times new roman,times; font-size: medium;">the circle. Angle AOB is 1.5 radians.</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the length of the chord [AB].</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of triangle AOB.</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><span style="font-family: times new roman,times; font-size: medium;">Angle BOC is 2.4 radians.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the length of arc ADC.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Angle BOC is 2.4 radians.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the shaded region.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Angle BOC is 2.4 radians.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is to be painted red. Red paint is sold in cans which cost \(\$ 32\)</span><span style="font-family: times new roman,times; font-size: medium;"> each. One can covers \(140{\text{ }}{{\text{m}}^2}\). How much does it cost to buy the paint?</span></p>
<p> </p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: In this question, do not penalise for missing or incorrect units. They are not included in the markscheme, to avoid complex answer lines. </span></p>
<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;">choosing cosine rule (must have cos in it) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({c^2} = {a^2} + {b^2} - 2ab\cos C\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution (into rhs) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({20^2} + {20^2} - 2(20)(20)\cos 1.5\) , \({\rm{AB}} = \sqrt {800 - 800\cos 1.5} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB = 27}}{\rm{.26555}} \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB}} = 27.3\) \([27.2{\text{, }}27.3]\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[3 marks]</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">choosing sine rule <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin A}}{a} = \frac{{\sin B}}{b}\) , \(\frac{{{\rm{AB}}}}{{\sin O}} = \frac{{{\rm{AO}}}}{{\sin B}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{AB}}}}{{\sin 1.5}} = \frac{{20}}{{\sin (0.5(\pi - 1.5))}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB}} = 27.26555 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB}} = 27.3\) \([27.2{\text{, }}27.3]\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into area formula <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}(20)(20)\sin 1.5\) , \(\frac{1}{2}(20)(27.2655504 \ldots )\sin(0.5(\pi - 1.5))\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 199.498997 \ldots \) (accept \(199.75106 = 200\) , from using 27.3) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 199\) \([199{\text{, }}200]\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate method to find angle AOC <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\pi - 1.5 - 2.4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into arc length formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \((2\pi - 3.9) \times 20\) , \(2.3831853 \ldots \times 20\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{arc length}} = 47.6637 \ldots \)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{arc length}} = 47.7\) \((47.6{\text{, }}47.7]\) (i.e. do <strong>not</strong> accept \(47.6\)) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Candidates may misread the question and use \({\rm{A}}\widehat {\rm{O}}{\rm{C}} = 2.4\) . If working shown, award <em><strong>M0</strong></em> then <em><strong>A0MRA1</strong></em> for the answer 48. Do not then penalize \({\rm{A}}\widehat {\rm{O}}{\rm{C}}\) in </span><span style="font-family: times new roman,times; font-size: medium;">part (d) which, if used, leads to the answer \(679.498 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>However</strong>, if they use the prematurely rounded value of 2.4 for \({\rm{A}}\widehat {\rm{O}}{\rm{C}}\) , penalise 1 mark </span><span style="font-family: times new roman,times; font-size: medium;">for premature rounding for the answer 48 in (c). Do not then penalize for this in (d). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">calculating sector area using <strong>their</strong> angle AOC <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}(2.38 \ldots )({20^2})\) , \(200(2.38 \ldots )\) , \(476.6370614 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">shaded area = <strong>their</strong> area of triangle AOB + <strong>their</strong> area of sector <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(199.4989973 \ldots + 476.6370614 \ldots \) , \(199 + 476.637\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{shaded area}} = 676.136 \ldots \) (accept \(675.637 \ldots = 676\) from using 199) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{shaded area}} = 676\) \([676{\text{, }}677]\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">dividing to find number of cans <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{676}}{{140}}\) , \(4.82857 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">5 cans must be purchased <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">multiplying to find cost of cans <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(5(32)\) , \(\frac{{676}}{{140}} \times 32\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">cost is 160 (dollars) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates generally handled the cosine rule, sectors and arcs well, but some candidates incorrectly treated triangle AOB as a right-angled triangle. A surprising number of candidates changed all angles to degrees and worked with those, often leading to errors in accuracy. </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;">Candidates generally handled the cosine rule, sectors and arcs well, but some candidates incorrectly treated triangle AOB as a right-angled triangle. A surprising number of candidates changed all angles to degrees and worked with those, often leading to errors in accuracy. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), some candidates misread the question and used 2.4 as the size of angle AOC while others rounded prematurely leading to the inaccurate answer of 48. In either case, marks were lost. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) proved to be straightforward and candidates were able to obtain full FT marks from errors made in previous parts. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates had a suitable strategy for part (e) and knew to work with a whole number of cans of paint. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = p\cos qx + r\) , for \( - 5 \le x \le 14\) , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/weather.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a minimum point at (0, −3) and a maximum point at (4, 7) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>r</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i), (ii) and (iii).</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 equation \(y = k\) has exactly <strong>two</strong> solutions. Write down the value of <em>k</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of finding the amplitude <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{7 + 3}}{2}\) , amplitude \(= 5\) <br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = - 5\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) period \(= 8\) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(q = 0.785\) \(\left( { = \frac{{2\pi }}{8} = \frac{\pi }{4}} \right)\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(r = \frac{{7 - 3}}{2}\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 2\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = - 3\) (accept \(y = - 3\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not recognize that the value of <em>p</em> was negative. The value of <em>q</em> was often </span><span style="font-family: times new roman,times; font-size: medium;">interpreted incorrectly as the period but most candidates could find the value of <em>r</em>, the vertical </span><span style="font-family: times new roman,times; font-size: medium;">translation.</span></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), candidates either could not find a solution or found too many.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of \(f(x) = a\sin bx + c\), for \(0 \leqslant x \leqslant 12\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_16.53.31.png" alt="N16/5/MATME/SP2/ENG/TZ0/10"></p>
<p class="p1" style="text-align: center;">The graph of \(f\) has a minimum point at \((3,{\text{ }}5)\) and a maximum point at \((9,{\text{ }}17)\).</p>
</div>
<div class="specification">
<p class="p1">The graph of \(g\) is obtained from the graph of \(f\) by a translation of \(\left( {\begin{array}{*{20}{c}} k \\ 0 \end{array}} \right)\). The maximum point on the graph of \(g\) has coordinates \((11.5,{\text{ }}17)\).</p>
</div>
<div class="specification">
<p class="p1">The graph of \(g\) changes from concave-up to concave-down when \(x = w\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(c\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(b = \frac{\pi }{6}\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find the value of \(a\).</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">(i) <span class="Apple-converted-space"> </span>Write down the value of \(k\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find \(g(x)\).</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 \(w\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find the maximum positive rate of change of \(g\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{5 + 17}}{2}\)</p>
<p class="p2"><span class="Apple-converted-space">\(c = 11\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)period is 12, per \( = \frac{{2\pi }}{b},{\text{ }}9 - 3\)</p>
<p class="p3"><span class="Apple-converted-space">\(b = \frac{{2\pi }}{{12}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="s2">\(b = \frac{\pi }{6}\) <span class="Apple-converted-space"> </span></span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(5 = a\sin \left( {\frac{\pi }{6} \times 3} \right) + 11\)</span>, substitution of points</p>
<p class="p2"><span class="s3">\(a = - 6\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{17 - 5}}{2}\)</span>, amplitude is 6</p>
<p class="p2"><span class="s2">\(a = - 6\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[6 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"> \(k = 2.5\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> \(g(x) = - 6\sin \left( {\frac{\pi }{6}(x - 2.5)} \right) + 11\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span><strong>METHOD 1 </strong>Using \(g\)</p>
<p class="p1">recognizing that a point of inflexion is required <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)s</span>ketch, recognizing change in concavity</p>
<p class="p1">evidence of valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(g''(x) = 0\)</span>, sketch, coordinates of max/min on \({g'}\)</p>
<p class="p1">\(w = 8.5\) (exact) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong>METHOD 2 </strong>Using \(f\)</p>
<p class="p1">recognizing that a point of inflexion is required <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)sketch, recognizing change in concavity</p>
<p class="p1">evidence of valid approach involving translation <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(x = w - k\)</span>, sketch, \(6 + 2.5\)</p>
<p class="p1"><span class="s3">\(w = 8.5\) </span>(exact) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach involving the derivative of \(g\) or \(f\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(g'(w),{\text{ }} - \pi \cos \left( {\frac{\pi }{6}x} \right)\)</span>, max on derivative, sketch of derivative</p>
<p class="p1">attempt to find max value on derivative <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\( - \pi \cos \left( {\frac{\pi }{6}(8.5 - 2.5)} \right),{\text{ }}f'(6)\), </span>dot on max of sketch</p>
<p class="p2">3.14159</p>
<p class="p1">max rate of change \( = \pi \) <span class="s3">(exact), 3.14 <span class="Apple-converted-space"> </span></span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows part of the graph of \(y = p\sin (qx) + r\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_09.06.35.png" alt></p>
<p class="p1">The point \({\text{A}}\left( {\frac{\pi }{6},{\text{ }}2} \right)\) is a maximum point and the point \({\text{B}}\left( {\frac{\pi }{6},{\text{ }}1} \right)\) is a minimum point.</p>
<p class="p1">Find the value of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(p\);</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">\(r\);</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">\(q\).</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">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\frac{{2 - 1}}{2},{\text{ }}2 - 1.5\)</p>
<p class="p1">\(p = 0.5\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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">valid approach <strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\frac{{1 + 2}}{2}\)</p>
<p class="p1">\(r = 1.5\) <strong><em>A1 <span class="Apple-converted-space"> </span>N2</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"><strong>METHOD 1</strong></p>
<p class="p1">valid approach (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(q = \frac{{2\pi }}{{{\text{period}}}},{\text{ }}\frac{{2\pi }}{{\left( {\frac{{2\pi }}{3}} \right)}}\)</p>
<p class="p1">period \( = \frac{{2\pi }}{3}\)\(\;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(q = 3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">attempt to substitute one point and <strong>their </strong>values for \(p\) and \(r\) into \(y\) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(2 = 0.5\sin \left( {q\frac{\pi }{6}} \right) + 1.5,{\text{ }}\frac{\pi }{2} = 0.5\sin (q1) + 1.5\)</p>
<p class="p1">correct equation in \(q\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(q\frac{\pi }{6} = \frac{\pi }{2},{\text{ }}q\frac{\pi }{2} = \frac{{3\pi }}{2}\)</p>
<p class="p1">\(q = 3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1">valid reasoning comparing the graph with that of \(\sin x\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)position of max/min, graph goes faster</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)max at \(\frac{\pi }{6}\) not at \(\frac{\pi }{2}\), graph goes \(3\) times as fast</p>
<p class="p1">\(q = 3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates found the correct value for the amplitude and vertical shift, but very few managed to find the correct value of the period and therefore of \(q\) in part (c). Some candidates substituted the coordinates of a point into the function but were not able to write a correct equation in terms of \(q\). Many candidates who found the correct answer did not show sufficient work to gain all three marks. The rubrics stress the need to show working.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates found the correct value for the amplitude and vertical shift, but very few managed to find the correct value of the period and therefore of \(q\) in part (c). Some candidates substituted the coordinates of a point into the function but were not able to write a correct equation in terms of \(q\). Many candidates who found the correct answer did not show sufficient work to gain all three marks. The rubrics stress the need to show working.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates found the correct value for the amplitude and vertical shift, but very few managed to find the correct value of the period and therefore of \(q\) in part (c). Some candidates substituted the coordinates of a point into the function but were not able to write a correct equation in terms of \(q\). Many candidates who found the correct answer did not show sufficient work to gain all three marks. The rubrics stress the need to show working.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>Note: In this question, distance is in millimetres.</strong></p>
<p>Let \(f(x) = x + a\sin \left( {x - \frac{\pi }{2}} \right) + a\), for \(x \geqslant 0\).</p>
</div>
<div class="specification">
<p>The graph of \(f\) passes through the origin. Let \({{\text{P}}_k}\) be any point on the graph of \(f\) with \(x\)-coordinate \(2k\pi \), where \(k \in \mathbb{N}\). A straight line \(L\) passes through all the points \({{\text{P}}_k}\).</p>
</div>
<div class="specification">
<p>Diagram 1 shows a saw. The length of the toothed edge is the distance AB.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_15.10.11.png" alt="N17/5/MATME/SP2/ENG/TZ0/10.d_01"></p>
<p>The toothed edge of the saw can be modelled using the graph of \(f\) and the line \(L\). Diagram 2 represents this model.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_15.11.17.png" alt="N17/5/MATME/SP2/ENG/TZ0/10.d_02"></p>
<p>The shaded part on the graph is called a tooth. A tooth is represented by the region enclosed by the graph of \(f\) and the line \(L\), between \({{\text{P}}_k}\) and \({{\text{P}}_{k + 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f(2\pi ) = 2\pi \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of \({{\text{P}}_0}\) and of \({{\text{P}}_1}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of \(L\).</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>Show that the distance between the \(x\)-coordinates of \({{\text{P}}_k}\) and \({{\text{P}}_{k + 1}}\) is \(2\pi \).</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>A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>substituting \(x = 2\pi \) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2\pi + a\sin \left( {2\pi - \frac{\pi }{2}} \right) + a\)</p>
<p>\(2\pi + a\sin \left( {\frac{{3\pi }}{2}} \right) + a\) <strong><em>(A1)</em></strong></p>
<p>\(2\pi - a + a\) <strong><em>A1</em></strong></p>
<p>\(f(2\pi ) = 2\pi \) <strong><em>AG N0</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>substituting the value of \(k\) <em>(<strong>M1)</strong></em></p>
<p>\({{\text{P}}_0}(0,{\text{ }}0),{\text{ }}{{\text{P}}_1}(2\pi ,{\text{ }}2\pi )\) <strong><em>A1A1 N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the gradient <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{2\pi - 0}}{{2\pi - 0}},{\text{ }}m = 1\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{y - 2\pi }}{{x - 2\pi }} = 1,{\text{ }}b = 0,{\text{ }}y - 0 = 1(x - 0)\)</p>
<p><em>y = x</em> <strong><em>A1 N3</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>subtracting \(x\)-coordinates of \({{\text{P}}_{k + 1}}\) and \({{\text{P}}_k}\) (in any order) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2(k + 1)\pi - 2k\pi ,{\text{ }}2k\pi - 2k\pi - 2\pi \)</p>
<p>correct working (must be in correct order) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2k\pi + 2\pi - 2k\pi ,{\text{ }}\left| {2k\pi - 2(k + 1)\pi } \right|\)</p>
<p>distance is \(2\pi \) <strong><em>AG N0</em></strong></p>
<p><strong><em>[2 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>recognizing the toothed-edge as the hypotenuse <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({300^2} = {x^2} + {y^2}\), sketch</p>
<p>correct working (using their equation of \(L\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({300^2} = {x^2} + {x^2}\)</p>
<p>\(x = \frac{{300}}{{\sqrt 2 }}\) (exact), 212.132 <strong><em>(A1)</em></strong></p>
<p>dividing their value of \(x\) by \(2\pi {\text{ }}\left( {{\text{do not accept }}\frac{{300}}{{2\pi }}} \right)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{212.132}}{{2\pi }}\)</p>
<p>33.7618 <strong><em>(A1)</em></strong></p>
<p>33 (teeth) <strong><em>A1 N2</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>vertical distance of a tooth is \(2\pi \) (may be seen anywhere) <strong><em>(A1)</em></strong></p>
<p>attempt to find the hypotenuse for one tooth <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({x^2} = {(2\pi )^2} + {(2\pi )^2}\)</p>
<p>\(x = \sqrt {8{\pi ^2}} \) (exact), 8.88576 <strong><em>(A1)</em></strong></p>
<p>dividing 300 by their value of \(x\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em></p>
<p>33.7618 <strong><em>(A1)</em></strong></p>
<p>33 (teeth) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3\sin x + 4\cos x\) , for \( - 2\pi \le x \le 2\pi \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the amplitude;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) the <em>x</em>-intercept that lies between \( - \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">and 0.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence write \(f(x)\) in the form \(p\sin (qx + r)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down one value of <em>x</em> such that \(f'(x) = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the two values of <em>k</em> for which the equation \(f(x) = k\) has exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of <em>x</em>, between \(0\) and \(1\), </span><span style="font-family: times new roman,times; font-size: medium;">for which the gradient of <em>f</em> is equal to the gradient of <em>g</em>. Find this value of <em>x</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/mike.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></p>
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>A1</strong></em> for approximately sinusoidal shape, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for end points approximately correct \(( - 2\pi {\text{, }}4)\) \((2\pi {\text{, }}4)\), </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for approximately correct position of graph, (<em>y</em>-intercept \((0{\text{, }}4)\), maximum to right of <em>y</em>-axis). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 5 <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(2\pi \) (6.28) <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \( - 0.927\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 5\sin (x + 0.927)\) (accept \(p = 5\) , \(q = 1\) , \(r = 0.927\) ) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. max/min, sketch of \(f'(x)\) indicating roots </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/wee.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">one 3 s.f. value which rounds to one of \( - 5.6\), \( - 2.5\), \(0.64\), \(3.8\) <em><strong>A1 N2 </strong></em></span></p>
<p> </p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = - 5\) , \(k = 5\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graphical approach (but must involve derivative functions) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/visit.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">each curve <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 0.511\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{1}{{x + 1}}\) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 3\cos x - 4\sin x\) \((5\cos (x + 0.927))\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to solve \(g'(x) = f'(x)\) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 0.511\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A2 N2</strong> </span></em></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[5 marks]</strong> </span></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some graphs in part (a) were almost too detailed for just a sketch but more often, the important features were far from clear. Some graphs lacked scales on the axes. </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;">A number of candidates had difficulty finding the period in part (b)(ii).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A number of candidates had difficulty writing the correct value of <em>q</em> in part (c). </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The most common approach in part (d) was to differentiate and set \(f'(x) = 0\) . Fewer students found the values of <em>x</em> given by the maximum or minimum values on their graphs. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) proved challenging for many candidates, although if candidates answered this part, they generally did so correctly. </span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (f), many candidates were able to get as far as equating the two derivatives but fewer used their GDC to solve the resulting equation. Again, many had trouble demonstrating their method of solution. </span></p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 5\cos \frac{\pi }{4}x\) and \(g(x) = - 0.5{x^2} + 5x - 8\) for \(0 \le x \le 9\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the same diagram, sketch the graphs of <em>f</em> and <em>g</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of \(f\) . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the <em>x</em>-intercept that lies between \(x = 0\) and \(x = 3\) ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the amplitude.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of <em>g</em> . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the two <em>x</em>-intercepts;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the equation of the axis of symmetry.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> . Find the area of <em>R</em>.</span></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><span style="font-family: times new roman,times;"><span style="font-size: medium;"><img src="images/luke.png" alt></span><em><span style="font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for <em>f</em> being of sinusoidal shape, with 2 maxima and one minimum, </span><span style="font-size: medium;"><em><strong>A1</strong></em> for <em>g</em> being a parabola opening down, </span><span style="font-size: medium;"><em><strong>A1</strong></em> for <strong>two</strong> intersection points in approximately correct position. </span></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks] </span></strong></em></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;">(i) \((2{\text{, }}0)\) (accept \(x = 2\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{period}} = 8\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \({\text{amplitude}} = 5\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \((2{\text{, }}0)\) , \((8{\text{, }}0)\) (accept \(x = 2\) , \(x = 8\) ) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(x = 5\) (must be an equation) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">intersect when \(x = 2\) and \(x = 6.79\) (may be seen as limits of integration) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {g - f} \) , \(\int {f(x){\rm{d}}x - \int {g(x){\rm{d}}x}}\) , \(\int_2^{6.79} {\left( {( - 0.5{x^2} + 5x - 8) - \left( {5\cos \frac{\pi }{4}x} \right)} \right)}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 27.6\) <em><strong>A2 N3</strong></em></span></p>
<p><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;">intersect when \(x = 2\) and \(x = 6.79\) (seen anywhere) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach using a sketch of <em>g</em> and <em>f</em> , or \(g - f\) . <em><strong>(M1)</strong></em></span></p>
<p><br><img src="images/going_out.png" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. area = \(A + B - C\) , \(12.7324 + 16.0938 - 1.18129 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 27.6\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Graph sketches were much improved over previous sessions. Most candidates graphed the two functions correctly, but many ignored the domain restrictions. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found parts (b) and (c) accessible, although quite a few did not know how to find the period of the cosine function.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found parts (b) and (c) accessible, although quite a few did not know how to find the period of the cosine function.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) proved elusive to many candidates. Some used creative approaches that split the area into parts above and below the <em>x</em>-axis; while this leads to a correct result, few were able to achieve it. Many candidates were unable to use their GDCs effectively to find points of intersection and the subsequent area. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A ship is sailing north from a point A towards point D. Point C is 175 km north of A. Point D is 60 km north of C. There is an island at E. The bearing of E from A is 055°. The bearing of E from C is 134°. This is shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.15.07.png" alt="M17/5/MATME/SP2/ENG/TZ2/09"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the bearing of A from E.</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>Finds CE.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find DE.</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>When the ship reaches D, it changes direction and travels directly to the island at 50 km per hour. At the same time as the ship changes direction, a boat starts travelling to the island from a point B. This point B lies on (AC), between A and C, and is the closest point to the island. The ship and the boat arrive at the island at the same time. Find the speed of the boat.</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>valid method <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(180 + 55,{\text{ }}360 - 90 - 35\)</p>
<p>235° (accept S55W, W35S) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find \({\rm{A\hat EC}}\) (may be seen in (a)) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\rm{A\hat EC}} = 180 - 55 - {\rm{A\hat CE}},{\text{ }}134 = {\text{E}} + 55\)</p>
<p>correct working to find \({\rm{A\hat EC}}\) (may be seen in (a)) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(180 - 55 - 46,{\text{ }}134 - 55\), \({\rm{A\hat EC}} = 79^\circ \)</p>
<p>evidence of choosing sine rule (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{a}{{\sin A}} = \frac{b}{{\sin B}}\)</p>
<p>correct substitution into sine rule <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{CE}}}}{{\sin 55^\circ }} = \frac{{175}}{{\sin {\rm{A\hat EC}}}}\)</p>
<p>146.034</p>
<p>\({\text{CE}} = 146{\text{ (km)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of choosing cosine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{D}}{{\text{E}}^2} = {\text{D}}{{\text{C}}^2} + {\text{C}}{{\text{E}}^2} - 2 \times {\text{DC}} \times {\text{CE}} \times \cos \theta \)</p>
<p>correct substitution into right-hand side <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({60^2} + {146.034^2} - 2 \times 60 \times 146.034\cos 134\)</p>
<p>192.612</p>
<p>\({\text{DE}} = 193{\text{ (km)}}\) <strong><em>A1</em></strong> <strong><em>N2</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>valid approach for locating B <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)BE is perpendicular to ship’s path, angle \({\text{B}} = 90\)</p>
<p>correct working for BE <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\sin 46^\circ = \frac{{{\text{BE}}}}{{146.034}},{\text{ BE}} = 146.034\sin 46^\circ ,{\text{ }}105.048\)</p>
<p>valid approach for expressing time <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(t = \frac{d}{s},{\text{ }}t = \frac{d}{r},{\text{ }}t = \frac{{192.612}}{{50}}\)</p>
<p>correct working equating time <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{146.034\sin 46^\circ }}{r} = \frac{{192.612}}{{50}},{\text{ }}\frac{s}{{105.048}} = \frac{{50}}{{192.612}}\)</p>
<p>27.2694</p>
<p>27.3 (km per hour) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[5 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 class="p1">The following diagram shows the quadrilateral \(ABCD\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_08.02.56.png" alt></p>
<p class="p1">\[{\text{AD}} = 6{\text{ cm}},{\text{ AB}} = 15{\text{ cm}},{\rm{ A\hat BC}} = 44^\circ ,{\rm{ A\hat CB}} = 83^\circ {\rm{ and D\hat AC}} = \theta \]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(AC\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the area of triangle \(ABC\).</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">The area of triangle \(ACD\) is half the area of triangle \(ABC\)<span class="s1">.</span></p>
<p class="p2">Find the possible values of \(\theta \).</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 class="p2">Given that \(\theta \) <span class="s2">is obtuse, find \(CD\)</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of choosing sine rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{{\text{AC}}}}{{\sin {\rm{C\hat BA}}}} = \frac{{{\text{AB}}}}{{\sin {\rm{A\hat CB}}}}\)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{{\text{AC}}}}{{\sin 44^\circ }} = \frac{{15}}{{\sin 83^\circ }}\)</p>
<p class="p1">\(10.4981\)</p>
<p class="p1">\({\text{AC}} = 10.5{\text{ }}{\text{ (cm)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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>finding \({\rm{C\hat AB}}\) (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;180^\circ - 44^\circ - 83^\circ ,{\rm{ C\hat AB}} = 53^\circ \)</p>
<p>correct substitution for area of triangle \(ABC\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2} \times 15 \times 10.4981 \times \sin 53^\circ \)</p>
<p>62.8813</p>
<p>\({\text{area}} = 62.9{\text{ }}{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution for area of triangle \(DAC\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2} \times 6 \times 10.4981 \times \sin \theta \)</p>
<p>attempt to equate area of triangle \(ACD\) to half the area of triangle \(ABC\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{\text{area ACD}} = \frac{1}{2} \times {\text{ area ABC; 2ACD}} = {\text{ABC}}\)</p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2} \times 6 \times 10.4981 \times \sin \theta = \frac{1}{2}(62.9),{\text{ }}62.9887\sin \theta = 62.8813,{\text{ }}\sin \theta = 0.998294\)</p>
<p>\(86.6531\), \(93.3468\)</p>
<p>\(\theta = 86.7^\circ {\text{ }},{\text{ }}\theta = 93.3^\circ {\text{ }}\) <strong><em>A1A1 N2</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Note: If candidates use an acute angle from part (c) in the cosine rule, award <strong><em>M1A0A0 </em></strong>in part (d).</p>
<p class="p2"> </p>
<p class="p1">evidence of choosing cosine rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p3"><span class="s1"><em>eg</em></span>\(\;\;\;{\text{C}}{{\text{D}}^2} = {\text{A}}{{\text{D}}^2} + {\text{A}}{{\text{C}}^2} - 2 \times {\text{AD}} \times {\text{AC}} \times \cos \theta \)</p>
<p class="p1">correct substitution into rhs <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p3"><span class="s1"><em>eg</em></span>\(\;\;\;{\text{C}}{{\text{D}}^2} = {6^2} + {10.498^2} - 2(6)(10.498)\cos 93.336^\circ \)</p>
<p class="p3">\(12.3921\)</p>
<p class="p3">\(12.4{\text{ }}{\text{ (cm)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<p class="p3"><span class="s1"><strong><em>Total [14 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a triangle ABC.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/ross.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The area of triangle ABC is \(80\) cm<sup>2</sup> , AB \( = 18\) cm , AC \( = x\) cm and \({\rm{B}}\hat {\rm{A}}{\rm{C}} = {50^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(x\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find BC.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into area formula <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{1}{2}(18x)\sin 50\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting <strong>their</strong> area expression equal to \(80\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(9x\sin 50 = 80\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 11.6\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({c^2} = {a^2} + {b^2} + 2ab\sin C\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into right hand side (may be in terms of \(x\)) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({11.6^2} + {18^2} - 2(11.6)(18)\cos 50\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">BC \( = 13.8\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The vast majority of candidates were very successful with this question. A small minority drew an altitude from C and used right triangle trigonometry. Errors included working in radian mode, assuming that the angle at C was \(90^\circ \), and incorrectly applying the order of operations when evaluating the cosine rule.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The vast majority of candidates were very successful with this question. A small minority drew an altitude from C and used right triangle trigonometry. Errors included working in radian mode, assuming that the angle at C was \(90^\circ \), and incorrectly applying the order of operations when evaluating the cosine rule.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram represents a large Ferris wheel at an amusement park.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P, Q and R represent different positions of a seat on the wheel.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/charmed.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The wheel has a radius of 50 metres and rotates clockwise at a rate of one revolution </span><span style="font-family: times new roman,times; font-size: medium;">every 30 minutes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A seat starts at the lowest point P, when its height is one metre above the ground.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the height of a seat above the ground after 15 minutes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">After six minutes, the seat is at point Q. Find its height above the ground at Q.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The height of the seat above ground after <em>t</em> minutes can be modelled by the function </span><span style="font-family: times new roman,times; font-size: medium;">\(h(t) = 50\sin (b(t - c)) + 51\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>b</em> and of <em>c</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The height of the seat above ground after <em>t</em> minutes can be modelled by the function </span><span style="font-family: times new roman,times; font-size: medium;">\(h(t) = 50\sin (b(t - c)) + 51\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Hence find the value of <em>t</em> the first time the seat is \(96{\text{ m}}\) above the ground.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 15 mins is half way, top of the wheel, \(d + 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">height \( = 101\) (metres) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of identifying rotation angle after 6 minutes <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{2\pi }}{5}\) , \(\frac{1}{5}\) </span><span style="font-family: times new roman,times; font-size: medium;">of a rotation, \({72^ \circ }\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. drawing a right triangle and using cosine ratio</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working (seen anywhere) <em><strong> A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \frac{{2\pi }}{5} = \frac{x}{{50}}\) , \(15.4(508 \ldots )\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate method <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. height \(= {\rm{radius}} + 1 - 15.45 \ldots \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">height \(= 35.5\) (metres) (accept 35.6) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">evidence of substituting into \(b = \frac{{2\pi }}{{{\rm{period}}}}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. period = 30 minutes, \(b = \frac{{2\pi }}{{30}}\) </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; font-size: medium;">\(b = 0.209\) \(\left( {\frac{\pi }{{15}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting into \(h(t)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(0) = 1\) , \(h(15) = 101\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(1 = 50\sin \left( { - \frac{\pi }{{15}}c} \right) + 51\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(c = 7.5\) <em><strong>A1 N2</strong></em></span></p>
<p><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;">evidence of setting up a system of equations <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">two correct equations</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 = 50\sin b(0 - c) + 51\) , \(101 = 50\sin b(15 - c) + 51\) <strong><em>A1A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve simultaneously <strong><em> (M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. evidence of combining two equations</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(b = 0.209\) \(\left( {\frac{\pi }{{15}}} \right)\) , \(c = 7.5\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1A1 N2N2</span></em></strong></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [6 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of solving \(h(t) = 96\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. equation, graph</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 12.8\) (minutes) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">d.</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;">Part (a) was well done with most candidates obtaining the correct answer. </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;">Part (b) however was problematic with most errors resulting from incorrect, missing or poorly drawn diagrams. Many did not recognize this as a triangle trigonometric problem while others used the law of cosines to find the chord length rather than the vertical height, but this was only valid if they then used this to complete the problem. Many candidates misinterpreted the question as one that was testing arc length and area of a sector and made little to no progress in part (b).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Still, others recognized that 6 minutes represented \(\frac{1}{5}\) of a rotation, but the majority then thought the height after 6 minutes should be \(\frac{1}{5}\) of the maximum height, treating the situation as linear. There were even a few candidates who used information given later in the question to answer part (b). Full marks are not usually awarded for this approach.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) was not well done. It was expected that candidates simply use the formula \(\frac{{2\pi }}{{{\rm{period}}}}\) to find the value of <em>b</em> and then substitute back into the equation to find the value of <em>c</em>. However, candidates often preferred to set up a pair of equations and attempt to solve them analytically, some successful, some not. No attempts were made to solve this system on the GDC indicating that candidates do not get exposed to many “systems” that are not linear. Confusing radians and degrees here did nothing to improve the lack of success. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), candidates were clear on what was required and set their equation equal to 96. Yet again however, solving this equation graphically using a GDC proved too daunting a task for most. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The depth of water in a port is modelled by the function \(d(t) = p\cos qt + 7.5\), for \(0 \leqslant t \leqslant 12\), where \(t\) is the number of hours after high tide.</p>
<p>At high tide, the depth is 9.7 metres.</p>
<p>At low tide, which is 7 hours later, the depth is 5.3 metres.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(q\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the model to find the depth of the water 10 hours after high tide.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{max}} - {\text{min}}}}{2}\), sketch of graph, \(9.7 = p\cos (0) + 7.5\)</p>
<p>\(p = 2.2\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(B = \frac{{2\pi }}{{{\text{period}}}}\), period is \(14,{\text{ }}\frac{{360}}{{14}},{\text{ }}5.3 = 2.2\cos 7q + 7.5\)</p>
<p>0.448798</p>
<p>\(q = \frac{{2\pi }}{{14}}{\text{ }}\left( {\frac{\pi }{7}} \right)\), (do not accept degrees) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(d(10),{\text{ }}2.2\cos \left( {\frac{{20\pi }}{{14}}} \right) + 7.5\)</p>
<p>7.01045</p>
<p>7.01 (m) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows triangle ABC .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/twins.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">AB = 7 cm, BC = 9 cm and \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {120^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find AC .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{B}}\widehat {\rm{A}}{\rm{C}}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({a^2} + {b^2} - 2ab\cos C\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({7^2} + {9^2} - 2(7)(9)\cos {120^ \circ }\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{AC}} = 13.9\) \(\left( { = \sqrt {193} } \right)\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing sine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin \widehat A}}{{{\rm{BC}}}} = \frac{{\sin \widehat B}}{{{\rm{AC}}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong> A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin \widehat A}}{9} = \frac{{\sin 120}}{{13.9}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\widehat A = {34.1^ \circ }\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \widehat A = \frac{{{\rm{A}}{{\rm{B}}^2} + {\rm{A}}{{\rm{C}}^2} - {\rm{B}}{{\rm{C}}^2}}}{{2({\rm{AB}})({\rm{AC}})}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \widehat A = \frac{{{7^2} + {{13.9}^2} - {9^2}}}{{2(7)(13.9)}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\widehat A = {34.1^ \circ }\) <em><strong>A1 N2</strong></em></span></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates were successful with this question. Most correctly used the cosine rule in part (a) and the sine rule in part (b). Some candidates did not check that their GDC was set in degree mode while others treated the triangle as if it were right angled. A large number of candidates were penalized for not leaving their answers exactly or to three significant figures. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates were successful with this question. Most correctly used the cosine rule in part (a) and the sine rule in part (b). Some candidates did not check that their GDC was set in degree mode while others treated the triangle as if it were right angled. A large number of candidates were penalized for not leaving their answers exactly or to three significant figures. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a quadrilateral <span class="s1">ABCD</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_17.33.03.png" alt="M16/5/MATME/SP2/ENG/TZ2/02"></p>
<p class="p1" style="text-align: center;">\[{\text{AD}} = {\text{7}}\;{\text{cm,}}\;{\text{BC}} = {\text{8}}\;{\text{cm,}}\;{\text{CD}} = {\text{12}}\;{\text{cm,}}\;{\rm{D\hat AB}} = {\text{1.75}}\;{\text{radians,}}\;{\rm{A\hat BD}} = {\text{0.82}}\;{\text{radians.}}\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find <span class="s1">BD</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 class="p1">Find \({\rm{D\hat BC}}\).</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">evidence of choosing sine rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{a}{{\sin A}} = \frac{b}{{\sin B}}\)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{a}{{\sin 1.75}} = \frac{7}{{\sin 0.82}}\)</p>
<p class="p2">9.42069</p>
<p class="p1"><span class="s1">\({\text{BD}} = 9.42{\text{ (cm)}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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">evidence of choosing cosine rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\cos B = \frac{{{d^2} + {c^2} - {b^2}}}{{2dc}},{\text{ }}{a^2} = {b^2} + {c^2} - 2bc\cos B\)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{8^2} + {{9.42069}^2} - {{12}^2}}}{{2 \times 8 \times 9.42069}},{\text{ }}144 = 64 + {\text{B}}{{\text{D}}^2} - 16{\text{BD}}\cos B\)</p>
<p class="p2">1.51271</p>
<p class="p2"><span class="s1">\({\rm{D\hat BC}} = 1.51\) </span>(radians) (accept <span class="s2">86.7°</span>) <strong><em>A1 N2</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;">
<p class="p1">Most candidates solved part (a) correctly, recognizing the need for the law of sines.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), some recognized they had to use cosine rule but substituted incorrectly. There were a few who used Pythagoras theorem or overly long approaches using the sine rule for 2(b).</p>
<p class="p1">Some used the calculator in degree mode instead of radian mode, not recognizing that the angles were given in radians.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows a circle with centre \(\rm{O}\) and radius \(5 \rm\,{cm}\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; min-height: 25px; text-align: center; margin: 0px;"><img src="images/maths_1_1.png" alt></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;">The points \(\rm{A}\), \(rm{B}\) and \(rm{C}\) lie on the circumference of the circle, and \({\rm{A\hat OC}} = 0.7\) radians.</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 length of the arc \({\text{ABC}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 perimeter of the shaded sector.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 area of the shaded sector.</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: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into arc length formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(0.7 \times 5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">arc length \(= 3.5\) (cm) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><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(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(3.5 + 5 + 5,{\text{ arc}} + 2r\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">perimeter \(= 13.5\) (cm) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into area formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{1}{2}(0.7){(5)^2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{area}} = 8.75{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[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;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a quadrilateral ABCD with obtuse angles \({\rm{A}}\widehat {\rm{B}}{\rm{C}}\) and \({\rm{A}}\widehat {\rm{D}}{\rm{C}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/footie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">AB = 5 cm, BC = 4 cm, CD = 4 cm, AD = 4 cm , \({\rm{B}}\widehat {\rm{A}}{\rm{C}} = {30^ \circ }\) , \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {x^ \circ }\) , \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = {y^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the cosine rule to show that \({\rm{AC}} = \sqrt {41 - 40\cos x} \) .</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;">Use the sine rule in triangle ABC to find another expression for AC.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Hence, find <em>x</em>, giving your answer to two decimal places.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find AC .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find <em>y</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, or otherwise, find the area of triangle ACD.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d(i) and (ii).</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;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(25 + 16 - 40\cos x\) , \({5^2} + {4^2} - 2 \times 4 \times 5\cos x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AC}} = \sqrt {41 - 40\cos x} \) <em><strong>AG</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{AC}}}}{{\sin x}} = \frac{4}{{\sin 30}}\) , \(\frac{1}{2}{\rm{AC}} = 4\sin x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AC}} = 8\sin x\) (accept \(\frac{{4\sin x}}{{\sin 30}}\)) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of appropriate approach using AC <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(8\sin x = \sqrt {41 - 40\cos x} \) , sketch showing intersection </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct solution \(8.682 \ldots \), \(111.317 \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">obtuse value \(111.317 \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 111.32\) to 2 dp (do <strong>not</strong> accept the radian answer 1.94 ) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) substituting value of <em>x</em> into either expression for AC <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{AC}} = 8\sin 111.32\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AC}} = 7.45\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of choosing cosine rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos B = \frac{{{a^2} + {c^2} - {b^2}}}{{2ac}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{4^2} + {4^2} - {{7.45}^2}}}{{2 \times 4 \times 4}}\) , \({7.45^2} = 32 - 32\cos y\) , \(\cos y = - 0.734 \ldots \)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(y = 137\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into area formula <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2} \times 4 \times 4 \times \sin 137\) , \(8\sin 137\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> area \(= 5.42\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">d(i) and (ii).</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;">Many candidates worked comfortably with the sine and cosine rules in part (a) and (b).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates worked comfortably with the sine and cosine rules in part (a) and (b). </span><span style="font-family: times new roman,times; font-size: medium;">Equally as many did not take the cue from the word "hence" and used an alternate method to </span><span style="font-family: times new roman,times; font-size: medium;">solve the problem and thus did not receive full marks. Those who managed to set up an </span><span style="font-family: times new roman,times; font-size: medium;">equation, again did not go directly to their GDC but rather engaged in a long, laborious </span><span style="font-family: times new roman,times; font-size: medium;">analytical approach that was usually unsuccessful.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Equally as many did not take the cue from the word "hence" and used an alternate method to solve the problem and thus did not receive full marks. Those who managed to set up an equation, again did not go directly to their GDC but rather engaged in a long, laborious analytical approach that was usually unsuccessful. No matter what values were found in (c) (i) most candidates recovered and earned follow through marks for the remainder of the question. A large number of candidates worked in the wrong mode and rounded prematurely throughout this question often resulting in accuracy penalties.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Equally as many did not take the cue from the word "hence" and used an alternate method to solve the problem and thus did not receive full marks. Those who managed to set up an equation, again did not go directly to their GDC but rather engaged in a long, laborious analytical approach that was usually unsuccessful. No matter what values were found in (c) (i) most candidates recovered and earned follow through marks for the remainder of the question. A large number of candidates worked in the wrong mode and rounded prematurely throughout this question often resulting in accuracy penalties.</span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a plan for a window in the shape of a trapezium.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tpc.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Three sides of the window are \(2{\text{ m}}\) long. The angle between the sloping sides of the </span><span style="font-family: times new roman,times; font-size: medium;">window and the base is \(\theta \) , where \(0 < \theta < \frac{\pi }{2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that the area of the window is given by \(y = 4\sin \theta + 2\sin 2\theta \) .</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><span style="font-family: times new roman,times; font-size: medium;">Zoe wants a window to have an area of \(5{\text{ }}{{\text{m}}^2}\). Find the two possible values of \(\theta \) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">John wants two windows which have the same area <em>A</em> but different values of \(\theta \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find all possible values for <em>A</em> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding height, <em>h</em> <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sin \theta = \frac{h}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(2\sin \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding base of triangle, <em>b</em> <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \theta = \frac{b}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(2\cos \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute valid values into a formula for the area of the window <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. two triangles plus rectangle, trapezium area formula</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression (must be in terms of \(\theta \) ) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\left( {\frac{1}{2} \times 2\cos \theta \times 2\sin \theta } \right) + 2 \times 2\sin \theta \) , \(\frac{1}{2}(2\sin \theta )(2 + 2 + 4\cos \theta )\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to replace \(2\sin \theta \cos \theta \) by \(\sin 2\theta \) <strong><em>M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4\sin \theta + 2(2\sin \theta \cos \theta )\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 4\sin \theta + 2\sin 2\theta \) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = 5\) , \(4\sin \theta + 2\sin 2\theta = 5\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to solve <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. a sketch, \(4\sin \theta + 2\sin \theta - 5 = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 0.856\) \(({49.0^ \circ })\) , \(\theta = 1.25\) \(({71.4^ \circ })\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognition that lower area value occurs at \(\theta = \frac{\pi }{2}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></em></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding value of area at \(\theta = \frac{\pi }{2}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></em></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4\sin \left( {\frac{\pi }{2}} \right) + 2\sin \left( {2 \times \frac{\pi }{2}} \right)\) , </span><span style="font-family: times new roman,times; font-size: medium;">draw square</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 4\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognition that maximum value of <em>y</em> is needed <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 5.19615 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(4 < A < 5.20\) (accept \(4 < A < 5.19\) ) <strong><em>A2 N5</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></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><span style="font-family: times new roman,times; font-size: medium;">As the final question of the paper, this question was understandably challenging for the majority of the candidates. Part (a) was generally attempted, but often with a lack of method or correct reasoning. Many candidates had difficulty presenting their ideas in a clear and organized manner. Some tried a "working backwards" approach, earning no marks. </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;">In part (b), most candidates understood what was required and set up an equation, but many did not make use of the GDC and instead attempted to solve this equation algebraically which did not result in the correct solution. A common error was finding a second solution outside the domain. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A pleasing number of stronger candidates made progress on part (c), recognizing the need for the end point of the domain and/or the maximum value of the area function (found graphically, analytically, or on occasion, geometrically). However, it was evident from candidate work and teacher comments that some candidates did not understand the wording of the question. This has been taken into consideration for future paper writing. </span></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;">The following diagram shows a triangle ABC.</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>
<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;">\({\rm{BC}} = 6\) , \({\rm{C}}\widehat {\rm{A}}{\rm{B}} = 0.7\) radians , \({\rm{AB}} = 4p\) , \({\rm{AC}} = 5p\) , where \(p > 0\) .</span></p>
<p> </p>
</div>
<div class="specification">
<p style="text-align: justify;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the circle with centre B that passes through the point C. The circle cuts the </span><span style="font-family: times new roman,times; font-size: medium;">line CA at D, and \({\rm{A}}\widehat {\rm{D}}{\rm{B}}\) is obtuse. Part of the circle is shown in the following diagram.</span></p>
<p style="text-align: justify;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/minion.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({p^2}(41 - 40\cos 0.7) = 36\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find <em>p</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the length of BD.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{A}}\widehat {\rm{D}}{\rm{B}}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({\rm{C}}\widehat {\rm{B}}{\rm{D}} = 1.29\) radians, correct to 2 decimal places.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find the area of the shaded region.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d(i) and (ii).</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;">(i) evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. choosing cosine rule </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({6^2} = {(5p)^2} + {(4p)^2} - 2 \times (4p) \times (5p)\cos 0.7\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplification <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(36 = 25{p^2} + 16{p^2} - 40{p^2}\cos 0.7\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({p^2}(41 - 40\cos 0.7) = 36\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(1.85995 \ldots \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 1.86\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A0</strong></em> for \(p = \pm 1.86\) , i.e. not rejecting the negative value. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{BD}} = 6\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. choosing sine rule </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin {\rm{A}}\widehat {\rm{D}}{\rm{B}}}}{{4p}} = \frac{{\sin 0.7}}{6}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{acute }}{\rm{A}}\widehat {\rm{D}}{\rm{B = 0}}{\rm{.9253166}} \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(\pi - 0.9253166 \ldots = 2.216275 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{D}}{\rm{B}} = 2.22\) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> [4 marks]</strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. recognize isosceles triangle, base angles equal </span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">\(\pi - 2(0.9253 \ldots )\) </span><em><span style="font-size: medium;"><strong>A1</strong> </span></em></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">\({\rm{C}}\widehat {\rm{B}}{\rm{D}} = 1.29\) </span><em><span style="font-size: medium;"><strong>AG N0</strong> </span></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) area of sector BCD <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.5 \times (1.29) \times {(6)^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of triangle BCD <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0.5 \times {(6)^2}\sin 1.29\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of subtraction <em><strong>M1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(5.92496 \ldots \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(5.937459 \ldots \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\text{area}} = 5.94\) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;"> [6 marks]</span></strong></em></span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were mixed results with this question. Most candidates could access part (a) and made the correct choice with the cosine rule but sloppy notation often led to candidates not being able to show the desired result. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were mixed results with this question. Most candidates could access part (a) and made the correct choice with the cosine rule but sloppy notation often led to candidates not being able to show the desired result. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), candidates again correctly identified an appropriate method but failed to recognize that their result of 0.925 was acute and not obtuse as required. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (d) (i), many attempted to use the sine rule under the incorrect assumption that DC was equal to 5<em>p</em>, rather than rely on some basic isosceles triangle geometry. Consequently, the result of 1.29 for \({\rm{C}}\widehat {\rm{B}}{\rm{D}}\) was not easy to show. There was a great deal of success with (d) (ii) with candidates using appropriate techniques to find the area of the shaded region although some stopped after finding the area of the sector. </span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the following circle with centre O and radius 6.8 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/gaston.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The length of the arc PQR is 8.5 cm.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\theta \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the shaded region.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(8.5 = \theta (6.8)\) , \(\theta = \frac{{8.5}}{{6.8}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 1.25\) (accept \({71.6^ \circ }\) ) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution into area formula (seen anywhere) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A = \pi {(6.8)^2}\) , \(145.267 \ldots \)<br></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution into area formula (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A = \frac{1}{2}(1.25)({6.8^2})\) </span><span style="font-family: times new roman,times; font-size: medium;">, 28.9</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi {(6.8)^2} - \frac{1}{2}(1.25)({6.8^2})\) </span><span style="font-family: times new roman,times; font-size: medium;">; \(145.267 \ldots - 28.9\) ; \(\pi {r^2} - \frac{1}{2}{r^2}\sin \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 116\) (\({\text{c}}{{\text{m}}^2}\)) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find reflex angle <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\pi - \theta \) , \(360 - 1.25\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct reflex angle <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{O}}{\rm{B}} = 2\pi - 1.25\) (\( = 5.03318 \ldots \))</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution into area formula <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A = \frac{1}{2}(5.03318 \ldots )({6.8^2})\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(A = 116\) (\({\text{c}}{{\text{m}}^2}\)) <em><strong>A1 N2</strong></em></span></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was almost universally done correctly. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many also had little trouble in part (b), with most subtracting from the circle's area, and a minority using the reflex angle. A few candidates worked in degrees, although some of these did so incorrectly by using the radian area formula. Some candidates only found the area of the unshaded sector. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram shows a parallelogram ABCD.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tired.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The coordinates of A, B and D are A(1, 2, 3) , B(6, 4,4 ) and D(2, 5, 5) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}<br>5\\<br>2\\<br>1<br>\end{array}} \right)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(\overrightarrow {{\rm{AD}}} \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) <strong>Hence</strong> show that \(\overrightarrow {{\rm{AC}}} = \left( {\begin{array}{*{20}{c}}<br>6\\<br>5\\<br>3<br>\end{array}} \right)\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the coordinates of point C.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} \).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong> find angle <em>A</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, or otherwise, find the area of the parallelogram.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of approach <em><strong>M1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{B}} - {\text{A}}\) , \(\overrightarrow {{\rm{AO}}} + \overrightarrow {{\rm{OB}}} \) , \(\left( {\begin{array}{*{20}{c}}<br>6\\<br>4\\<br>4<br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br>1\\<br>2\\<br>3<br>\end{array}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}<br>5\\<br>2\\<br>1<br>\end{array}} \right)\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{D}} - {\text{A}}\) , \(\overrightarrow {{\rm{AO}}} + \overrightarrow {{\rm{OD}}} \) , \(\left( {\begin{array}{*{20}{c}}<br>2\\<br>5\\<br>5<br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br>1\\<br>2\\<br>3<br>\end{array}} \right)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\rm{AD}}}= \left( {\begin{array}{*{20}{c}}<br>1\\<br>3\\<br>2<br>\end{array}} \right)\) <em><strong> A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) evidence of approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e</span><span style="font-family: times new roman,times; font-size: medium;">.g. \(\overrightarrow {{\rm{AC}}} = \overrightarrow {{\rm{AB}}} + \overrightarrow {{\rm{AD}}} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\overrightarrow {{\rm{AC}}} = \left( {\begin{array}{*{20}{c}}<br>5\\<br>2\\<br>1<br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br>1\\<br>3\\<br>2<br>\end{array}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\rm{AC}}} = \left( {\begin{array}{*{20}{c}}<br>6\\<br>5\\<br>3<br>\end{array}} \right)\) <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[5 marks]</strong> </span></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of combining vectors (there are at least 5 ways) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\overrightarrow {{\rm{OC}}} = \overrightarrow {{\rm{OA}}} + \overrightarrow {{\rm{AC}}} \) , \(\overrightarrow {{\rm{OC}}} = \overrightarrow {{\rm{OB}}} + \overrightarrow {{\rm{AD}}} \), \(\overrightarrow {{\rm{AB}}} = \overrightarrow {{\rm{OC}}} - \overrightarrow {{\rm{OD}}} \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\rm{OC}}} = \left( {\begin{array}{*{20}{c}}<br>1\\<br>2\\<br>3<br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br>6\\<br>5\\<br>3<br>\end{array}} \right)\left( { = \left( {\begin{array}{*{20}{c}}<br>7\\<br>7\\<br>6<br>\end{array}} \right)} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. coordinates of C are \((7{\text{, }}7{\text{, }}6)\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[3 marks]</strong> </span></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of using scalar product on \(\overrightarrow {{\rm{AB}}} \) and \(\overrightarrow {{\rm{AD}}} \) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} = 5(1) + 2(3) + 1(2)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} = 13\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(\left| {\overrightarrow {{\rm{AB}}} } \right| = 5.477 \ldots \) , \(\left| {\overrightarrow {{\rm{AD}}} } \right| = 3.741 \ldots \) <em><strong>(A1)(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(\cos A = \frac{{\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} }}{{\left| {\overrightarrow {{\rm{AB}}} } \right|\left| {\overrightarrow {{\rm{AD}}} } \right|}}\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos A = \frac{{13}}{{20.493}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\widehat A = 0.884\) \((50.6^\circ )\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[</span></strong><span style="font-family: times new roman,times; font-size: medium;"><strong>7 marks]</strong> </span></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">evidence of using \({\rm{area}} = 2\left( {\frac{1}{2}\left| {\overrightarrow {{\rm{AD}}} } \right|\left| {\overrightarrow {{\rm{AB}}} } \right|\sin {\rm{D}}\widehat {\rm{A}}{\rm{B}}} \right)\) <strong><em> (M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{area}} = 2\left( {\frac{1}{2}(3,741 \ldots )(5.477 \ldots )\sin 0.883 \ldots } \right)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 15.8\) <em><strong>A1 N2</strong> </em></span></p>
<p><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;">evidence of using \({\rm{area}} = b \times h\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding height of parallelogram <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h = 3.741 \ldots \times \sin 0.883 \ldots ( = 2.892 \ldots )\) , \(h = 5.477 \ldots \times \sin 0.883 \ldots ( = 4.234 \ldots )\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 15.8\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[3 marks]</strong> </span></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates performed very well in this question, showing a strong ability to work with the algebra and geometry of vectors. </span></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates performed very well in this question, showing a strong ability to work with the algebra and geometry of vectors. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some candidates were unable to find the scalar product in part (c), yet still managed to find the correct angle, able to use the formula in the information booklet without knowing that the scalar product is a part of that formula. </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few candidates considered that the area of the parallelogram is twice the area of a triangle, which is conveniently found using \({\rm{B}}\widehat {\rm{A}}{\rm{D}}\) . In an effort to find base \(\times \) height , many candidates multiplied the magnitudes of \(\overrightarrow {{\rm{AB}}} \) and \(\overrightarrow {{\rm{AD}}} \) , missing that the height of a parallelogram is perpendicular to a base. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows triangle PQR. The length of [PQ] is 7 cm , the length of [PR] is 10 cm , and \({\rm{P}}\widehat {\rm{Q}}{\rm{R}}\) is \(75^\circ \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/park.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\rm{P}}\widehat {\rm{R}}{\rm{Q}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of triangle PQR.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">choosing sine rule <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution \(\frac{{\sin R}}{7} = \frac{{\sin 75^\circ }}{{10}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\sin R = 0.676148 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{P}}\widehat {\rm{R}}{\rm{Q = 42}}{\rm{.5}}^\circ \) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 180 - 75 - R\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 62.5\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substitution into any correct formula <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{area }}\Delta {\text{PQR}} = \frac{1}{2} \times 7 \times 10 \times \sin ({\text{their }}P)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(= 31.0\) (cm<sup>2</sup>) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done with most students using the law of sines to find the angle.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In </span><span style="font-family: times new roman,times; font-size: medium;">part (b), the most common error occurred when angle <em>R</em> or 75 degrees was used to find the </span><span style="font-family: times new roman,times; font-size: medium;">area. This particular question was the most common place to incur an accuracy penalty.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A ship leaves port A on a bearing of \(030^\circ \) . It sails a distance of \(25{\text{ km}}\) to point B. At B, the ship changes direction to a bearing of \(100^\circ \) . It sails a distance of \(40{\text{ km}}\) to </span><span style="font-family: times new roman,times; font-size: medium;">reach point C. This information is shown in the diagram below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/barking.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">A second ship leaves port A and sails directly to C.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the distance the second ship will travel.</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><span style="font-family: times new roman,times; font-size: medium;">Find the bearing of the course taken by the second ship.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">finding \({\text{A}}\widehat {\rm{B}}{\rm{C}} = 110^\circ \) (\( = 1.92\) radians) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{A}}{{\rm{C}}^2} = {\rm{A}}{{\rm{B}}^2} + {\rm{B}}{{\rm{C}}^2} - 2({\rm{AB}})({\rm{BC}})\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{A}}{{\rm{C}}^2} = {25^2} + {40^2} - 2(25)(40)\cos 110^\circ \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}{{\rm{C}}^{}} = 53.9\) (km) <em><strong>A1</strong> </em></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>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into the sine rule <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin {\rm{B}}\widehat {\rm{A}}{\rm{C}}}}{{40}} = \frac{{\sin 110^\circ }}{{53.9}}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{B}}\widehat {\rm{A}}{\rm{C}} = 44.2^\circ \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">bearing \( = 074^\circ \) <em><strong>A1 N1</strong> </em></span></p>
<p><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;">correct substitution into the cosine rule <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos {\rm{B}}\widehat {\rm{A}}{\rm{C}} = \frac{{{{40}^2} - {{25}^2} - {{53.9}^2}}}{{ - 2(25)(53.9)}}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{B}}\widehat {\rm{A}}{\rm{C}} = 44.3^\circ \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">bearing \( = 074^\circ \) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates found this question very accessible, although some attempted to use Pythagoras' theorem to find AC.</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;">Often candidates correctly found \({\rm{B}}\widehat {\rm{A}}{\rm{C}}\) in part (b), but few added the \(30^\circ \) to obtain the required bearing. Some candidates calculated \({\rm{B}}\widehat {\rm{C}}{\rm{A}}\) , misinterpreting that the question required the course of the second ship. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a circle, centre <span class="s1">O </span>and radius \(r\) <span class="s1">mm</span>. The circle is divided into five equal sectors.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_15.31.24.png" alt="N16/5/MATME/SP2/ENG/TZ0/03"></p>
<p class="p1">One sector is <span class="s1">OAB</span>, and \({\rm{A\hat OB}} = \theta \).</p>
</div>
<div class="specification">
<p class="p1">The area of sector <span class="s1">AOB </span>is \(20\pi {\text{ m}}{{\text{m}}^2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Write down the </span><strong>exact </strong>value of \(\theta \) in radians.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(r\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find AB.</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="s1">\(\theta = \frac{{2\pi }}{5}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct expression for area <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(A = \frac{1}{2}{r^2}\left( {\frac{{2\pi }}{5}} \right),{\text{ }}\frac{{\pi {r^2}}}{5}\)</p>
<p class="p3">evidence of equating their expression to \(20\pi \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}{r^2}\left( {\frac{{2\pi }}{5}} \right) = 20\pi ,{\text{ }}{r^2} = 100,{\text{ }}r = \pm 10\)</p>
<p class="p2"><span class="Apple-converted-space">\(r = 10\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">evidence of choosing cosine rule <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({a^2} = {b^2} + {c^2} - 2bc\cos A\)</p>
<p class="p1">correct substitution of <strong>their</strong> \(r\) and \(\theta \) into RHS <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({10^2} + {10^2} - 2 \times 10 \times 1{\text{0}}\cos \left( {\frac{{2\pi }}{5}} \right)\)</p>
<p class="p3">11.7557</p>
<p class="p2"><span class="s2">\({\text{AB}} = 11.8{\text{ (mm)}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">evidence of choosing sine rule <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\sin A}}{a} = \frac{{\sin B}}{b}\)</p>
<p class="p1">correct substitution of <strong>their</strong> \(r\) and \(\theta \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\sin \frac{{2\pi }}{5}}}{{{\text{AB}}}} = \frac{{\sin \left( {\frac{1}{2}\left( {\pi - \frac{{2\pi }}{5}} \right)} \right)}}{{10}}\)</p>
<p class="p3">11.7557</p>
<p class="p2"><span class="s2">\({\text{AB}} = 11.8{\text{ (mm)}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><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 class="p1">The following diagram shows triangle \(ABC\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_10.08.48.png" alt></p>
<p class="p1">\[{\text{BC}} = 10{\text{ cm}},{\rm{ A\hat BC}} = 80^\circ \;{\text{and}}\;{\rm{B\hat AC}} = 35^\circ .\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(AC\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the area of triangle \(ABC\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of choosing sine rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{{\text{AC}}}}{{\sin ({\rm{A\hat BC)}}}} = \frac{{{\text{BC}}}}{{\sin ({\rm{B\hat AC)}}}}\)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{{\text{AC}}}}{{\sin 80^\circ }} = \frac{{10}}{{\sin 35^\circ }}\)</p>
<p class="p1">\({\text{AC}} = 17.1695\)</p>
<p class="p1">\({\text{AC}} = 17.2{\text{ (cm)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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">\({\rm{A\hat CB}} = 65^\circ \;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{2} \times 10 \times 17.1695 \times \sin 65^\circ \)</p>
<p class="p1">\({\text{area}} = 77.8047\)</p>
<p class="p1">\({\text{area}} = 77.8{\text{ (c}}{{\text{m}}^{\text{2}}}{\text{)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates found this question straightforward and accessible.</p>
<p class="p1">Most recognized the need for the sine rule in part (a) to solve the problem. Occasionally, the setup had an incorrect match of angle and side. Some used radians instead of degrees, thus losing a mark.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was also well done by most of the candidates. Some right triangle trigonometry correct approaches were seen to find the area. A few candidates used the cosine rule or right angled trigonometry, which were less efficient methods and often wasted valuable time.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = p\cos \left( {q(x + r)} \right) + 10\), for \(0 \leqslant x \leqslant 20\). The following diagram shows the graph of \(f\).</span></p>
<p style="margin: 0px; font-style: normal; font-variant: normal; font-weight: normal; font-size: 21px; line-height: normal; font-family: 'Times New Roman'; text-align: center;"><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; background-color: #f7f7f7;"><img src="images/maths_6.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;">The graph has a maximum at \((4, 18)\) and a minimum at \((16, 2)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(r\).</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 \(p\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 \(q\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Solve \(f(x) = 7\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = - 4\) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f; min-height: 18.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1</em> </strong>for \(r = 4\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f; min-height: 18.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{\max y{\text{ value -- }}y{\text{ value}}}}{2}\), distance from \(y = 10\)</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;">\(p = 8\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> period is \(24\), \(\frac{{360}}{{24}}\), substitute a point into <strong>their</strong> \(f(x)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = \frac{{2\pi }}{{24}}\left( {\frac{\pi }{{12}},{\text{ exact}}} \right)\), \(0.262\) (do not accept degrees) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> line on graph at \(y = 7,{\text{ }}8\cos \left( {\frac{{2\pi }}{{24}}(x - 4)} \right) + 10 = 7\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 11.46828\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 11.5\) (accept \((11.5, 7)\)) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not award the final <strong><em>A1 </em></strong>if additional values are given. If an incorrect value of \(q\) leads to multiple solutions, award the final <strong><em>A1 </em></strong>only if <strong>all</strong> solutions within the domain are given.</span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a circle with centre O and radius 4 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/yes.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The points A, B and C lie on the circle. The point D is outside the circle, on (OC).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Angle ADC = 0.3 radians and angle AOC = 0.8 radians.</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find AD.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find OD.</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><span style="font-family: times new roman,times; font-size: medium;">Find the area of sector OABC.</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><span style="font-family: times new roman,times; font-size: medium;">Find the area of region ABCD.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">choosing sine rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{AD}}}}{{\sin 0.8}} = \frac{4}{{\sin 0.3}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{AD}} = 9.71{\text{ (cm)}}\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding angle \({\rm{OAD}} = \pi - 1.1 = (2.04)\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">choosing cosine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{O}}{{\rm{D}}^2} = {9.71^2} + {4^2} - 2 \times 9.71 \times 4 \times \cos (\pi - 1.1)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{OD}} = 12.1{\text{ (cm)}}\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding angle \({\rm{OAD}} = \pi - 1.1 = (2.04)\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">choosing sine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{OD}}}}{{\sin (\pi - 1.1)}} = \frac{{9.71}}{{\sin 0.8}} = \frac{4}{{\sin 0.3}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{OD}} = 12.1{\text{ (cm)}}\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into area of a sector formula <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{area}} = 0.5 \times {4^2} \times 0.8\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 6.4{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substitution into area of triangle formula OAD <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A{\rm{ = }}\frac{1}{2} \times 4 \times 12.1 \times \sin 0.8\) , \(A{\rm{ = }}\frac{1}{2} \times 4 \times 9.71 \times \sin 2.04\) , \(A{\rm{ = }}\frac{1}{2} \times 12.1 \times 9.71 \times \sin 0.3\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">subtracting area of sector OABC from area of triangle OAD <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\text{area ABCD}} = 17.3067 - 6.4\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{area ABCD}} = 10.9{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was generally quite well done, and it was pleasing to note that candidates could come up with multiple methods to arrive at the correct answers. Many candidates worked comfortably with the sine and cosine rules to find sides of triangles. Some candidates chose alternative right-angled triangle methods, often with success, although this proved a time-consuming approach. Some unnecessarily converted the radian values to degrees, which sometimes led to calculation errors that could have been avoided. A large number of candidates accrued the accuracy penalty in this question. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was generally quite well done, and it was pleasing to note that candidates could come up with multiple methods to arrive at the correct answers. Many candidates worked comfortably with the sine and cosine rules to find sides of triangles. Some candidates chose alternative right-angled triangle methods, often with success, although this proved a time-consuming approach. Some unnecessarily converted the radian values to degrees, which sometimes led to calculation errors that could have been avoided. A large number of candidates accrued the accuracy penalty in this question. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was generally quite well done, and it was pleasing to note that candidates could come up with multiple methods to arrive at the correct answers. Many candidates worked comfortably with the sine and cosine rules to find sides of triangles. Some candidates chose alternative right-angled triangle methods, often with success, although this proved a time-consuming approach. Some unnecessarily converted the radian values to degrees, which sometimes led to calculation errors that could have been avoided. A large number of candidates accrued the accuracy penalty in this question. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was generally quite well done, and it was pleasing to note that candidates could come up with multiple methods to arrive at the correct answers. Many candidates worked comfortably with the sine and cosine rules to find sides of triangles. Some candidates chose alternative right-angled triangle methods, often with success, although this proved a time-consuming approach. Some unnecessarily converted the radian values to degrees, which sometimes led to calculation errors that could have been avoided. A large number of candidates accrued the accuracy penalty in this question. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A circle centre O and radius \(r\) is shown below. The chord [AB] divides the area of the </span><span style="font-family: times new roman,times; font-size: medium;">circle into two parts. Angle AOB is \(\theta \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/crop.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for the area of the shaded region.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The chord [AB] divides the area of the circle in the ratio 1:7. Find the </span><span style="font-family: times new roman,times; font-size: medium;">value of \(\theta \) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substitution into formula for area of triangle <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}r \times r\sin \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of subtraction <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}{r^2}\theta - \frac{1}{2}{r^2}\sin \theta \) , \(\frac{1}{2}{r^2}(\theta - \sin \theta )\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing that shaded area is \(\frac{1}{8}\) </span><span style="font-family: times new roman,times; font-size: medium;">of area of circle <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{8}\) </span><span style="font-family: times new roman,times; font-size: medium;">seen anywhere</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up correct equation <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}{r^2}(\theta - \sin \theta ) = \frac{1}{8}\pi {r^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">eliminating 1 variable <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}(\theta - \sin \theta ) = \frac{1}{8}\pi \) , \(\theta - \sin \theta = \frac{\pi }{4}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve <em><strong> M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. a sketch, writing \(\sin x - x + \frac{\pi }{4} = 0\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\theta = 1.77\) </span><span style="font-family: times new roman,times; font-size: medium;"> (do not accept degrees) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The population of deer in an enclosed game reserve is modelled by the function \(P(t) = 210\sin (0.5t - 2.6) + 990\), where \(t\) is in months, and \(t = 1\) corresponds to 1 January 2014.</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 number of deer in the reserve on 1 May 2014.</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;">Find the rate of change of the deer population on 1 May 2014.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = 5\) <span style="font: 21.0px 'Times New Roman';"><strong><em>(A1)</em></strong></span></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;">correct substitution into formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\(210\sin (0.5 \times 5 - 2.6) + 990,{\text{ }}P(5)\)</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;">\(969.034982 \ldots \)</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;">969 (deer) (must be an integer) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of considering derivative <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(P'\)</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;">\(104.475\)</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;">\(104\) (deer per month) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(i).</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;">(the deer population size is) <strong>increasing</strong> <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following graph shows the depth of water, <em>y</em> metres , at a point P, during one day. The time <em>t</em> is given in hours, from midnight to noon. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ffs.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the graph to write down an estimate of the value of <em>t</em> when </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) the depth of water is minimum; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the depth of water is maximum; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (iii) the depth of the water is increasing most rapidly.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i), (ii) and (iii).</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 depth of water can be modelled by the function \(y = \cos A(B(t - 1)) + C\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(A = 8\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the value of <em>C</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (iii) Find the value of <em>B</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A sailor knows that he cannot sail past P when the depth of the water is less than 12 m . Calculate the values of <em>t</em> between which he cannot sail past P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 7 <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) 1 <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) 10 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of appropriate approach <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A = \frac{{18 - 2}}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 8\) <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(C = 10\) <em><strong>A2 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii)<strong> METHOD 1</strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{period}} = 12\) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(B \times {\rm{period}} = 2\pi \) (accept \(360^\circ \) ) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(12 = \frac{{2\pi }}{B}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(B = \frac{\pi }{6}\) </span><span style="font-family: times new roman,times; font-size: medium;">(accept 0.524 or 30) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(10 = 8\cos 3B + 10\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">simplifying <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos 3B = 0\) \(\left( {3B = \frac{\pi }{2}} \right)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(B = \frac{\pi }{6}\) </span><span style="font-family: times new roman,times; font-size: medium;">(accept 0.524 or 30) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct answers <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(t = 3.52\) , \(t = 10.5\) , between 03:31 and 10:29 (accept 10:30) <em><strong>N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For part (a), most candidates correctly used the graph to identify the times of maximum and </span><span style="font-family: times new roman,times; font-size: medium;">minimum depth. Most failed to consider that the depth of water is increasing most rapidly at a </span><span style="font-family: times new roman,times; font-size: medium;">point of inflexion and often answered with the interval \(t = 9\) to \(t = 11\) . A few candidates </span><span style="font-family: times new roman,times; font-size: medium;">answered with the depth instead of time, misinterpreting which axis to consider.</span></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times;"><span style="font-size: medium;">A substantial number of candidates showed difficulty finding parameters of a trigonometric </span><span style="font-size: medium;">function with many only making superficial attempts at part (b), often leaving it blank entirely.</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times;"><span style="font-size: medium;">Some divided \(2\pi \) by the period of 12, while others substituted an ordered pair such as \((4{\text{, }}10)\)</span><span style="font-size: medium;">and solved for <em>B</em>, often correctly. Many found that \(c = 17\) , thus confusing the vertical </span><span style="font-size: medium;">translation with a <em>y</em>-intercept.</span></span></p>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For (c), many candidates simply read approximate values from the graph where \(y = 12\) and </span><span style="font-family: times new roman,times; font-size: medium;">thus answered with \(t = 3.5\) and \(t = 10.5\) . Although the latter value is correct to three </span><span style="font-family: times new roman,times; font-size: medium;">significant figures, \(t = 3.5\) incurs the accuracy penalty as it was expected that candidates </span><span style="font-family: times new roman,times; font-size: medium;">calculate this value in their GDC to achieve a result of \(t = 3.52\) . Those who attempted an </span><span style="font-family: times new roman,times; font-size: medium;">analytic approach rarely achieved correct results.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a triangle ABD with AB =13 cm and AD = 6.5 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let C be a point on the line BD such that BC = AC = 7 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/blaine.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the size of angle ACB.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the size of angle CAD.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the cosine formula <em><strong>(M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos {\rm{A}}\widehat {\rm{C}}{\rm{B}} = \frac{{{7^2} + {7^2} - {{13}^2}}}{{2 \times 7 \times 7}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{C}}{\rm{B}} = 2.38\) radians \(( = 136^\circ )\) <em><strong>A1 N2</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of <strong>appropriate</strong> approach involving right-angled triangles <em><strong>(M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sin \left( {\frac{1}{2}{\rm{A}}\widehat {\rm{C}}{\rm{B}}} \right) = \frac{{6.5}}{7}\)</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;">\({\rm{A}}\widehat {\rm{C}}{\rm{B}} = 2.38\) radians \(( = 136^\circ )\) <em><strong>A1 N2</strong> </em></span></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;"> \({\rm{A}}\widehat {\rm{C}} {\rm{D}} = \pi - 2.381\) \((180 - 136.4)\) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> evidence of choosing the sine rule in triangle ACD <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> e.g. \(\frac{{6.5}}{{\sin 0.760 \ldots }} = \frac{7}{{\sin {\rm{A}}\widehat {\rm{D}} {\rm{C}}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = 0.836 \ldots \) \(( = 47.9 \ldots ^\circ )\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{C}}\widehat {\rm{A}}{\rm{D}} = \pi - (0.760 \ldots + 0.836 \ldots )\) \((180 - (43.5 \ldots + 47.9 \ldots ))\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \( = 1.54\) \(( = 88.5^\circ )\) <em><strong>A1 N3</strong> </em></span></p>
<p><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;"> \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = \frac{1}{2}(\pi - 2.381)\) \(\left( {\frac{1}{2}(180 - 136.4)} \right)\) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> evidence of choosing the sine rule in triangle ABD <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> e.g. \(\frac{{6.5}}{{\sin 0.380 \ldots }} = \frac{{13}}{{\sin {\rm{A}}\widehat {\rm{D}}{\rm{C}}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = 0.836 \ldots \) \(( = 47.9 \ldots ^\circ )\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\rm{C}}\widehat {\rm{A}}{\rm{D}} = \pi - 0.836 \ldots - (\pi - 2.381 \ldots )\) \(( = 180 - 47.9 \ldots - (180 - 136.4))\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \( = 1.54\) \(( = 88.5^\circ )\) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Note</strong>: Two triangles are possible with the given information. If candidate finds \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = 2.31\) \((132^\circ )\) leading to \({\rm{C}}\widehat {\rm{A}}{\rm{D}} = 0.076\) \((4.35^\circ )\) , award marks as per markscheme. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was generally well done. Even the weakest candidates often earned marks. Only a very few candidates used a right-angled triangle approach. </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;">Almost no candidates realized there was an ambiguous case of the sine rule in part (b). Those who did not lose the mark for accuracy in the previous question often lost it here. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a triangle ABC.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_10.20.56.png" alt="N17/5/MATME/SP2/ENG/TZ0/01"></p>
<p style="text-align: center;">\({\text{AB}} = 5{\rm{ cm, C\hat AB}} = \) 50° and \({\rm{A\hat CB}} = \) 112°</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find BC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle ABC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of choosing sine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\sin A}}{a} = \frac{{\sin B}}{b}\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{BC}}}}{{\sin 50}} = \frac{5}{{\sin 112}}\)</p>
<p>4.13102</p>
<p>\({\text{BC}} = 4.13{\text{ (cm)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\rm{\hat B}} = 180 - 50 - 112\), 18°, \({\text{AC}} = 1.66642\)</p>
<p>correct substitution into area formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2} \times 5 \times 4.13 \times \sin 18,{\text{ }}0.5(5)(1.66642)\sin 50,{\text{ }}\frac{1}{2}(4.13)(1.66642)\sin 112\)</p>
<p>3.19139</p>
<p>\({\text{area}} = 3.19{\text{ (c}}{{\text{m}}^2}{\text{)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a circle with centre O and radius 8 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/lone.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The points A, B, C, D, E and F are on the circle, and [AF] is a diameter. The length </span><span style="font-family: times new roman,times; font-size: medium;">of arc ABC is 6 cm.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the size of angle AOC .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence find the area of the shaded region.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The area of sector OCDE is \(45{\text{ c}}{{\text{m}}^2}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the size of angle COE .</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><span style="font-family: times new roman,times; font-size: medium;">Find EF .</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6 = 8\theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A}}\widehat {\rm{O}}{\rm{C}} = 0.75\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substitution into formula for area of triangle <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{area}} = \frac{1}{2} \times 8 \times 8 \times \sin (0.75)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area \(= 21.8 \ldots \) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substitution into formula for area of sector <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{area}} = \frac{1}{2} \times 64 \times 0.75\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area of sector \(= 24\) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting areas <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}{r^2}\theta - \frac{1}{2}ab\sin C\) </span><span style="font-family: times new roman,times; font-size: medium;">, \({\text{area of sector}} - {\text{area of triangle}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area of shaded region \( = 2.19{\text{ c}}{{\text{m}}^2}\) <em><strong>A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to set up an equation for area of sector <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(45 = \frac{1}{2} \times {8^2} \times \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{C}}\widehat {\rm{O}}{\rm{E}} = 1.40625\) (1.41 to 3 sf) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to find angle EOF <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi - 0.75 - 1.41\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}\widehat {\rm{O}}{\rm{F}} = 0.985\) (seen anywhere) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{EF}} = \sqrt {{8^2} + {8^2} - 2 \times 8 \times 8 \times \cos 0.985} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">EF \(= 7.57{\text{ cm}}\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to find angles that are needed <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. angle EOF and angle OEF</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}\widehat {\rm{O}}{\rm{F}} = 0.9853 \ldots \) <strong>and</strong> \({\text{O}}\widehat {\rm{E}}{\text{F (or O}}\widehat {\text{F}}{\text{E)}} = 1.078 \ldots \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing sine rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{EF}}}}{{\sin 0.985}} = \frac{8}{{\sin 1.08}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">EF \(= 7.57{\text{ cm}}\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 3</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to find angle EOF <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi - 0.75 - 1.41\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{E}}\widehat {\rm{O}}{\rm{F}} = 0.985\)</span> (seen anywhere) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using half of triangle EOF <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = 8\sin \frac{{0.985}}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct calculation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = 3.78\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">EF \(= 7.57{\text{ cm}}\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates demonstrated understanding of trigonometry on this question. They generally </span><span style="font-family: times new roman,times; font-size: medium;">did well in parts (a) and (c), and even many of them on part (b). Fewer candidates could do </span><span style="font-family: times new roman,times; font-size: medium;">part (d).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many opted to work in degrees rather than in radians, which often introduced multiple </span><span style="font-family: times new roman,times; font-size: medium;">inaccuracies. Some worked with an incorrect radius of 6 or 10.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A pleasing number knew how to find the area of the shaded region.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Inability to work in radians and misunderstanding of significant figures were common </span><span style="font-family: times new roman,times; font-size: medium;">problems, though. Weaker candidates often made the mistake of using triangle formulae for </span><span style="font-family: times new roman,times; font-size: medium;">sectors or used degrees in the formulas instead of radians.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For some candidates there were many instances of confusion between lines and arcs. In (a) </span><span style="font-family: times new roman,times; font-size: medium;">some treated 6 as the length of AC . In (d) some found the length of arc EF rather than the </span><span style="font-family: times new roman,times; font-size: medium;">length of the segment.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Several students seemed to confuse the area of sector in (b) with the shaded region.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates demonstrated understanding of trigonometry on this question. They generally </span><span style="font-family: times new roman,times; font-size: medium;">did well in parts (a) and (c), and even many of them on part (b). Fewer candidates could do </span><span style="font-family: times new roman,times; font-size: medium;">part (d).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many opted to work in degrees rather than in radians, which often introduced multiple </span><span style="font-family: times new roman,times; font-size: medium;">inaccuracies. Some worked with an incorrect radius of 6 or 10.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A pleasing number knew how to find the area of the shaded region.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Inability to work in radians and misunderstanding of significant figures were common </span><span style="font-family: times new roman,times; font-size: medium;">problems, though. Weaker candidates often made the mistake of using triangle formulae for </span><span style="font-family: times new roman,times; font-size: medium;">sectors or used degrees in the formulas instead of radians.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For some candidates there were many instances of confusion between lines and arcs. In (a) </span><span style="font-family: times new roman,times; font-size: medium;">some treated 6 as the length of AC . In (d) some found the length of arc EF rather than the </span><span style="font-family: times new roman,times; font-size: medium;">length of the segment.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Several students seemed to confuse the area of sector in (b) with the shaded region.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates demonstrated understanding of trigonometry on this question. They generally </span><span style="font-family: times new roman,times; font-size: medium;">did well in parts (a) and (c), and even many of them on part (b). Fewer candidates could do </span><span style="font-family: times new roman,times; font-size: medium;">part (d).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many opted to work in degrees rather than in radians, which often introduced multiple </span><span style="font-family: times new roman,times; font-size: medium;">inaccuracies. Some worked with an incorrect radius of 6 or 10.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A pleasing number knew how to find the area of the shaded region.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Inability to work in radians and misunderstanding of significant figures were common </span><span style="font-family: times new roman,times; font-size: medium;">problems, though. Weaker candidates often made the mistake of using triangle formulae for </span><span style="font-family: times new roman,times; font-size: medium;">sectors or used degrees in the formulas instead of radians.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For some candidates there were many instances of confusion between lines and arcs. In (a) </span><span style="font-family: times new roman,times; font-size: medium;">some treated 6 as the length of AC . In (d) some found the length of arc EF rather than the </span><span style="font-family: times new roman,times; font-size: medium;">length of the segment.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Several students seemed to confuse the area of sector in (b) with the shaded region.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates demonstrated understanding of trigonometry on this question. They generally </span><span style="font-family: times new roman,times; font-size: medium;">did well in parts (a) and (c), and even many of them on part (b). Fewer candidates could do </span><span style="font-family: times new roman,times; font-size: medium;">part (d).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many opted to work in degrees rather than in radians, which often introduced multiple </span><span style="font-family: times new roman,times; font-size: medium;">inaccuracies. Some worked with an incorrect radius of 6 or 10.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A pleasing number knew how to find the area of the shaded region.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Inability to work in radians and misunderstanding of significant figures were common </span><span style="font-family: times new roman,times; font-size: medium;">problems, though. Weaker candidates often made the mistake of using triangle formulae for </span><span style="font-family: times new roman,times; font-size: medium;">sectors or used degrees in the formulas instead of radians.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For some candidates there were many instances of confusion between lines and arcs. In (a) </span><span style="font-family: times new roman,times; font-size: medium;">some treated 6 as the length of AC . In (d) some found the length of arc EF rather than the </span><span style="font-family: times new roman,times; font-size: medium;">length of the segment.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Several students seemed to confuse the area of sector in (b) with the shaded region.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{3x}}{2} + 1\) , \(g(x) = 4\cos \left( {\frac{x}{3}} \right) - 1\) . Let \(h(x) = (g \circ f)(x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for \(h(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the period of \(h\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \(h\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to form any composition (even if order is reversed) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct composition \(h(x) = g\left( {\frac{{3x}}{2} + 1} \right)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(h(x) = 4\cos \left( {\frac{{\frac{{3x}}{2} + 1}}{3}} \right) - 1\) \(\left( {4\cos \left( {\frac{1}{2}x + \frac{1}{3}} \right) - 1,4\cos \left( {\frac{{3x + 2}}{6}} \right) - 1} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3 </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">period is \(4\pi (12.6)\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">range is \( - 5 \le h(x) \le 3\) \(\left( {\left[ { - 5,3} \right]} \right)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient. </span></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;">The circle shown has centre O and radius 3.9 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/1234.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Points A and B lie on the circle and angle AOB is 1.8 radians.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find AB.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the shaded region.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><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;">choosing cosine rule <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting correctly <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{AB}} = \sqrt {{{3.9}^2} + {{3.9}^2} - 2(3.9)(3.9)\cos 1.8} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB}} = 6.11\) (cm) <em><strong>A1 N2</strong></em></span></p>
<p><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;">evidence of approach involving right-angled triangles <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting correctly <em><strong> A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sin 0.9 = \frac{x}{{3.9}}\) , \(\frac{1}{2}{\rm{AB}} = 3.9\sin 0.9\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB}} = 6.11\) (cm) <em><strong>A1 N2</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 3</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">choosing the sine rule <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting correctly <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\sin 0.670 \ldots }}{{3.9}} = \frac{{\sin 1.8}}{{{\rm{AB}}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AB}} = 6.11\) (cm) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">reflex \({\rm{A}}\widehat {\rm{O}}{\rm{B}} = 2\pi - 1.8\) \(( = 4.4832)\) <em><strong>(A2)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution \(A = \frac{1}{2}{(3.9)^2}(4.4832 \ldots )\) </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;">area =34.1 (cm<sup>2</sup>) <em><strong>A1 N2</strong></em></span></p>
<p><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;">finding area of circle \(A = \pi {(3.9)^2}\) \(( = 47.78 \ldots )\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding area of (minor) sector \(A = \frac{1}{2}{(3.9)^2}(1.8)\)</span><span style="font-family: times new roman,times; font-size: medium;"> \(( = 13.68 \ldots )\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">subtracting <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi {(3.9)^2} - 0.5{(3.9)^2}(1.8)\) , \(47.8 - 13.7\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area = 34.1 (cm<sup>2</sup>) <em><strong>A1 N2</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 3</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding reflex \({\rm{A}}\widehat {\rm{O}}{\rm{B}} = 2\pi - 1.8\) \(( = 4.4832)\) <em><strong>(A2)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding proportion of total area of circle <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{2\pi - 1.8}}{{2\pi }} \times \pi {(3.9)^2}\) , \(\frac{\theta }{{2\pi }} \times \pi {r^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area = 34.1 (cm<sup>2</sup>) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well answered by the majority of candidates. Full solutions were common in both parts, and a variety of successful approaches were used. Radians were well handled with few candidates working with the angle in degrees. Some candidates incorrectly found the length of the arc subtended by the central angle rather than the length of segment [AB]. </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;">In part (b), some candidates incorrectly subtracted the area of the triangle or even a length. Many candidates failed to give answers to 3 significant figures and therefore lost an accuracy mark. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a circle with centre O and radius \(r\) cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/scooby.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Points A and B are on the circumference of the circle and \({\rm{A}}\hat {\rm{O}}{\rm{B}} = 1.4\) radians .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point C is on [OA] such that \({\rm{B}}\hat {\rm{C}}{\rm{O}} = \frac{\pi }{2}\) radians</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \({\rm{OC}} = r\cos 1.4\) .</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 area of the shaded region is \(25\) cm<sup>2</sup> . Find the value of \(r\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">use right triangle trigonometry <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\cos 1.4 = \frac{{{\rm{OC}}}}{r}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{OC}} = r\cos 1.4\) <em><strong>AG N0 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct value for BC </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{BC}} = r\sin 1.4\) , \(\sqrt {{r^2} - {{(r\cos 1.4)}^2}} \) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of \(\Delta {\rm{OBC}} = \frac{1}{2}r\sin 1.4 \times r\cos 1.4\) \(\left( { = \frac{1}{2}{r^2}\sin 1.4 \times \cos 1.4} \right)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of sector \({{\rm{OAB}} = \frac{1}{2}{r^2} \times 1.4}\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to subtract in any order <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> sector – triangle, \({\frac{1}{2}{r^2}\sin 1.4 \times \cos 1.4 - 0.7{r^2}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({0.7{r^2} - \frac{1}{2}r\sin 1.4 \times r\cos 1.4 = 25}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve <em><strong>their</strong></em> equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> sketch, writing as quadratic, \(\frac{{25}}{{0.616 \ldots }}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 6.37\) <strong><em>A1 N4 </em></strong></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[7 marks]</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Exception to <em><strong>FT</strong></em> rule. Award <strong><em>A1FT</em></strong> for a correct <strong><em>FT</em></strong> answer from a quadratic equation involving two trigonometric functions. </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;">As to be expected, candidates found this problem challenging. In part (a), many were able to use right angle trigonometry to find the length of OC.</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;">As to be expected, candidates found this problem challenging. Those who used a systematic approach in part (b) were more successful than those whose work was scattered about the page. While a pleasing number of candidates successfully found the area of sector AOB, far fewer were able to find the area of triangle BOC. Candidates who took an analytic approach to solving the resulting equation were generally less successful than those who used their GDC. Candidates who converted the angle to degrees generally were not very successful.<br></span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a circle with centre \(O\) and radius \(3\) cm.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_15.51.50.png" alt></p>
<p class="p1">Points A, B, and C lie on the circle, and \({\rm{A\hat OC}} = 1.3{\text{ radians}}\)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the length of arc \(ABC\)<span class="s1">.</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 class="p1">Find the area of the shaded region.</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>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;l = 1.3 \times 3\)</p>
<p>\(l\) = \(3.9\) (cm) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)finding reflex angle, \(2\pi - {\rm{C\hat OA}}\)</p>
<p>correct angle <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2\pi - 1.3,{\text{ }}4.98318\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2}(2\pi - 1.3){3^2}\)</p>
<p>\(22.4243\)</p>
<p>\({\text{area}} = 9\pi - 5.85{\text{ (exact), }}22.4{\text{ }} {\text{ }}({\text{c}}{{\text{m}}^2})\) <strong><em>A1 N3</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>correct area of small sector <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2}(1.3){3^2},{\text{ }}5.85\)</p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)circle − small sector, \(\pi {r^2} - \frac{1}{2}\theta {r^2}\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\pi ({3^2}) - \frac{1}{2}(1.3){3^2}\)</p>
<p>\(22.4243\)</p>
<p>\({\text{area}} = 9\pi - 5.85{\text{ }}({\text{exact}}),{\text{ }}22.4{\text{ }}{\text{ }}({\text{c}}{{\text{m}}^2})\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows a circle with centre O and radius 40 cm.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_17.31.00.png" alt="M17/5/MATME/SP2/ENG/TZ2/01"></p>
<p>The points A, B and C are on the circumference of the circle and \({\rm{A\hat OC}} = 1.9{\text{ radians}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the length of arc ABC.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the perimeter of sector OABC.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of sector OABC.</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>correct substitution into arc length formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\((40)(1.9)\)</p>
<p>\({\text{arc length}} = 76{\text{ (cm)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{arc}} + 2r,{\text{ }}76 + 40 + 40\)</p>
<p>\({\text{perimeter}} = 156{\text{ (cm)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into area formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}(1.9){(40)^2}\)</p>
<p>\({\text{area}} = 1520{\text{ (c}}{{\text{m}}^{\text{2}}}{\text{)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a waterwheel with a bucket. The wheel rotates at a </span><span style="font-family: times new roman,times; font-size: medium;">constant rate in an anticlockwise (counter-clockwise) direction.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bucket.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres </span><span style="font-family: times new roman,times; font-size: medium;">above the water level. After <em>t</em> seconds, the height of the bucket above the water level </span><span style="font-family: times new roman,times; font-size: medium;">is given by \(h = a\sin bt + 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(a = 4\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The wheel turns at a rate of one rotation every 30 seconds.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(b = \frac{\pi }{{15}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the first rotation, there are two values of <em>t</em> when the bucket is <strong>descending</strong> at a rate </span><span style="font-family: times new roman,times; font-size: medium;">of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find these values of <em>t</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the first rotation, there are two values of <em>t</em> when the bucket is <strong>descending</strong> at a rate </span><span style="font-family: times new roman,times; font-size: medium;">of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Determine whether the bucket is underwater at the second value of <em>t</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing the amplitude is the radius <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. amplitude is half the diameter</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = \frac{8}{2}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 4\) <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing the maximum height <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h = 6\) , \(a\sin bt + 2 = 6\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct reasoning</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a\sin bt = 4\) and \(\sin bt\) has amplitude of 1 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 4\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">period = 30 <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{{2\pi }}{{30}}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{\pi }{{15}}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2 = 4\sin 30b + 2\) , \(\sin 30b = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(30b = 2\pi \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{\pi }{{15}}\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing \(h'(t) = - 0.5\) (seen anywhere) <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to solve <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(h'\) , finding \(h'\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct work involving \(h'\) <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(h'\) showing intersection, \( - 0.5 = \frac{{4\pi }}{{15}}\cos \left( {\frac{\pi }{{15}}t} \right)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 10.6\) , \(t = 19.4\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning for <strong>their</strong> conclusion (seen anywhere) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(t) < 0\) so underwater; \(h(t) > 0\) so not underwater</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into <em>h</em> <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(19.4)\) , \(4\sin \frac{{19.4\pi }}{{15}} + 2\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct calculation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(19.4) = - 1.19\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct statement <em><strong>A1 N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. the bucket is underwater, yes</span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning for <strong>their</strong> conclusion (seen anywhere) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(t) < 0\) so underwater; \(h(t) > 0\) so not underwater</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. solving \(h(t) = 0\) , graph showing region below <em>x</em>-axis</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct roots <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(17.5\), \(27.5\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct statement <em><strong>A1 N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. the bucket is underwater, yes</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done. </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;">Parts (a) and (b) were generally well done, however there were several instances of candidates working backwards from the given answer in part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (c) and (d) proved to be quite challenging for a large proportion of candidates. Many did not attempt these parts. The most common error was a misinterpretation of the word "descending" where numerous candidates took \(h'(t)\) to be 0.5 instead of \( - 0.5\) but incorrect derivatives for <em>h</em> were also widespread. The process required to solve for <em>t</em> from the equation \( - 0.5 = \frac{{4\pi }}{{15}}\cos \left( {\frac{\pi }{{15}}t} \right)\) overwhelmed those who attempted algebraic methods. Few could obtain both correct solutions, more had one correct while others included unreasonable values including \(t < 0\) . </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), not many understood that the condition for underwater was \(h(t) < 0\) and had trouble interpreting the meaning of "second value". Many candidates, however, did recover to gain some marks in follow through. </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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">In triangle \(\rm{ABC}\), \(\rm{AB} = 6\,\rm{cm}\) and \(\rm{AC} = 8\,\rm{cm}\). The area of the triangle is \(16\,\rm{cm}^2\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the two possible values for \(\hat A\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\hat A\) is obtuse, find \({\text{BC}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into area formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{1}{2}(6)(8)\sin A = 16,{\text{ }}\sin A = \frac{{16}}{{24}}\)</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;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(A = \arcsin \left( {\frac{2}{3}} \right)\)</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;">\(A = 0.729727656…, 2.41186499…\); \((41.8103149^\circ, 138.1896851^\circ)\)</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;">\(A = 0.730\); </span><span style="font-family: 'times new roman', times; font-size: medium;">\(2.41\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(accept degrees <em>ie </em>\(41.8^\circ\);</span><span style="font-family: 'times new roman', times; font-size: medium;"> \(138^\circ\))</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of choosing cosine rule <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{B}}{{\text{C}}^2} = {\text{A}}{{\text{B}}^2} + {\text{A}}{{\text{C}}^2} - 2({\text{AB)(AC)}}\cos A,{\text{ }}{a^2} + {b^2} - 2ab\cos C\)</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;">correct substitution into RHS (angle must be obtuse) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({\text{B}}{{\text{C}}^2} = {6^2} + {8^2} - 2(6)(8)\cos 2.41,{\text{ }}{6^2} + {8^2} - 2(6)(8)\cos 138^\circ \),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \({\text{BC}} = \sqrt {171.55} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{BC}} = 13.09786\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{BC}} = 13.1{\text{ cm}}\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="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;">Let \(f(x) = \cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right),{\text{ for }} - 4 \leqslant x \leqslant 4.\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(f\).</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;">Find the values of \(x\) where the function is decreasing.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(a\);</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(c\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><br><img src="images/maths_9a_markscheme.png" alt> <strong><em> A1A1A1 N3</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for approximately correct sinusoidal shape.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> <strong>Only</strong> if this <strong><em>A1</em></strong> is awarded, award the following:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1</em></strong> for correct domain,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1</em></strong> for approximately correct range.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p> </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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizes decreasing to the left of minimum or right of maximum,</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;"><em>eg</em> \(f'(x) < 0\) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>x-</em>values of minimum and maximum (may be seen on sketch in part (a)) <strong><em>(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(x = - 3,{\text{ (1, 1.4)}}\)</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;">two correct intervals <strong><em>A1A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \( - 4 < x < - 3,{\text{ }}1 \leqslant x \leqslant 4;{\text{ }}x < - 3,{\text{ }}x \geqslant 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizes that \(a\) is found from amplitude of wave <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>y-</em>value of minimum or maximum <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>(−3, −1.41) , (1, 1.41)</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;">\(a = 1.41421\)</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;">\(a = \sqrt 2 {\text{, (exact), 1.41,}}\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognize that shift for sine is found at <em>x</em>-intercept <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find <em>x</em>-intercept <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right) = 0,{\text{ }}x = 3 + 4k,{\text{ }}k \in \mathbb{Z}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = - 1\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(c = 1\) <strong><em>A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to use a coordinate to make an equation <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\sqrt 2 \sin \left( {\frac{\pi }{4}c} \right) = 1,{\text{ }}\sqrt 2 \sin \left( {\frac{\pi }{4}(3 - c)} \right) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to solve resulting equation <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>sketch, \(x = 3 + 4k,{\text{ }}k \in \mathbb{Z}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = - 1\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(c = 1\) <strong><em>A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a circle with centre \(O\) and radius \(8\) cm.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_08.25.41.png" alt></p>
<p class="p1">The points \(A\), \(B\) and \(C\) are on the circumference of the circle, and \({\rm{A\hat OB}}\) radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the length of arc \(ACB\).</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 \(AB\).</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">Hence, find the perimeter of the shaded segment \(ABC\).</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">correct substitution into formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(l = 1.2 \times 8\)</p>
<p class="p1">\(9.6{\text{ (cm)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</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><strong>METHOD 1</strong></p>
<p>evidence of choosing cosine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(2{r^2} - 2 \times {r^2} \times \cos ({\rm{A\hat OB)}}\)</p>
<p>correct substitution into right hand side <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\({8^2} + {8^2} - 2 \times 8 \times 8 \times \cos (1.2)\)</p>
<p>\(9.0342795\)</p>
<p>\({\text{AB}} = 9.03{\text{ }}[9.03,{\text{ }}9.04]{\text{ (cm)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>evidence of choosing sine rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(\frac{{{\text{AB}}}}{{\sin ({\rm{A\hat OB)}}}} = \frac{{{\text{OB}}}}{{\sin ({\rm{O\hat AB)}}}}\)</p>
<p>finding angle \(OAB\) or \(OBA\) (may be seen in substitution) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(\frac{{\pi - 1.2}}{2},{\text{ }}0.970796\)</p>
<p>\({\text{AB}} = 9.03{\text{ }}[9.03,{\text{ }}9.04]{\text{ (cm)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(P = 9.6 + 9.03\)</p>
<p class="p1">\(18.6342\)</p>
<p class="p1">\(18.6{\text{ }}[18.6,{\text{ }}18.7]{\text{ (cm)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were well done, but it was not uncommon to see students finding area instead of perimeter in part (c). Most candidates recognized the need to use the cosine rule in part (b), and other candidates chose to use the sine rule to find the length of AB.</p>
<p class="p1">There are candidates who do not seem comfortable working with radians and transform the angles into degrees. Other candidates used an angle of \(1.2\pi \) instead of 1.2, supposing that angles in radians always should have \(\pi \).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were well done, but it was not uncommon to see students finding area instead of perimeter in part (c). Most candidates recognized the need to use the cosine rule in part (b), and other candidates chose to use the sine rule to find the length of AB.</p>
<p class="p1">There are candidates who do not seem comfortable working with radians and transform the angles into degrees. Other candidates used an angle of \(1.2\pi \) instead of 1.2, supposing that angles in radians always should have \(\pi \).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were well done, but it was not uncommon to see students finding area instead of perimeter in part (c). Most candidates recognized the need to use the cosine rule in part (b), and other candidates chose to use the sine rule to find the length of AB.</p>
<p class="p1">There are candidates who do not seem comfortable working with radians and transform the angles into degrees. Other candidates used an angle of \(1.2\pi \) instead of 1.2, supposing that angles in radians always should have \(\pi \).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is a vertical tower TA of height 36 m at the base A of a hill. A straight </span><span style="font-family: times new roman,times; font-size: medium;">path goes up the hill from A to a point U. This information is represented by the </span><span style="font-family: times new roman,times; font-size: medium;">following diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/squeak.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The path makes a \({4^ \circ }\) angle with the horizontal.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point U on the path is \(25{\text{ m}}\) away from the base of the tower.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The top of the tower is fixed to U by a wire of length \(x{\text{ m}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Complete the diagram, showing clearly all the information above.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>x</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/baseball.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for labelling \({4^ \circ }\) with horizontal, <em><strong>A1</strong></em> for labelling [AU] 25 metres, </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for drawing [TU].</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{T}}\widehat {\rm{A}}{\rm{U}} = {86^ \circ }\) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing cosine rule <strong><em> (M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({x^2} = {25^2} + {36^2} - 2(25)(36)\cos {86^ \circ }\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 42.4\) <strong><em>A1 N3</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></em></strong></p>
<div class="question_part_label">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;">This question was attempted in a satisfactory manner. Even the weakest candidates earned some marks here, showing some clear working. In part (a) the diagram was completed fairly well, with some candidates incorrectly labelling the angle with the vertical as \({4^ \circ }\) . The cosine rule was applied satisfactory in part (b), although some candidates incorrectly used their calculators in radian mode. Approaches using a combination of the sine rule and/or right-angled triangle trigonometry were seen, especially when candidates incorrectly labelled the \(25{\text{ m}}\) path as being the distance from the horizontal to U. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was attempted in a satisfactory manner. Even the weakest candidates earned some marks here, showing some clear working. In part (a) the diagram was completed fairly well, with some candidates incorrectly labelling the angle with the vertical as \({4^ \circ }\) . The cosine rule was applied satisfactory in part (b), although some candidates incorrectly used their calculators in radian mode. Approaches using a combination of the sine rule and/or right-angled triangle trigonometry were seen, especially when candidates incorrectly labelled the \(25{\text{ m}}\) path as being the distance from the horizontal to U. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a square \(ABCD\), and a sector \(OAB\) of a circle centre \(O\), radius \(r\). Part of the square is shaded and labelled \(R\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_15.28.33.png" alt></p>
<p>\[{\rm{A\hat OB}} = \theta {\text{, where }}0.5 \ \le \ \theta < \pi .\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the area of the square \(ABCD\) is \(2{r^2}(1 - \cos \theta )\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>When \(\theta = \alpha \), the area of the square \(ABCD\) is equal to the area of the sector \(OAB\).</p>
<p>(i) Write down the area of the sector when \(\theta = \alpha \).</p>
<p>(ii) Hence find \(\alpha \).</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>When \(\theta = \beta \), the area of \(R\) is more than twice the area of the sector.</p>
<p>Find all possible values of \(\beta \).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>area of \({\text{ABCD}} = {\text{A}}{{\text{B}}^2}\) (seen anywhere) <strong><em>(A1)</em></strong></p>
<p>choose cosine rule to find a side of the square <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{a^2} = {b^2} + {c^2} - 2bc\cos \theta \)</p>
<p>correct substitution (for triangle \(AOB\)) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;{r^2} + {r^2} - 2 \times r \times r\cos \theta ,{\text{ O}}{{\text{A}}^2} + {\text{O}}{{\text{B}}^2} - 2 \times {\text{OA}} \times {\text{OB}}\cos \theta \)</p>
<p>correct working for \({\text{A}}{{\text{B}}^2}\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;2{r^{\text{2}}} - 2{r^2}\cos \theta \)</p>
<p>\({\text{area}} = 2{r^2}(1 - \cos \theta )\) <strong><em>AG N0</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award no marks if the only working is \(2{r^2} - 2{r^2}\cos \theta \).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(\frac{1}{2}\alpha {r^2}\;\;\;\left( {{\text{accept }}2{r^2}(1 - \cos \alpha )} \right)\) <strong><em>A1 N1</em></strong></p>
<p>(ii) correct equation in one variable <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2(1 - \cos \alpha ) = \frac{1}{2}\alpha \)</p>
<p>\(\alpha = 0.511024\)</p>
<p>\(\alpha = 0.511\;\;\;({\text{accept }}\theta = 0.511)\) <strong><em>A2 N2</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for \(\alpha = 0.511\) and additional answers.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note: </strong>In this part, accept \(\theta \) instead of \(\beta \), and the use of equations instead of inequalities in the working.</p>
<p> </p>
<p>attempt to find \(R\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)subtraction of areas, \({\text{square}} - {\text{segment}}\)</p>
<p>correct expression for segment area <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2}\beta {r^2} - \frac{1}{2}{r^2}\sin \beta \)</p>
<p>correct expression for \(R\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2{r^2}(1 - \cos \beta ) - \left( {\frac{1}{2}\beta {r^2} - \frac{1}{2}{r^2}\sin \beta } \right)\)</p>
<p>correct inequality <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2{r^2}(1 - \cos \beta ) - \left( {\frac{1}{2}\beta {r^2} - \frac{1}{2}{r^2}\sin \beta } \right) > 2\left( {\frac{1}{2}\beta {r^2}} \right)\)</p>
<p>correct inequality in terms of angle only <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;2(1 - \cos \beta ) - \left( {\frac{1}{2}\beta - \frac{1}{2}\sin \beta } \right) > \beta \)</p>
<p>attempt to solve their inequality, must represent \(R > \) twice sector <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)sketch, one correct value</p>
<p> </p>
<p><strong>Note: </strong>Do not award the second <strong><em>(M1) </em></strong>unless the first <strong><em>(M1) </em></strong>for attempting to find \(R\) has been awarded.</p>
<p> </p>
<p><strong>both </strong>correct values \(1.30573\) and \(2.67369\) <strong><em>(A1)</em></strong></p>
<p>correct inequality \(1.31 < \beta < 2.67\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<p><strong><em>Total [16 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">Those who attempted part (a) could in general show what was required by using the cosine rule. On rare occasions some more complicated approaches were seen using half of angle theta. In some cases, candidates did not show all the necessary steps and lost marks for not completely showing the given result.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">A number of candidates correctly answered part (bi) and created a correct equation in (bii), but did not solve the equation correctly, usually attempting an analytic method where the GDC would do. For many a major problem was to realize the need to reduce the equation to one variable before attempting to solve it. Occasionally, an answer would be written that was outside the given domain.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">When part (c) was attempted, many candidates did not recognize that the area in question requires the subtraction of a segment area, and often set the square area greater than twice the sector. Many candidates made mistakes when trying to eliminate brackets or just did not use them. Of those who created a correct inequality, few reached a fully correct conclusion.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider a circle with centre \(\rm{O}\) and radius \(7\) cm. Triangle \(\rm{ABC}\) is drawn such that its vertices are on the circumference of the circle.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_8.png" alt></p>
<p style="font-stretch: normal; margin: 0px; text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'Times New Roman';"><span style="line-height: normal;">\(\rm{AB}=12.2\) cm, \(\rm{BC}=10.4\) cm and \(\rm{A}\hat{\rm{C}}\rm{B}=1.058\) radians.</span></span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\rm{B\hat AC}}\).</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;">Find \({\text{AC}}\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence or otherwise, find the length of arc \({\text{ABC}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="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>Notes: </strong>In this question, there may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</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;">Candidates may have their GDCs in degree mode, leading to incorrect answers. If working shown, award marks in line with the markscheme, with <strong><em>FT </em></strong>as appropriate.</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;">Ignore missing or incorrect units.</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;">evidence of choosing sine rule <strong><em>(M1)</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;"><em>eg</em> \(\frac{{\sin \hat A}}{a} = \frac{{\sin \hat B}}{b}\)</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;">correct substitution <strong><em>(A1)</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;"><em>eg</em> \(\frac{{\sin \hat A}}{{10.4}} = \frac{{\sin 1.058}}{{12.2}}\)</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;">\({\rm{B\hat AC}} = 0.837\) <strong><em>A1 N2</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>[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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes: </strong>In this question, there may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</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;">Candidates may have their GDCs in degree mode, leading to incorrect answers. If working shown, award marks in line with the markscheme, with <strong><em>FT </em></strong>as appropriate.</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;">Ignore missing or incorrect units.</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>METHOD 1</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;">evidence of subtracting angles from \(\pi \) <strong><em>(M1)</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;"><em>eg</em> \({\rm{A\hat BC}} = \pi - A - C\)</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;">correct angle (seen anywhere) <strong><em>A1</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;">\({\rm{A\hat BC}} = \pi - 1.058 - 0.837,{\text{ }}1.246,{\text{ }}71.4^\circ \)</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;">attempt to substitute into cosine or sine rule <strong><em>(M1)</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;">correct substitution <strong><em>(A1)</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;"><em>eg</em> \({12.2^2} + {10.4^2} - 2 \times 12.2 \times 10.4\cos 71.4,{\text{ }}\frac{{{\text{AC}}}}{{\sin 1.246}} = \frac{{12.2}}{{\sin 1.058}}\)</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;">\({\text{AC}} = 13.3{\text{ (cm)}}\) <strong><em>A1 N3</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>METHOD 2</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;">evidence of choosing cosine rule <strong><em>M1</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;"><em>eg</em> \({a^2} = {b^2} + {c^2} - 2bc\cos A\)</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;">correct substitution <strong><em>(A2)</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;"><em>eg</em> \({12.2^2} = {10.4^2} + {b^2} - 2 \times 10.4b\cos 1.058\)</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;">\({\text{AC}} = 13.3{\text{ (cm)}}\) <strong><em>A2 N3</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>[5 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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes: </strong>In this question, there may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</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;">Candidates may have their GDCs in degree mode, leading to incorrect answers. If working shown, award marks in line with the markscheme, with <strong><em>FT </em></strong>as appropriate.</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;">Ignore missing or incorrect units.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"><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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</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;"><em>eg</em> \(\cos {\rm{A\hat OC}} = \frac{{{\text{O}}{{\text{A}}^2} + {\text{O}}{{\text{C}}^2} - {\text{A}}{{\text{C}}^2}}}{{2 \times {\text{OA}} \times {\text{OC}}}}\), \({\rm{A\hat OC}} = 2 \times {\rm{A\hat BC}}\)</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;">correct working <strong><em>(A1)</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;"><em>eg</em> \({\text{13.}}{{\text{3}}^2} = {7^2} + {7^2} - 2 \times 7 \times 7\cos {\rm{A\hat OC}},{\text{ }}O = 2 \times 1.246\)</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;">\({\rm{A\hat OC}} = 2.492{\text{ }}(142.8^\circ )\) <strong><em>(A1)</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>EITHER</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;">correct substitution for arc length (seen anywhere) <strong><em>A1</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;"><em>eg</em> \(2.492 = \frac{l}{7},{\text{ }}l = 17.4,{\text{ }}14\pi \times \frac{{142.8}}{{360}}\)</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;">subtracting arc from circumference <strong><em>(M1)</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;"><em>eg</em> \(2\pi r - l,{\text{ }}14\pi = 17.4\)</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>OR</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;">attempt to find \({\rm{A\hat OC}}\) reflex <strong><em>(M1)</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;"><em>eg</em> \(2\pi - 2.492,{\text{ }}3.79,{\text{ }}360 - 142.8\)</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;">correct substitution for arc length (seen anywhere) <strong><em>A1</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;"><em>eg</em> \(l = 7 \times 3.79,{\text{ }}14\pi \times \frac{{217.2}}{{360}}\)</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>THEN</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;">\({\text{arc ABC}} = 26.5\) <strong><em>A1 N4</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>METHOD 2</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;">valid approach to find \({\rm{A\hat OB}}\) or \({\rm{B\hat OC}}\) <strong><em>(M1)</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;"><em>eg</em> choosing cos rule, twice angle at circumference</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;">correct working for finding <strong>one </strong>value, \({\rm{A\hat OB}}\) or \({\rm{B\hat OC}}\) <strong><em>(A1)</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;"><em>eg</em> \(\cos {\rm{A\hat OB}} = \frac{{{7^2} + {7^2} - {{12.2}^2}}}{{2 \times 7 \times 7}}\), \({\rm{A\hat OB}} = 2.116,{\rm{B\hat OC}} = 1.6745\)</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>two </strong>correct calculations for arc lengths </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;"><em>eg</em> \({\text{AB}} = 7 \times 2 \times 1.058{\text{ }}( = 14.8135),{\text{ }}7 \times 1.6745{\text{ }}( = 11.7216)\) <strong><em>(A1)(A1)</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;">adding <strong>their </strong>arc lengths (seen anywhere)</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;"><em>eg</em> \(r{\rm{A\hat OB}} + r{\rm{B\hat OC}},{\text{ }}14.8135 + 11.7216,{\text{ }}7(2.116 + 1.6745)\) <strong><em>M1</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;">\({\text{arc ABC}} = 26.5{\text{ (cm)}}\) <strong><em>A1 N4</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;"><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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Candidates may work with other interior triangles using a similar method. Check calculations carefully and award marks in line with markscheme.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"><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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows the chord [AB] in a circle of radius 8 cm, where \({\text{AB}} = 12{\text{ cm}}\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_13.20.17.png" alt="M17/5/MATME/SP2/ENG/TZ1/05"></p>
<p>Find the area of the shaded segment.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to find the central angle or half central angle <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)<img src="images/Schermafbeelding_2017-08-14_om_13.40.22.png" alt="M17/5/MATME/SP2/ENG/TZ1/05/M">, cosine rule, right triangle</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta = \frac{{{8^2} + {8^2} - {{12}^2}}}{{2 \bullet 8 \bullet 8}},{\text{ }}{\sin ^{ - 1}}\left( {\frac{6}{8}} \right),{\text{ }}0.722734,{\text{ }}41.4096^\circ ,{\text{ }}\frac{\pi }{2} - {\sin ^{ - 1}}\left( {\frac{6}{8}} \right)\)</p>
<p>correct angle \({\rm{A\hat OB}}\) (seen anywhere) </p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{1.69612, }}97.1807^\circ ,{\text{ 2}} \times {\text{si}}{{\text{n}}^{ - 1}}\left( {\frac{6}{8}} \right)\) <strong><em>(A1)</em></strong></p>
<p>correct sector area</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}(8)(8)(1.70),{\text{ }}\frac{{97.1807}}{{360}}(64\pi ),{\text{ }}54.2759\) <strong><em>(A1) </em></strong></p>
<p>area of triangle (seen anywhere) <strong><em>(A1) </em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}(8)(8)\sin 1.70,{\text{ }}\frac{1}{2}(8)(12)\sin 0.722,{\text{ }}\frac{1}{2} \times \sqrt {64 - 36} \times 12,{\text{ }}31.7490\)</p>
<p>appropriate approach (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({A_{{\text{triangle}}}} - {A_{{\text{sector}}}}\), their sector-their triangle</p>
<p>22.5269</p>
<p>area of shaded region \( = 22.5{\text{ }}({\text{c}}{{\text{m}}^2})\) <strong><em>A1</em></strong> <strong><em>N4</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0A0A0A0A1 </em></strong>then <strong><em>M1A0 </em></strong>(if appropriate) for correct triangle area without any attempt to find an angle in triangle OAB.</p>
<p> </p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>At an amusement park, a Ferris wheel with diameter 111 metres rotates at a constant speed. The bottom of the wheel is <em>k</em> metres above the ground. A seat starts at the bottom of the wheel.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The wheel completes one revolution in 16 minutes.</p>
</div>
<div class="specification">
<p>After <em>t</em> minutes, the height of the seat above ground is given by \(h\left( t \right) = 61.5 + a\,{\text{cos}}\left( {\frac{\pi }{8}t} \right)\), for 0 ≤ <em>t</em> ≤ 32.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After 8 minutes, the seat is 117 m above the ground. Find <em>k</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find when the seat is 30 m above the ground for the third time.</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>valid approach to find <em>k</em> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 8 minutes is half a turn, <em>k </em>+ diameter, <em>k </em>+ 111 = 117</p>
<p><em>k</em> = 6 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg</em> \(\frac{{{\text{max}}\,\, - \,\,{\text{min}}}}{2}\) <em>a</em> = radius</p>
<p>\(\left| a \right| = \frac{{117 - 6}}{2},\,\,55.5\) <em><strong>(A1)</strong></em></p>
<p><em>a</em> = −55.5 <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to substitute valid point into equation for <em>f</em> <em><strong>(M1)</strong></em><br><em>eg</em> <em>h</em>(0) = 6, <em>h</em>(8) = 117</p>
<p>correct equation <em><strong>(A1)</strong></em><br><em>eg</em> \(6 = 61.5 + a\,{\text{cos}}\left( {\frac{\pi }{8} \times 0} \right),\,\,117 = 61.5 + a\,{\text{cos}}\left( {\frac{\pi }{8} \times 8} \right),\,\,6 = 61.5 + a\)</p>
<p><em>a</em> = −55.5 <em><strong>A1 N2</strong></em></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg</em> sketch of <em>h</em> and \(y = 30,\,\,h = 30,\,\,61.5 - 55.5\,{\text{cos}}\left( {\frac{\pi }{8}t} \right) = 30,\,\,t = 2.46307,\,\,t = 13.5369\)</p>
<p>18.4630</p>
<p><em>t</em> = 18.5 (minutes) <em><strong>A1 N3</strong></em></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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows quadrilateral ABCD.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p style="text-align: center;">\({\text{AB}} = 11\,{\text{cm,}}\,\,{\text{BC}} = 6\,{\text{cm,}}\,\,{\text{B}}\mathop {\text{A}}\limits^ \wedge {\text{D = 100}}^\circ {\text{, and C}}\mathop {\text{B}}\limits^ \wedge {\text{D = 82}}^\circ \)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find DB.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find DC.</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>evidence of choosing sine rule <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(\frac{a}{{{\text{sin }}A}} = \frac{b}{{{\text{sin }}B}} = \frac{c}{{{\text{sin }}C}}\)</p>
<p>correct substitution <em><strong>(A1)</strong></em><br><em>eg </em>\(\frac{{{\text{DB}}}}{{{\text{sin }}59^\circ }} = \frac{{{\text{11}}}}{{{\text{sin }}100^\circ }}\)</p>
<p>9.57429</p>
<p>DB = 9.57 (cm) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of choosing cosine rule <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\({a^2} = {b^2} + {c^2} - 2bc\,\,{\text{cos}}\,\,\left( A \right),\,\,\,{\text{D}}{{\text{C}}^2} = \,\,{\text{D}}{{\text{B}}^2}{\text{ + B}}{{\text{C}}^2}{\text{ }} - {\text{ 2DB}} \times \,\,{\text{BC}} \times \,{\text{cos}}\,\left( {{\text{D}}\mathop {\text{B}}\limits^ \wedge {\text{C}}} \right)\) </p>
<p>correct substitution into RHS <em><strong>(A1)</strong></em><br><em>eg </em> \({9.57^2} + {6^2} - 2 \times 9.57 \times 6 \times \,\,{\text{cos}}\,\,82^\circ ,\,\,\,111.677\)<em> </em></p>
<p>10.5677</p>
<p>DC = 10.6 (cm) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A Ferris wheel with diameter \(122\) metres rotates clockwise at a constant speed. </span><span style="font-family: times new roman,times; font-size: medium;">The wheel completes \(2.4\) rotations every hour. The bottom of the wheel is \(13\) metres </span><span style="font-family: times new roman,times; font-size: medium;">above the ground.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/open.png" alt></span></p>
<p> <span style="font-family: times new roman,times; font-size: medium;">A seat starts at the bottom of the wheel.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">After <em><strong>t</strong> </em>minutes, the height \(h\) metres above the ground of the seat is given by\[h = 74 + a\cos bt {\rm{ .}}\]<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum height above the ground of the seat.</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-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that the period of \(h\) is \(25\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the <strong>exact</strong> value of \(b\) .</span></p>
<p> </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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) (i) Show that the period of \(h\) is \(25\) minutes.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down the <strong>exact</strong> value of \(b\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) </span><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) </span><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(h\) , for \(0 \le t \le 50\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">bcd.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(h\) , for \(0 \le t \le 50\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In one rotation of the wheel, find the probability that a randomly selected seat is </span><span style="font-family: times new roman,times; font-size: medium;">at least \(105\) metres above the ground.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(13 + {\rm{diameter}}\) , \(13 + 122\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum height \( = 135\) (m) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) period \( = \frac{{60}}{{2.4}}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">period \( = 25\) minutes <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(b = \frac{{2\pi }}{{25}}\) \(( = 0.08\pi )\) <strong><em>A1 N1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong><em>[2 marks]<br></em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) (i) period \( = \frac{{60}}{{2.4}}\) <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">period \( = 25\) minutes <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(b = \frac{{2\pi }}{{25}}\) \(( = 0.08\pi )\) <strong><em>A1 N1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong><em>[2 marks]<br></em></strong></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;">(b) <strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{max}} - 74\) , \(\left| a \right| = \frac{{135 - 13}}{2}\) , \(74 - 13\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| a \right| = 61\) (accept \(a = 61\) ) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em><span style="font-family: times new roman,times; font-size: medium;"><em> N2</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute valid point into equation for <em><strong>h</strong></em> <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos \left( {\frac{{2\pi \times 12.5}}{{25}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos (\pi )\) , \(13 = 74 + a\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 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)<br></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1A1 N4</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct domain, <em><strong>A1</strong></em> for approximately correct range,</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for approximately correct sinusoidal shape with \(2\) cycles. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this last <em><strong>A1</strong></em> awarded, award <em><strong>A1</strong></em> for max/min in approximately </span><span style="font-family: times new roman,times; font-size: medium;">correct positions. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 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 [9 marks]<br></span></strong></em></p>
<div class="question_part_label">bcd.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">valid approach <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{max}} - 74\) , \(\left| a \right| = \frac{{135 - 13}}{2}\) , \(74 - 13\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left| a \right| = 61\) (accept \(a = 61\) ) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em><span style="font-family: times new roman,times; font-size: medium;"><em> N2</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute valid point into equation for <em><strong>h</strong></em> <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos \left( {\frac{{2\pi \times 12.5}}{{25}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(135 = 74 + a\cos (\pi )\) , \(13 = 74 + a\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = - 61\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdIAAAEFCAIAAADYOfH2AAAgAElEQVR4nO2dZ0BTSdi2nxQ6EnoREKkKgohiAwsWsKJixS4qq2vvrmtBsWABBVFExd5WXCsqiCJ2qQIKSFcUlCK9l5Dvx7xvPl5AFyHJ5IS5fik55U5yzp05M08BDoFAIBAECOAWQCAQCB0LYrsEAoEgUIjtEggEgkAhtksgEAgChdgugUAgCBRiuwQCgSBQhMt2k5OTy8rKcKsgEAgEPiIstltbW+vi4qKqquri4oJbC4FAIPARYbFdNpu9fv16AJgwYQJuLQQCgcBHhMV2ORxOVlaWoaGhhobGhw8fcGshEAgEfiFEtsvhcOzt7QHg7t27uIUQCAQCvxAu250wYQIA3L59G7cQAoFA4BfCZbvTp08HgFu3buEWQiAQCPxCuGz30KFDALBr1y42m41bC4FAIPAF4bLdu3fvMhgMW1vbqqoq3FoIBAKBLwiX7SYmJsrKyh4+fBi3EAKBQOAXwmW7KSkp8vLytra2dXV1uLUQCAQCXxAu23379q2srKyioqK/v7+7u3tYWBhuRQQCgcBjhMt2Q0ND5eXlAUBcXBwA/v77b9yKCAQCgccIl+1+//7dyMgI/pfNmzfjVkQgEAg8Ruhst3v37lzbNTAwIPMMBAJBxBAu283PzzczMwMAWVnZrl27AoCtrW1xcTFuXQQCgcAzhMt2i4qKLCwsAMDKyur169f6+vo6OjqfP3/GrYtAIBB4hnDZbmlpab9+/QDAxMQkLS1t6NChampqHz9+xK2LQCAQeIZw2W5FRYW1tTUAODo6VldXDx48GACeP3+OWxeBQCDwDOGy3bq6uuHDhwPA+vXr2Ww2+vfTp09x6yIQCASeIVy2W1BQYGlpCQBPnjzhcDjIdseNG3fmzBky1UAgEEQD4bLdR48eodCxgIAADoezatWqgQMHGhoaiomJubq64lZHIBAIPEC4bDcqKqpTp056enqvX7/mcDjFxcUZGRlv37598+ZNQUEBbnUEAoHAA4TLdtlsdkhISFJSUkNDA24tBAKBwBeEy3YJBAJB5CG2SyAQCAKF2C6BQCAIFGK7BAKBIFCI7RIIBIJAIbZLIBAIAgWn7dbX15eWlmIUQCAQCIIHm+3GxcVNmTLFxcUFlwACgUDAgqBtNzExMTw8PCYmxsDAAACWLFkiYAEEAoGAFwHZbkJCwt69e69fv967d29ZWVllZWVUe2HlypVtPubLly/Pnj3LQ5EEAoEgANpuu2w2Ozk5uaCg4MOHDxwOJz4+Pj09/Wcbnz59mtshTUpKavz48StXrly/fn1CQkLbzt7Q0ODo6Dhw4MCqqqo2vgECgUDAwW/bbnp6+oEDBwIDA11cXHR0dCwsLLp27bpjxw5jY+N58+bV1dU136Wurm7atGmoDfvSpUvfvXtXU1PTTt0FBQU9e/ZkMBikCDqBQKAWv2e7tbW1U6ZMAQAGgwGNoNPpAMBisd69e9d8r+fPn0tKSs6aNev58+eVlZU80R0YGMhkMlksVkREBE8OSCAQCILhp7br7++fkZHR5I+1tbUTJkzgum2XLl2GDx8+ceLEwMDAvn37AsDNmzeb7NLQ0DBv3jwAePHiBQ91P3z4EAA6d+4cExPDw8MSCAQCv2nZdj9+/KilpXX9+vXmL718+XLkyJFjxoyZPHnykydP0HRBZGSkrKwsABw4cKDJ9vHx8fLy8jxviRYaGoqG2Nu3b+fhYQkEAoHftGy7AQEBADBlypQWX62pqamvr+f+Nzg42MrKCgAcHByarKpVVFQcOXIEDY23b9/+/ft3Xul+/fo1jUZTVFQkc7sEAoFatGy7ISEhADB48OBflxt//fr19OnTJSUlkbEGBQU12SAsLExGRoY7KdGjR489e/bwpCVlbGyshISEmJjY7du3uX8MDg729fWtra1t//EJBAKBT7Rsuy9evAAAKyurX4QcJCcna2hocC11/vz5zTN9b968SaPRuGtuCG1t7bi4uLbJramp+f79+/fv3wMDA5GhDxo0yNnZedasWTNmzNDU1GQwGHfv3m3bwQkEAkEAtGy70dHRNBrN0tKyoqLiZ3tevXqV66SOjo5lZWXNt/n7778BoGvXrjdv3rx27ZqxsTFy4ZEjR7ZhTPrjx4+ZM2dqaWlpaWkpKyujQzVBXFz83r17v3tkAqFD8f3791/c2gR+07LtXrhwgUaj9ezZs7i4+Gd7zpkzh2t2GzdubHGb3NzcK1euBAcHo/+mp6ffuXPHwMBAXl6+xVCzX7N9+/bmPtsYBoNBp9MPHTr0u0cWVerq6nJycvLz83ELIQgLxcXFa9eutbCw2L17d2RkJMk2wkLLtvvHH38AQL9+/UpKSn6256FDh7h+p6Cg4OrqGh8f35pTnjt3jsFgzJ0793eTJvz8/LhnFBMT4/6je/fuEhISK1eufPjwYUBAwC+S5ToI1dXV5eXl4eHhjo6OWlpaxsbGq1evvnfv3tWrVxcsWODt7Z2Tk4NbI0HQfP78eefOnUOHDuXeREpKSsOHD79x40Z5eTludR2Llm13w4YNAODq6vqLPWtqag4cOLB79+4tW7ZYWFgAwIgRI8LDw//zlG/evJGSklJRUcnMzPwtreXl5WPHjrW2tj569GhAQMCYMWMAYN68eTk5OcHBwdXV1b91NJGksrLyxo0bI0aM6N27t4qKys8eC3r16jV16tR9+/aRD62DEBkZ2b17d/Tt02g0c3Nz7sWgqKi4Zs2aEydOXLt2DbfMjkILtltaWopyIpoH4TY0NLi5ue3fv5/7l7q6uqqqKkdHR/QVbtmy5RcnS09Pv379uoODAwCMHj36FzMYP6OgoKCoqAgpQQnH7SmmI0rU19cfOXLE2tqaRqMxGAxNTU30jcjLy2toaGhpaXXt2tXY2FhVVbWx/9rZ2b169Qq3dgIfKSkpOXHihJ6eHvdLl5GR8fHx2bx5s5WVVZN0U2dn598dDBHaQAu2m5qaikIUvLy8mrwUEREhLS2N7DUoKGjTpk0DBgwwMzND35m0tHSLUQS1tbUvX74MDAzkTgfLycm1M6n327dv+vr6ANC5c+fk5OT2HEoEqK6udnJyQreQurp6p06dunTpMnPmzE2bNkVHR2dlZX379i0vL6+oqCgyMvLUqVPnz5+3t7dHEzVKSkorVqyIiIgggXeiR0lJyZAhQ7iuqqWldfbs2cePH6NXy8vL/f39nZycdHV1udtMmDAhNDS0xeIqBF7Rgu1++vRJR0eHRqOdP3++yUuurq7cr6dxTBiLxVJVVT19+jR3y+Dg4MWLFzs5Oc2bN8/e3h7lsNHpdCkpKQCwtLQsLCxsj+78/Hxk90pKSu/fv2/PoUSAo0ePcr8LNze3HTt2BAQE/HqXmpqaoKAgR0dHFBAiJyfn4eEhGLUEwRASEuLs7My9MFRUVDw9PVvcMiIiYufOnTNnzkRbSkpKbtmyxdvb+9dh+4Q204Ltfv36VV9fn8lkXr58uclLHz9+3LRpE3qSBQB9ff1OnTrt2rUrISEhNTW18Zbu7u5N5hPV1dU7d+7ctWtXAPDx8Wmn7sLCQjShzGKxoqOj23k06hIbG7tixQr0CKmsrLxjx47fWqhsaGi4fPly7969AcDCwuLJkydkaVsEYLPZWVlZqEwKd67pP9ddiouLvby8UP8BAGAymStWrCAL1PygBdvNyckxNDRksVhpaWkt7lNZWXngwIEdO3Z8+vQpPDy8caIworq6Gk28NkZBQUFFRWXo0KHu7u7tf4QpLS3t168fANjY2LRz4Exd7ty5g9bNtLW11dTUQkJC2nac79+/m5qaAoCEhMSaNWt4K5IgePLy8riLZioqKlu2bGn9DH5BQcHp06fRzQUAjRdyCLyiBdvNy8tDv3htXtksKytDVRrQz+zatWv37du3bNmyp0+f8ipIu6KiwtraGgAWLlzIkwNSjri4OEVFRe4kT/PM7N/i48eP6BlTTEzs2LFjbDabVzoJAqakpOT48ePowpCQkHj9+nUbDlJVVbVp0yYA0NDQ2L17N89FdnBasN3q6uo5c+ZYWlomJia2+bixsbHe3t4nT558+fJlO+T9lOrqahsbGwAYMWLEL4KLRZWUlBRbW1t0a8nKyjYuTNFmqqur0VQgnU7nSd0MAhYOHz6MFldZLFY7n1127dqFZhvevHnDK3kEzs/idktKSvLy8gQs5beoq6uzs7NDS2qRkZG45QgaFFiNVj+aT8G3mbq6OhROb2pqSgoZU5Hy8vJRo0ah3872j1Jra2vRY6utrS25HngItobt7aShoWH06NHQ1vIOlObq1avcIkTTp0/n7cFjYmL69OkDAH369OmAjxGUpq6ubuHChejCkJKS+vjxY/uPeffuXRQzunz58vYfjYCgqu1yOJzFixcDwPDhw3ELESgNDQ0DBgxAt9bQoUP5sdAcGxuL4vxmz57N84MT+Ed+fj7393jChAnNKwK2jZ07d6I4mZs3b5JJf55AYdu9ffs2qnTecSaeqqqqtmzZwmQyAWDQoEH8mwi6efOmvLy8mJjYuXPn+HQKAm+pqqpavXo1mtW1srLKzc3l1ZFRaBMAKCoqNokTJbQNCtvun3/+iX7Y/fz8cGsREB8+fJCQkEDxBo8ePeLrubZs2YIWsn8WR0gQKi5evMgN1vxZRcA24+XlhdKjOs69xlcobLtokgEAfH19cWsRED4+Pujqt7Oz43cVmy9fvqDcFhLJK/xw10JR0Njbt295e/yioiJUScfU1FTIF9spAYVtd+XKlQDQpUuXlJQU3FoEQVpaWs+ePdGt1bxcBj94/PixhISEvLx8R84DpAQHDx5EU09iYmIHDx7kR1LvkiVL0LVHksjbD4VtFwVRaWlptblFELXYunUruu5NTEw+fPgggDOy2ezZs2cDgJGRUQf5kCmKpaUlujZWr17Np1MkJiaiijm6urpk3qmdUNh2Uccg+K+6wKJBSUnJsGHD0Pvds2ePwM6bmJjYo0cPFKlGCqMIIQ0NDSdOnECRJwBw+PBh/p3r2rVr6CwPHjzg31k6AhS2XRTXAgAuLi64tfCd1NRUBQUFlJP27NkzQZ7a399fXFxcQkKCpK4JIW/fvuW27ra3t+dr5ZqMjAxjY2MA+Pvvv/l3lo4AhW1337596Gr7dW11EaCqqmrWrFnozU6ZMqV57SF+s3nzZgAYNWpUi41KCRjh5kfo6OgIILorNDRUSkpKWVmZJ7kYHRYK266Hhwe64ES+wcSXL1+4XSGwVITKzMxEcfhkmCNUfPjwQV5eHl0Y7S+m2hoaGhpQ6xlnZ2cBnE5UobDtent7AwCDwXB3d8ethb8cPXoUFTi2sbFBPY0EDJvNRq2YWCzWrVu3BC+A0CIbN25EnisvLy+wyiR+fn50Ol1bW5vkjrcZCtvuyZMnkRHExsbi1sJHamtr+/fvj+4ujIPN169fo8nlGTNm4NJAaEx9fT13ldXCwkJgbXg+ffqEcnYWLFhAWqC2DQrb7vnz59ESU2vaFVOX+Ph4JSUlAFBQUMD7A+Pl5YXu8OzsbIwyCIigoCDUDQ8ABFkSt7S0FDXtVlVVJXP9bYPCtnv16lUAkJSUFO3et6tXr0a31uTJk/EqiY+PV1NTQwNeUhIFL0VFRd26dUMXxpAhQwScOZaRkSEjIyMuLn7hwgXBL/CKABS23Zs3bwIAk8kMDQ3FrYVf1NbWokZnADBnzhzccjjXr19nMBhMJrPNDYQIPCE8PJwbN8bXWN0Wqa+vt7e3RwNekivcBihsu/fv3wcAGo0WHByMWwu/ePTokbi4OADo6+sLQ1/6uro6VPfazs6ONLvERVJSEjdNXE5ODkvn7BcvXkhJSTEYjDNnzgj+7FSHwrYbHByMrrz79+/j1sIvuOG6kyZNwq3lf9i9ezcA0On0x48f49bSQXFzc0NXBZPJdHNzwyVj0aJFQMqftwkK2+7z589ROa6ZM2eK5ARTdnY2SgoCgH///Re3nP8hJyfHxMQEAFasWIFbSweFGzfGYrE+f/6MS8bp06cBwNzc/MePH7g0UBQK2+7bt2/RA7ilpaVIBrK8efMG/a4sXbq0pqYGt5z/z/Lly9G8R35+Pm4tHY7MzEwdHR1kuyYmJt++fcOl5MWLFywWCwA2bdpUWVmJSwYVobDtRkVFoS5P+/btw62FL/j6+iLb/eeff3Br+T9ERkZKSUnRaLRr167h1tLh4ObE6+np4Q2dzMzMNDc3RylLPK/wK9pQ2HZjY2PRj629vX15eTluOTzmw4cPKFyXwWDwsDcwrzh27BgATJgwoaCgALeWDkReXh4qNw4A8+fPxy2Hc/fuXSkpKTqdHhYWhlsLlaCw7SYkJCBjMjQ05GHnKCFh06ZN3BInX79+xS2nKdXV1ba2tgBw/Phx3Fo6ED4+PvC//PXXX7jlcOrr61FXCycnpw7SbYAnUNh2k5OTUfS+oaEhxhkuPrFixQp0d02fPl1geZ+/xbZt2wBg9OjRvOpQS/g1xcXFffv2RVfFtGnThORjHz58OJI0atQokkTTSihsu+np6ZqamgCwZs0aEYtkqKmpQXWeAODo0aO45bTM/fv3paSkFBUVRe9RQzgJDAxEl4SSkpJg2ou0htOnT6Mi60OHDhWx25B/UNh2MzMzUY9FZ2dnoVrobyc1NTUoVACEu1NcQUGBpqYmk8n09PQUyUgSYcPf3x9dFba2tri1/B9QAO+gQYPIaLeVUNh2s7OzDQwMAEBNTS0pKQm3HJ7x5csXNGeNQsdwy/kpNTU106ZNQ0+XxHb5TVpampGREboqDh06hFvO/yEsLIxGo+nr62dlZeHWQg0obLs5OTkom0BNTS0+Ph63HJ5x9uxZ1AW2U6dOwhY61oSIiAgFBYXevXvn5OTg1iLi3L59G3mutLT0ixcvcMv5P8TGxqIr9ty5c7i1UAMK2+6PHz9QZrqSklJMTAxuObwhISGBm5km/Nl3tbW1vXr1AoB3797h1iLioAd5ADA1NRW2KbXy8nK0sDZq1Chh0yacUNh2i4qK+vTpAwDy8vIRERG45fAGFA+LGDlypJDHI9fU1KACaU+ePMGtRZTJzc01NTUFAAkJiQMHDghhC+fHjx+jAF4hfz4TEihsu6Wlpajtwrhx4yoqKnDL4Q0oKgsAjI2NhX/mhM1mo+hde3t7kh7KP8LDw1FFcz09PeH8nHNzcxUVFQFg165duLVQAArbbkVFxaBBgwBg7ty5uLXwhuzsbG1tbQAwMDDAUs2vDdy7d09SUpJGo7158wa3FtGkvr6eG9liaWkpnA9AtbW1s2fPBgAXFxfcWigAhW23pqYG9ZKaMmUKbi28gVvQz9HREbeW1sJms1Ge0po1a3BrEU3y8/NRfDoA7Ny5E7ecnxIXFycvL9+9e3dS+Pw/obDt1tfXjxo1Ck0yCOFsVxsYMmQIuruEp7pua1iwYAEA9OvXD7cQ0eT+/fuo0h6DwRDa3BkOh1NWVqanpwcABw8exK1F2KGw7XI4HNRZRFFRUdhCatpAcXExigqwtLQU2hSJFnnx4gWDwZCTkxNYz/COQ0FBARpbAACLxRKGDiM/o6Kiws7ODgD++OMP3FqEHWrb7pQpU9AVeeXKFdxa2svLly/Re/Hz88Ot5feIiYlBwzE3NzfReOwQHo4ePcqNbLGzsxPypWNfX1+UWpmWloZbi1BDbdtFPW/odPrVq1dxa2kX5eXl6FEdAO7du4dbzu+RkpKioqICAObm5sK54ENdvLy80FUhKSl569Yt3HL+g6CgIGlpaTqdHh0djVuLUENt23VycgIABQWFp0+f4tbSLiIjI2k0GgDo6+tnZ2fjlvPbrF+/HgC0tbVFKUtbGFiyZAmyXTk5OeEPKKyrq7O0tASACxcu4NYi1FDbdtFFKSUldffuXdxa2oWbmxuy3cmTJ1Oxnkhpaem4ceMAwMPDA7cW0SE6OppbnUNRUVH4bZfNZs+ZMwcAHBwccGsRaqhtu6tXrwaAHj16ULqJXkxMDFoClpSUfPjwIW45bWT79u0AsHz5cjK9yysaFzX39PSkxO9xVFQUi8UifS1/DbVtF7VgoHqZc/TjAQDr1q2jrmdt2bIFTfjgbfAlSqBAHQCwtramUPOksWPHAsD+/ftxCxFeqG27KJVWXFzcx8cHt5Y2UlRU1K9fP/QuhKcrexsICwvT1dUF4Wg2IwKkpKSgDsESEhLPnj3DLec3mDRpEvqpwC1EeKG27e7evRsNBzZt2oRbSxs5e/YsmtXV0dGhevnEs2fPiomJTZ06lZTfbSdfv37l9u8ZO3askBeia4K/vz+NRuvatSu1ws8FCbVt98CBA+jS3LBhA24tbcTFxQW9hblz59bW1uKW0y6KioqMjIwYDEZiYiJuLdTmwYMH6KqQl5en3HJxTEwMnU6nYgS6wKC27R45cgRdncLcheEXVFZWjhkzBr0FEWjBW1ZWhuoTCnPpAOGHzWYvXLgQZQOfPXuWWkNdDoeTmZlpaGgoGpc0n6C27R4/fhxdnRSdv09LS1NVVQUAAwOD79+/45bTXthsNpptV1FRCQ4Oxi2HqqSkpLBYLJR+IpxlHv+TP//8EwAGDBgg5Gl1uKC27Z4+fRpFklM0K+b8+fNozSQgIAC3Ft6Qk5Ojr69PIjfbw+XLl1FMCHV/unx8fOh0uoqKCtWXK/gEtW33woULACAtLf369WvcWn6b4uLiyZMno3QPUZoM9fT0RAV9CgsLcWuhJCj30sTEpK6uDreWNpKXl6empkan04W5ZBpGqG27165dQ6PF0NBQ3Fp+G25TQmVlZVFa833//r2cnBwABAYG4tZCPZ49e6ahoQEAxsbGZWVluOW0kfr6+jVr1lA6xIivUNt2b926heZ2Kfc4lpWVhYa61F0P/AVnzpwBgOnTp1MisUqoWLlyJTcmkrq5M5z/rZ22atUqSr8LPkFt271//z66Ru/fv49by+/x7NmzTp06oaF6VFQUbjk85vPnzxoaGkwmUwTqIAuSwsJCMzMzdEnfvn0bt5x2cfjwYVQdiUzvNofatvvkyRN0jT548AC3lt/DwcEBKXdzc8OthS/MnDkTZayRwU7ryc7OVldXB4CRI0dSPQbg0aNHEhISNBrt7Nmz5KGnCdS23ZcvX6LA7N27d1MoM+rjx4+o9g2TyaTiYmBr2LNnDwAYGRmRCryt59y5c2JiYnJycrGxsbi18IC5c+cCAIvF+vDhA24twgW1bTcsLExCQgIANDQ0KPTVTps2DQ11N2/eTN3V6l+TkpIiKysrJiZ2+fJl3FqoQXl5uZWVFfIp0VhivX79OgDQ6fRXr17h1iJcUNt2o6OjZWVlAUBLS4sqtnvq1Cn0U0Gj0ahb5vE/qampQaVX+/TpU1JSglsOBbhz5w6DwQAACwsLSlfU43Lz5k0ajUan08kUfxOobbvv37+Xl5dHM/cfP37ELee/qa+vnzhxIhrqLl68uKqqCrciPhIdHS0jI6OpqUmhooUY+euvv9CFsXDhQtxaeENJSQkq6PP48WPcWoQLattuYmIi6uLVvXt3SpRVTk1NRWsm1M2saz3Z2dlKSkoyMjLkGfPXNDQ0nDlzRlpaGtnurl27cCviGajniKWlZWpqKm4tQgS1bTclJQXFljMYjJMnT+KW89/8888/DAaDwWAsW7ZM5Jf4i4uLbWxsUKIw5eq5CJKamhpumUd9ff3Pnz/jVsQzUMSOvLz8y5cvcWsRIqhtuxkZGV26dEGJXm/fvsUt5z8oKysbMWIEAHTp0kU0Ju/+E5QoLCYmdufOHdxahBc2mz1s2DBkuxMmTMAth5egyHpJSckZM2aI6upxG6C27X758gV1NPjjjz+Ef/CYmpqKZqJNTU07yHRneno6+l1ctmwZbi3CS1xcnKKiIrLdFStW4JbDS+rr67du3QoAvXr1qqmpwS1HWKC27X779s3IyAio0Eimurra2dkZNZJYuXIlbjmCY9++fQCwaNEi3EKElNLSUkdHR251jvfv3+NWxGM+fvxIp9O7d+9eWlqKW4uwQG3bzc3N7datGwB069bt06dPuOX8iszMTGVlZbSYRrl+Ae3h3LlzKBb10qVLZIa3Oa9fv4b/RSTrgicmJkpLS9PpdEqsvggGattuWVkZWioFgK1bt+KW8yt8fX1RVOaaNWtwaxEor169kpGRAQAZGRnRSL7iLffu3UMXhp6enkiWyvz06ZOSkhIAmJmZkZRFBLVtt7q6mlvcYM6cObjl/JSSkpLevXt32ETJJUuWoO+IivU5+Up9fT3qyi4lJXXhwgXccvhCfX39qlWrAEBFRSUzMxO3HKGA2rZbWlrKjbyZOXMmbjk/BaXrAMC8efNwa8HAkydP0ICORBE14dChQ6ioiLGxsQgv9Kenp6PF5EOHDuHWIhRQ23YrKyuHDBmCbHfatGm45fyUTZs2AUCPHj1EI9f+d6moqBg4cKAIl1trM0uXLkVXb5cuXYqLi3HL4RelpaVo6XvFihVkfp9Dddutra1FkbB0Ov3SpUu45bRMVlaWpqamkE+D8BtU/adz585CvvIpSCoqKvr16wcACgoKd+7cEeHRLofD2bVrFwCoqamRC4BDddtls9kjR45E44VFixYJZ+guKoEIANOnT8etBRvBwcHoQ1i5cqVwfk2Cx9vbG30mzs7OuLXwHdTkW0ZGhmQJc6huuxwOZ/ny5ejatba2Fs7nl8GDByOF165dw60FG2w229XVFaWK4NYiFCQnJ6NyIgDg4uKCWw7fOX/+PJPJZDAYPj4+uLXgh/K2++bNG7RcM3DgwNraWtxymnL9+nXUz9HOzi4tLQ23HJwkJSXR6XR5eXkSz8DhcLy8vLjhuu7u7rjl8J3CwsIePXp08Kk2LpS33R07dqAggT59+lRWVuKW83+Ii4tDngsAAQEBuOVg5sePH3369EG5WM+ePcMtByf5+fljxozhLqZ9/foVtyK+U1dXh+YDJ0yYINr1TpVmXq0AACAASURBVFsD5W3X09MTjXbNzc2FrZx2eHg4Cg9SUFAIDw/HLQc/V65c6Tjju19w9uxZ9DkoKioGBQXhliMg/Pz8AEBCQkL4q1bxG8rbLorEBgATExOhKrlbWVnp7e3NZDKlpKRu3LiBW45QUFVVNXr0aADYtm0bbi04WbBgARr1d6g08dTUVFS4yt/fH7cWzFDedtesWYNs18jIKDc3F7ec/w83gGHy5Mm4tQgR27dvBwAtLa0Ou6L99OlTBQUFAHB0dMStRdBMmDABAMaPH9/Bo1kob7sbNmxA7mZgYCBURWwXLVqEhO3duxe3FiHC3d0dABgMRocd8mzYsAGtRnS06hwcDmfSpEkAYGNjg1sIZihvu3///Tdyt65du3758gW3nP8hPz8frdtqa2uT+PDGZGVl6evrA0C/fv3y8/NxyxE01dXVZmZm6HKNiIjALUfQoAoqffr0EeGUvNZAedtF2S/ouTU9PR23HA6Hw2Gz2Tt27ECq1q5di1uO0HHkyBE6nS4jIyMk35cg2bdvH5PJpNPpDx48wK0FA+gRkEaj3bp1C7cWnFDednfv3o0MTkNDIykpCbccDofDiY2NRW3kaTTa9evXccsROvLz8zt37kyj0fbs2VNdXY1bjuDIysrS0tJCl+ujR49wy8HA8+fP0dv38PDoyNO7lLfdS5cuiYuLA4CqqmpCQgJuOZy8vLzFixcDAJPJ3LVrl3AmzuGFzWaj+U0mk9mhymCiRhsA0LNnz4459VRUVGRpaYnCPTty7V3K225ERARKslRUVIyJicEt53+aNiI9QrXEJ1RUV1ejnOmHDx/i1iI4Ro0ahQJXO3JDT39/fzqdrqenR2yXwiQlJXXt2hUAWCxWZGQkXjH5+fmoBayent6///5Levb9gqtXrwKAra1tB3kguHjxorS0NBrqCls6pSDJysqSkZGRkJDomLPbCMrbLmrVBQCysrJv3rzBK+bgwYNIzNKlS/EqEX5evHiBZocWL17cEZobotKXANC3b18hLB4iMKqqqsaPHw8AgwcPrqiowC0HD5S3XW6+qaSk5PPnzzEqiYqK4rbd7uBZWK2hvr5+ypQp6OMKDg7GLYe/3Lx5E6VIqKqqkkpAPj4+ACAlJRUdHY1bCx4ob7s3b95Ety6NRgsMDMQlo66ubsaMGUiJvLx8x+wi8buEh4ejUkEi2TGXS35+vrq6OvJcjJeo8PDjx4+ePXsCwIYNG3BrwQPlbTcgIIAbt4trWbyqqmrlypVIhoyMDKnA0HpQtminTp1ev36NWwu/CAgIEBMTA4AtW7bg1iIs7N+/HwBWr16NWwgeKG+7wcHBKNXS3NwcV+SAv78//C+zZ8/GooGi3Lt3DzU33LFjh0gGcubl5RkaGqJJsPv37+OWIyzExcWpqKjo6+t3zHkGytvu8+fPUeHHHj165OTkYNEwceJE5LmGhoYdoXYqb9m2bRsAdOvWTSQDP1AzGxRQmJWVhVuOsFBfX4/C6UaPHt0BI8kob7tv375FC+LW1tZYBCQnJ6NydgDg4OCARQOlSUpKkpaWFhcXF70ivAkJCWioy2AwRo0aVVRUhFuREIGKDjMYjA5YfpfythsVFSUjIwMAOjo6gs9Sy8nJQc1fAcDMzEwY0uQoR1VV1eTJk1Hk9fv373HL4SXLli1D18aCBQs6ePGX5nz+/BlF3L969Qq3FkFDeduNjY3lNs45evSogM/Obf4KAKT8QpuJiYnp3r07ACxcuBC3Fp6RkJCAAhikpKSwR5QLJ4cOHQKAFStW4BYiaChvu/Hx8dxo2f379wvy1CUlJag5GIPBWL16tUhOTQoM1FdYVlZWNEIa2Gw26qOBbDc+Ph63ImGkoKDAxMREWlr6n3/+wa1FoFDedquqqubMmYNS3QXcEf306dMoiKJfv34dIc+Kr8TGxqKQhilTpojAD9iZM2dQ0BgAbNq0qUMVWvstli9fDgADBgzoIDniCMrbLofD2blzJwB07dpVkCvF6enpqqqq6L5yc3MT2HlFGE9PT3FxcRkZGSEp4Nlm8vLyUCl3BoOxcuXKDmUov8uLFy9QwnRdXR1uLYJDFGzXxcUFpUsIrJheRUXF7Nmzkeeqqal1wDUBflBdXY3W/V1cXHBraRdHjx5F18bAgQM7cvmF1lBWVjZw4EAxMbEOVYCY8rZbVVU1c+ZMZH8CGyUlJSV16tQJ3Vq7d+8WzElFHjabjZ7NFRQUAgICcMtpO7a2tujaWLduHW4tFOD58+fq6uq9e/fuOCn1lLddDofj4eGBHugE5oCoLQ0ADB06VKi6xIsAU6dOBYCpU6fiFtJGEhISUAHoqVOnkhn/VoJy648dO4ZbiICgvO2WlZWhdBcAWLx4sQDOGBERoaOjAwBMJvP27dsCOGOHIiwsTEpKqlOnThcvXmSz2bjl/B7Pnz+3srJCVyOlB+wCZt26dQCwdOlSEVhNbQ2Ut93a2totW7agC93JyYnfp6uoqLCwsECn09HRyc7O5vcZOyB//vknKmJAueyJ6dOno2tjxIgReXl5uOVQhrCwMFlZWTqdfu7cOdxaBAHlbZfD4Vy7dg1d63PmzOHridhstqurKwoa69Gjx7t378giNT+oqKjo1asXAFArXTguLg6liWtoaJDf498F/dZSd3Lpt6C87dbW1qJCt2JiYnfv3uX3ufT09JDFnzp1iq/n6uBs2rQJPU98/PgRt5bWgvpHMJlMLy8v3FqoR0pKirKycgdpZU95262urh40aBBaUuOrFebm5rq6ukpKSqIAeMpNO1KLzMxM1Nt8zJgxuLX8N0VFRc7Ozig/YuXKlbjlUJLKyko0plFSUgoKCsIth79Q3nbZbPbYsWPRCHTkyJH8q9l6/fp1dBZdXV1cFSY7FKgxnYqKSmpqKm4t/0F8fDyqPiorK9uhWtDzkIaGhvPnz6O4zPnz5+OWw18ob7uc/y3YCgDDhg3j32TrkSNH0FmmTJnCp1MQGpOcnIzGjwsWLCgpKcEt56fk5OSgQF0mk7lx40bccqgN+q1VVlam0ORSGxAF27169SqKorW2tuZTiqGLiwv6HVZQUAgJCeHHKQhNqKur8/DwYDKZAHD58mXccn7K9u3b0e8xi8UiK2ntpLS0FPV5GjFihAiXJxYF242KikJTrn379q2qquL58SsrK1HHPQBYvnw5z49P+AUo0Xb8+PFC2Ny7oaHh4MGDKDlCU1PT19eXH5dfR4NbylWEG5uKiO2ioWivXr3Kysp4fvwtW7agoLFx48aJ8C+wcMJdMp04caJQTanX1dW5u7tLSEig3+M///wTtyLRYf369QBgZGSUnJyMWwtfEAXbjYuLU1BQAABTU9PCwkIeHrmuru7Ro0caGhooer8Ddh8RBu7cuYMWrIQqjDcjI0NbWxt5rqys7L1793ArEh1qampQYw57e3vcWviCKNhuQkICqsHYrVs33qYGHTlyBM0tiouLnz59modHJrSeuro6VFK5V69eQjLJ++7du8GDB6MfAwDw9fXFrUjUePPmDQDQ6fR58+ZlZmbilsNjRMF2U1JSOnfuDAD6+vo8XNNISkpCdQhR9KhIthOnClVVVQ4ODig6293dHW/mfm1t7YgRI1BdDhQ/I8yBFhSlurp648aN6BNesmSJiK1VioLtZmRkoHtAR0eHVz+Mnz9/trS0RN+6uro6acqCnYiICNSrVFZW9vPnz7hk1NfX79u3D0XOSEhI2NnZhYWF4RIj2uTm5i5duhR91IMHDxalMa8o2O6XL19QMX8tLa20tLT2H/Dbt2+DBw+G/2XQoEGkKYswwB3+bNy4UZCdRBrz+PFjZAQAsHXr1qqqKlKXg68cPXoUTeZoaWlFRETglsMbRMF2s7OzUd9ZdXX19lc6r6qqGj9+PNdznZ2dhT9LqoPAZrO3b9+O5vG1tLS2bt0q4N4Njx8/HjhwILowDAwMcFl/R8PPz09cXBwARo8ezY9QJcEjCrabm5tramqKklvan5p55coVFC6GxrnFxcU8EUngFXfu3EGxJQAwb968oKAgAVSaLy4unjVrFrr5UWxTZGQkv09K4HLt2jUUzGtlZXX+/HmqT6aLgu3++PED1cBVUFB49+5dew718OFD1L8WJYZT/dsVVdLT0xvPAvXo0YPf9eb9/Py4p3Nycvr27RtfT0dozp49e7j1/0xNTZ8+fYpbUdsRBdstLi7u168fAMjJybVn9iclJYXbDFhDQ+P79+88FEngLQUFBZ6enmZmZuj76tatW25uLp/OFRQUpKSkhE7k6OhIJvqxUF9fX1BQsHTpUllZWRS25OjoeP369YcPH1KuT6go2G5ZWZm1tTUAyMjItLmJb15e3qxZs9Ct1blzZ39/f96KJPCD4uJiVG0ZLXbzw3lfvnyprKyMTmFvby8ac4uU5tmzZ2pqatyHD2Vl5RkzZixevPjcuXPl5eW41bUKobDd4uLi2NjYd+/etS2lvbKy0sbGBgX0hIaGtuEIQUFB8+fPR9+ihYVFRkZGGw5CwEJ1dfWiRYu4oQU8LN2Qk5MTEBCgqamJDj5ixAje5kAS2kx4eLidnd3EiRONjIygEVQp4yAUthsQEMBgMJhMprW19YYNGwIDA+/evZufn9/K3Wtra+3s7ABATEwsODj4d89eV1c3ZMgQKSkpNMMgJHlQhNZTV1e3atUqAKDRaNbW1u0PIiwsLPTy8uImyyD8/Px4opbAQ9LS0qZNm8ZgMGRlZZWUlO7cuYNbUavAb7sNDQ0rVqyQkZFBUzZc7t+/38ojsNnscePGobuu9Xshnj9/7uDgICUlJS0tbWFhcebMmd9/BwT8NDQ0bN26FUXUTpgwISYmBv3xtw7y/fv3zMzMy5cvczNlEJaWlvfu3SPLaMJJcXGxn5/fhw8f2h88KjAEarsZGRl37txpEl7e0NAwcOBASUlJXV1dKyurESNGsFgsAOjevfvr169beWSUOQoAv7Wiffv2bVS6DADMzMww5j4ReAJ3pkhXV3fJkiVLlixJSUlJTExsPuVXX19fU1NTWVlZXl7+9OnTQ4cObdmyxcDAQF1dnVtpAQBUVVXHjRsnqnWwCLgQqO0+fPhQTEys+YDU19fXxMTk+PHjaHjy+PFjdXV1ALh+/Xorj8xdV/nnn39aucvBgwe542sNDY2HDx+2/o0QhJPCwkIPDw9UABehqKgoIyMzZMgQJycnR0dHBweHsWPHjhgxYvDgwQMGDOjTp4+FhQX6mW+CpKSkgYGByKRFEYQKgdruy5cvAcDW1rZ5PmXjxbSnT58i27169Worj8wd5ly4cOHXW7LZ7FOnTp09e5Ybcj9+/HgKPZ4Q/pO3b98aGxujhkC/i56e3t69e52dnf/55x+SnUjgEwK13aioKBqN9oseECkpKVOnTlVUVAQAY2Pj1idfOjs7o9vm182DQ0JC/Pz8UCsKxPDhw3lbK5IgDOTk5Dx8+HDSpEldunRp0V51dXUlJCQMDQ07d+7MYrGUlJS0tbX/+uuvT58+4dZOEH0EartXr16l0WiHDh362QaRkZHcOiOurq6tP/KKFSvQXt7e3s1fRXUCIyMjudH1TCZz6NChTk5OX79+bcMbIVCCurq6L1++XLp0adGiRT169HBycpo+ffo///xz+PDhd+/eXblyJSYmJj09PT4+Pjk5mczsEwSGQG03ODhYXFz85MmTP9sgMTERDXUBQEdHZ8+ePa3MP0FdQACguadXV1fPnz9/2rRp3CMDgLW1NYl771BUVlbilkAg/A8Ctd2UlBQWi7V3796fbXD48OHGT4I6OjqtrHLy999/o1327dvX+O8lJSUbN27kjqABYMmSJbdv3+ZJfUgCgUBoAwK13cTERHl5+V+kz+/evdvc3Pzhw4fz5s0DACUlpVZmne3atQsAVFVVufXIKysrfXx8xo4dyzXckSNH/vvvv3gbExAIBIJAbffIkSMAMGrUqJ9V9qqqqkLdYZOTk1Eww5YtW1pzZDc3NwBQUFB48uRJcXHx1atXp0+fzjVce3v7ZcuWPXr0iJdvhkAgENqEQG33w4cPysrKGzdubJ4+VFBQsHnz5o8fPxYVFQUGBl68eBG1R/vrr79ac2R3d3fksIqKiqjTBAAoKyuvWrXKw8ODxCoQCAThQaC2+/btWwUFBRMTk+YN6eLj41ksloGBATfYAABYLFYrE9W8vb3RLmJiYnZ2dtra2k5OTnFxcXx4EwQCgdAuBGq7+/btA4C+ffs2r+TE9c3GzJ07l81mt+bIiYmJXbt2BYAZM2bU19d//fqVNPolEAjCiUBtNyAggMlk7tixo66urslLXl5eXLfV0NCYMGGCi4tL6wuNJyQkoMD4xYsX81o1gUCgGCUlJRkZGUI79hKo7VZWVu7fv//Lly/NX/rw4cOIESOcnZ0PHz4cEhLyu0dOT09HZfpmzZrFC6UEAoGqVFZWDhs2zNDQUGiDtfEXfuTSyvmEFvH390cj5alTp/JQEoFAoBzfv38HAHl5eaHthShEttse7t27x40Vw62FQCDgJC8vT0ZGRkFBofWtEgSMiNjuo0ePUCraqFGjhHZChyA8oJI3d+/edXFxac9jFkEI+fHjh4KCgpycnNBWphcR23327BmqTm1jY9O8qiSB0Jj6+vpp06Y5OTlpaGgoKSmhDB2CyFBUVKSuri4tLS209eRExHbfvHmD6qsOHDiQct2bCQImNDSUW/yTwWBQpe8hoZWUlZV17dpVQkJCaNuCiIjtRkVFSUtLA4ClpaXQLl8ShIS//vqrcXj49u3bcSsi8JLy8nJdXV0Gg9Fi+4Jv375FRUWdOXMG41OOiNhubGysnJwcAJibmwvt8iVBSPj69auHh4e5uTmy3VbW/SBwKSgo+Fk1Ky4NDQ11dXWVlZUoSL+ysjInJ+fbt29fvnzJyMhITU2Nj49vUu26oaGhtra2qqqqrKyspKSEzWbn5eWhbWpra2/cuNH4QTYsLKyoqIjD4bDZ7KSkpBMnThw/fvzw4cPu7u5z586l0WgSEhJ37txJSkqKj48/f/781atXX758+fDhw6FDh8rLywOAs7Mz7z+a1iEitpuQkIA+ShMTk4KCAtxyCBRg9erVyHY3bNjQmu3ZbHZtbW1lZWVpaWkrT8FtlFlaWlpaWlpfX5+Tk5OVlZWZmZmenp6cnJyYmPj+/fs3b95wK5GiXSoqKkpLSwsLC/Pz879///7169fPnz+npaUlJSUlJCTU19cnJyc/e/YsMDDww4cPWVlZ586de/r0aUVFRWxs7MGDB3fs2HHx4kXuLu/fv3/37l14ePjz588fPHhw7969f//998qVK+fOnfP19fXy8jp06NCePXt27Njx119/rV27dtmyZYsXL547d+706dNHjx69fPny+/fvo10uXrz4119/GRkZ6enprVixYufOnZs3b16zZs2yZcsWLVo0Z86cP//8c/78+XZ2dsOGDRs4cKC5ufnEiRO9vb1tbGy6dOmio6OjpqamqqqqqKgoJSWlp6c3Z86cqVOnjh8/3s7ObujQoQMGDOjbt6+pqamJicnMmTMNDAw0NTXnz58/Z84cABgxYsSCBQscHR3t7e1lZGR69OhhZWXVt29fZWXl5jmuNBpNXFxcXFxcSkqq+asAsHPnzjZeOu1GRGy3pqZm0aJFAGBkZNTKEr3CTEVFRUZGBrpdi4qK8vPzc3JyMjMzf1G1ssntmpOTw2aza2pqUlJSfjbrUl9fX11dXVFRUVJSUlhYmJubW1BQgE6dlJTUYhn4uro61JCpoqKioKAgLy8PjV8+ffqExi/FxcUcDqeoqCgvL6+2tjYtLe3ixYsfPnyIiIh49uxZUVFRbm5u4yEPGozExMS8ffv22bNnQUFB9+7du379+qVLl86cOXPs2DEfH5+srKynT59u2rRp69atMTExaJdXr145OzuHhYWFh4ffvn3b398f7eLl5RUYGHjmzBkXF5dt27atWbPm9OnT27dvX7RoUUhISHh4eGhoaHBw8OXLl1GJOwBYtWpVXFzc3r17d+3atXXr1o0bN65atWrp0qVOTk5TpkxxcXF5+/btgQMHrK2traysLCwsBg0adOvWLU9PTzc3N7TLhg0bVq1atWTJEicnp8mTJ+/atevEiRPDhg0bMmTIgAEDLC0tzczM+vTpM336dF1dXW1tbQ0NDWVlZQUFBVlZWQkJCRqNpq2tPX78+JEjR3J7a/bs2dPExMTQ0FBXV1dLS0tdXV1ZWblTp04KCgqXLl0aMGAAEq+mpta9e3cAkJGR6d27N/dN0el0dBYWi6Wtra2qqspgMJhMJrdQFM9BnUPpdDqNRmvykqSkpIyMDPp3a3rc0en0xl6JGhTQaDQLCwsAYDKZaGq+f//+pqamvXr1Wr58+cyZM9HG48ePHzVqFPq3kZERyqKSk5MzNzfv27evo6Ojk5PTokWLhgwZkpGRwaP79bdpu+2y2ezo6Ojq6ur/3BI9blRXV5eXl6Pngt/apaSkBD3UfPv27dOnT+iMqampN2/e5HA4kZGRqBXxnTt3mEymjo5Oenp6SUnJjx8/cnNzs7Ozv3z5kpqaWlZWVlxcHBcXFx0d/fbt29DQ0KCgoDt37ly/fv3s2bPh4eGlpaXcXTIzMzMyMlJSUj5+/Pjhw4fo6Ojnz5+HhIQEBgbevn37+vXrZ86ciYuLKy0tLSgoSExM3L1797p16wIDAyMiInbs2BEZGfnu3btXr16FhIQ8fPjw1q1bly5dOn/+/OnTp48dO+bh4eHm5rZz585Nmza5uromJyefPHly9uzZjo6OY8eOPXjw4NOnTw8fPtytW7c+ffqYm5sbGxvr6+vr6uqqqalNnDhx06ZNK1as+OOPP+bPn+/o6Dhp0qQxY8Y0boWLbtcuXbrMmTNn8uTJLBZr1KhRq1atcnZ2njdv3owZMyZOnIh2GTRoUP/+/Xv37t2zZ8/u3bt37drVwsLCzs5OSUmpU6dOQ4cOXbhwoYeHx9KlS8ePHz969Ojhw4cPGjSoT58+8+fPHzJkiJGRkYGBQdeuXTU1NdXU1JSUlCQkJIYPH75nzx5zc3MzM7PBgwdra2sDgLq6uqqqqoyMTLdu3QwMDHR0dNAuioqKcnJyUlJSYmJijUvRN26ZDgBaWlrcHs9ycnIsFgvtAgCdO3dGc/qNYTKZjf/LdQE6nd74LFzk5eVbbB78C9rWH/NnaGpqcl2p/TTXxmKx0KdEo9GWLFmio6MjJSUlJyfH7beiq6sLAEwmc9GiRQYGBuiPysrKysrK1tbWdDqdyWSiXZSUlNTV1QcNGrRhw4bu3bubmppaWFj07dvXyspqyJAh58+fX7Vq1apVqwICApycnIKCgk6dOnXu3LnY2Njk5OQ1a9YMHz781atXM2bM6Nu379GjR1esWGFjY2Nra7thw4Zt27Zt27btxYsX69ate/HiRXx8vIeHR3BwsJ+fH+rMNG/ePA6Hk5CQkJycnJaWFhYWVltbW15eXl5ejkzj5MmT586da2hoQE0PunbtmpaWVl5eHhISkpSUVFlZ2Xj8UVFR8fuexzPaYrslJSV79uyZMmWKvLz8/Pnzjxw5snfvXu4TyvLly7lPKBMmTBg9erSNjQ16FkC394IFCzw8PPbs2bN9+/bNmzejhxr0hDJ16lR7e/tRo0YNGzbM2tq6X79+FhYWZmZmhoaG+vr6mpqaKioqY8eOXb9+fe/evSUkJFatWmVubo5GCnp6egAgLi5uYmJibGysp6eHhhWdO3eWlZWdP3/+vHnz0AXU/N5TVFQ0MzMzMjLS1dXt0qWLhoaGvLy8goKCpKQkk8kUExPT0NBocjOrqamhXXr16oXMRVJSEs0vi4mJNbn0URHLFpGXl28+OqAuaBjCHapwB1+tQUVFxd7e3sDAoEuXLjIyMugz7NSpE3qV+48WQV8TALBYLK6HNrHvxjT+zLW0tPT09JSVlfX09Ozs7PT09JD16OnpofpKCDqdLiYmpqysvHfv3vHjxysqKqqqqnbu3LlLly56enqGhobGxsampqZ9+vSxsrIaNmzY6NGjZ86ceeDAgQULFixevHj58uWbN2+eM2dOjx49zM3NLS0tBwwYMHv2bNTqbdy4cVOmTHF0dNy1a9fu3bvXrl27devWvXv3enh4HD9+/MyZM1euXPn3338DAgIeP3788OFDDw+P169fb968ecaMGYsWLVqxYsX69eu3bdt26NCh4OBgX19f7i73798PDAwMCQl5+fLlmzdvnj9/Hh0dnZCQkJKSkpGR4evre/r06YKCgtu3b9+7d4/D4eTl5d2/f//p06efPn1KT09Ho6vY2Fi0y6dPn75+/Yqcrry8vLq6+rfiNQUTZTR8+HAAMDU1FdrQwLbYbkJCgri4eOtvp/aD7p/mczTI5pojKSkpKysrLy+PJnzl5eUNDAyMjY0b9wz+NWggwGKx0N3ev39/NB6h0WhNXJXBYDT+NLgilZSUGAwGGshoamrKycnJysqiJrVqamqamppNvFhZWXnDhg2mpqby8vK9e/eeNWvW/PnzFy1a5OrqunTpUh0dnSFDhjS+w83MzPr27Tto0KCRI0eOGDGCxWI5ODj0799fQUFh5syZXbp0GThwoLGxcZcuXfT19Y2MjExMTHr27NmnT5+VK1c6OjqOGTNm8uTJM2fOnD9//uLFi318fA4dOjRq1CgtLa1Lly6tWrXK0NDQxMTEzMysX79+VlZW69atc3JymjRp0qxZsxYsWLBkyZKVK1du2LBh69atu3bt2r9/v4eHx7Vr17y8vCwsLO7fv79w4UIHB4ewsLClS5c6OjrOmTPn1KlTd+/eXbt27bZt21xdXffv33/kyJHjx4/7+fldunTp2rVrqFBncHDw+PHjFyxYEBgYuHPnzrdv3965c+fChQsvX750c3M7cuSIj48P2uXGjRv37t179OhRaGgoMpQzZ87EsE0aNQAADgVJREFUx8d//PgRPf7fvHnz4MGDnp6eJ06cOHPmzOXLl7nOFRISYmxsjL7QI0eO5ObmpqSkZGVlVVZW5ufn//jxw9/fHz0q+fv7BwQEBAcHP3v27PXr15GRkWjlJykpKT09PTMzMzs7Oy8vr7CwsKysrKqqqq6u7tcJO9xVJgKfWLZsGQAMGjQIt5Cf0hbbzcrK2rFjx+TJk/v37z948GBNTc0FCxaYmppu3759xowZZmZmXl5ePj4+Dg4Ox44dW7Bgwd27dx8+fHju3LlXr15t2LDBzc1t4MCBFhYWAwcOHDhw4LBhw+bOnbt06VJra2tvb28HB4cRI0bY29tPmTJlz54969evX7t2bWhoqLu7+9u3b728vPbt2+fn5xcaGurs7NyvX7+jR49OmjRp586d69atAwAmk7l79+7k5OSkpKSMjIyPHz8eP348Njb25MmTp06devHihb+//9atWwcPHmxra+vo6Hjz5s21a9fOmzfvyJEju3fv9vT03LNnz9GjRxMTE729vSMjI6Oios6fPx8ZGRkSEhIUFPTs2bPw8PDHjx9v377d09Nz5cqV586d279/v6urK7rDQ0JC/P39L168GBsb+/Tp06SkJDStGRERERsbi5rUpqenZ2Vlpaene3t7e3p6+vr6+vn5xcbGcjic1NTUuLi4JquCJSUlmZmZZWVl2dnZ+fn5hYWFaKDBvcPr6+vfvXvH4XAKCgo+fPjA4XA+f/6MpmvRLkVFReXl5TU1Nb++4SsrK9HCMZvNLiwsrKioqKmpqa+vb33iX0NDA1XaJg0ePBjZLonbFTGOHTsGAKampkL789b2ud3c3NwfP36UlpaimWm0kFVfX9+a+K3v378XFhYWFxeXlpZWVlai7Mw2PIA03uX8+fNojJmSkvKfO5IE4g4Om822sbFBzxk3btzALYfASyIjI9FoV2gzp0QkkoHNZqN5dBaLFR0djVsOgQLY2dmhOaI2FBolCDOpqakAcOTIEdxCfoqI2G5paamlpSUAdOrUKSwsDLccAgUYOXIkANja2qKQOILIUFZWdv78+RZT1IQEEbHdiooKKysrAJCWln758iVuOQQK4OLiAgCrVq3CLYTQ4RAR262pqRkxYgQASEhIPH36FLccAgXIy8vr0aOHl5cXbiEEarNv374TJ0781i4iYrscDufq1asAQKfTHz16hFsLgRoUFBRwg+0JhDaQkZEBAH/++edv7SU6thsaGooWpgMCAnBrIRAIHQJPT08AWLt27W/tJVK2KykpKSkp+e+//+LWQiAQOgQnTpwAAAcHh996bBId2128eLG6urqhoeHFixdxayEQCB0ClC5Ap9N/K25VdGx37ty5aJLh/PnzuLUQCIQOwbVr15DtoixTRH19fVVVFaoF2GLNCtGx3RcvXqD6Cb6+vri1EAiEDsGDBw/QaG/WrFmhoaEeHh59+/a1sLAwNTXt1q2bvr5+cHBw871Ex3aTkpJQ/RoSEkQgdBxqa2tRHWcfH5+NGzemp6cL7NTv37/39vb+dVGtcePGNS9FIDq26+rqigqVHTp0CLcWAoHAd9hs9v3794cNG4aKhSKbe/LkCb/PW1dX5+3tPXbsWFSivkW35VYltLGxQTVnGiM6trt582b0Pvfu3YtbC4FA4C9Xr17t3bt389rZAogfDQwMbHJSKSkp1KLNxcUlPDz8+PHj79698/b2njhxop+fX/MjiI7turq6oo9gx44duLUQqEddXV1mZubatWs9PT1xayH8itTU1KCgIFNTU67rMZlMU1PTHTt2uLm5paamcrfMyMi4cuXKsGHDLl26xEMBhw8fRuflFtQfO3YsElZYWNhk4xbbWIiO7e7fvx99BJs3b8athUAlsrKyDhw4MGzYMC0tLQCYNGkSbkWEnxIVFaWmptZ4pDl06NDAwMDmHRRramq4TdW0tbVfvHjBKw2urq66uroDBgyIiYlxcHAAANSYo/WIju1yf4J+N2OE0GH59OnTH3/8oaGh0fg2trOzw62L0DIfPnzQ0dHhflP9+/efM2cOakrSnFevXjXp6mRnZ9eaYtz/CWrVWF1dXVNTY2xs3Lt3798t7Cs6tnv8+HH04S5fvhy3FgJOWtndi81mo6BLhKqq6tixY7dv337q1Cl+KyS0gbKysh49egCAg4PDixcv3rx584vEsIyMjG7dukEz+vTpw0NJvr6+0KbQKdGx3VOnTqFP1tnZGbcWAgZu3Lhx4MCB7du3T5w48ezZs25ubleuXPmFBYeHh6NZBQBQUVG5f/9+8xVngvDg7u4OAJ07d/7Pr6msrMzMzAx9s/b29uvWrbOxsUF9ZlksVkxMDK8kbdy4EQDasBggOrZ74cIF9EHPnz8ftxYCBqytrZuPbmxsbKKioppv3NDQgG5jxJ07dwQvmNBKsrOz582bh0K15OXlr1+//u3bt19sj6zAwMCg8ZRrdnb21q1bAcDIyKi6urr9qioqKvr16wcAS5cuff36dWho6PPnz9FLbDa7pqampKTkZ83cRMd2UeFHAHB0dMSthYCBCRMmNLddAPjnn3+ab/zu3bvG/Z6nT5++fv36xYsX37x5s/FSOEEY2LJlS5PvVFNTc+/evS12RHz48KG+vj4ANI9eKC8vFxMTY7FYPKn2+ePHDwkJiSbC+vXr169fv169epmamhoaGs6bN69F5xUd2719+zaaQZ8yZQpuLQQMTJs2DU3RLly48Pjx47a2tuhOuHLlSvONIyIiWvRoAOjWrVtr2rASBMPnz59RKoSCgsLy5cunTp2qq6uLvqnm1VcqKiosLCwAYPXq1c39rqCgQFpamsViFRUVtV9YaWkpV8nPWLlypYjXZCgqKho6dCgAjBs3DrcWAgZiYmIOHjyYmZmJ/hsWFobGsy0ukSUkJKCgy86dO//xxx9N7hZbW9u8vDzByie0DCrxBY0mggoKCmbPng0A2traTcIY8vPzJSQkWCxWaWlp80P5+vrS6XRFRUVefbknT57s2bMn+rFHIRZjx469ffv2pUuXTp8+7ePjU1xc3OKOomO7b968kZeXR/cMWRvp4Hz69AnNu9Hp9KtXrzbfoKam5sGDB6hSSX19vbu7++HDh319fa9cuYJ+vP/++2/ByyY0p7CwcMKECe7u7o0nZHNychYtWoTGWJWVldy/Z2ZmslgsOTm5nJycJsdJSkpCkw9//vnnz6Zc20B1dXVAQEBaWlpeXh53bvc/ER3bDQ0NRZPuQ4YMaWUIEUFUWbp0KRoiLV68+GcxlQ0NDdnZ2c3//vLlyy5duqioqMTHx/NZJqFVtOiSqK1Dr169CgoKuH88cuQIALBYrMa2Gx0dvW/fPhUVFQDo3r37p0+fBKD514iO7XIfKvv3719TU4NbDgEbFy9elJGRAYCuXbv+LJb+16xYsQIAdu3axXNtBF6B2jps2rSp8R8PHDgAALKyso8fP46OjnZ2dkadbdHU8OjRo3mbJdxmRMd2o6Oj0c3Wu3fvFvOgCR2BwMBASUlJABAXFz979iyHw6msrIyKivL09GzlMCctLQ2ln5LiHsLDjx8/mnx969evp9PpTVJ+37x5o6CggCaXuClqnTp1srGx8fb2FqjiXyI6thsXFycnJwcAPXv2/NlMNkG0qa2tHTduHLrZxMXFT506de7cud69e6urqwPAmjVrGm9cXV197dq1xrn8lZWV586d40baN9megJHr168rKyt//PiRw+GUlJQ4ODjQ6fQJEyY0n39Yvnw5AKARGAAcOHCAJwnBvEV0bDc5OVlTUxMATExMmtfFIHQE7t69Ky4urqCggNbTGjNo0KAmeRNPnz6VkJAYNmyYu7t7QkJCamrq5MmTudtra2u/fv0a1xshNOHFixeysrKWlpYXL150dHRE39GDBw+ab5mamnrq1KmoqKirV6/eunWr8YKb8CA6tltUVIRCNQ0NDb9//45bDgED165dk5GRERcX51bkQyxZsqR5cIubmxt3A3l5+Sa77Ny5E8tbIPwMbjkxNG+wadOmqqoq3KLaiOjYblpaGnqW1NXV/fr1K245BAxERESoqqo2ds/u3bv7+fm1GMyA6pi0yIwZM8jygLBx48YNCQkJLS2tYcOGBQYG4pbTLkTHdr98+YKSRrS1tTMyMnDLIWCgtLS0V69eAGBtbe3j43PhwoVfTDcVFBTs3LlTWlpaVlZWR0fHwMDA1dV1zZo1U6dOJZ4rnERGRrYY80c5RMd2s7OzDQ0NAaBz585COIlOEABlZWWPHj3y8fH58uVLK3c5duxYWFhYXl4eTxJGCYTWIDq2m5ub2717dwBQVVVNTEzELYdAIBBaRnRst7CwsH///gCgqKjYtiB5AoFAEACiY7vV1dULFiwAABaL1WKJVQKBQBAGRMd2q6qqkO3Kysq+ffsWtxwCgUBoGdGxXQ6Hc+vWLQCQkJDgYZdQAoFA4C2iY7u1tbWzZs1CWaEhISG45RAIBELLiI7tNjQ0oI5ydDqd6tHUBAJBhBEp212zZg3KMmrcuo5AIBCECtGxXQ6Hc/z4cVTp/MaNG7i1EAgEQsuIlO1GRUWhSuf+/v64tRAIBELLiJTtvnv3DtXZXLVqFWmnRiAQhBORst3Y2FgpKSlU27ihoQG3HAKBQGgBkbLdb9++WVlZAYC7uzuxXQKBIJyIlO1yOJzNmzejBhP5+fm4tRAIBEILiJrtnj9/fvTo0QEBAWRul0AgCCeiZrsNDQ1keoFAIAgzoma7BAKBIOQQ2yUQCASBQmyXQCAQBAqxXQKBQBAoxHYJBAJBoPw/+x56PaMDPLQAAAAASUVORK5CYII=" alt><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1A1 N4</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct domain, <em><strong>A1</strong></em> for approximately correct range,</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for approximately correct sinusoidal shape with \(2\) cycles. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this last <em><strong>A1</strong></em> awarded, award <em><strong>A1</strong></em> for max/min in approximately </span><span style="font-family: times new roman,times; font-size: medium;">correct positions. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">setting up inequality (accept equation) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(h > 105\) , \(105 = 74 + a\cos bt\) , sketch of graph with line \(y = 105\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">any <strong>two</strong> correct values for <em><strong>t</strong></em> (seen anywhere) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(t = 8.371 \ldots \) , \(t = 16.628 \ldots \) , \(t = 33.371 \ldots \) , \(t = 41.628 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{16.628 - 8.371}}{{25}}\) , \(\frac{{{t_1} - {t_2}}}{{25}}\) , \(\frac{{2 \times 8.257}}{{50}}\) , \(\frac{{2(12.5 - 8.371)}}{{25}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.330\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were successful with part (a).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">bcd.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A surprising number had difficulty producing enough work to show that the period was \(25\); writing down the exact value of \(b\) also overwhelmed a number of candidates. In part (c), candidates did not recognize that the seat on the Ferris wheel is a minimum at \(t = 0\) thereby making the value of a negative. Incorrect values of \(61\) were often seen with correct follow through obtained when sketching the graph in part (d). Graphs again frequently failed to show key features in approximately correct locations and candidates lost marks for incorrect domains and ranges.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was very poorly done for those who attempted the question and most did not make the connection between height, time and probability. The idea of linking probability with a real-life scenario proved beyond most candidates. That said, there were a few novel approaches from the strongest of candidates using circles and angles to solve this part of question 10.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows a circle, centre O, with radius 4 cm. Points A and B lie on the circumference of the circle and AÔB = <em>θ</em> , where 0 ≤ <em>θ</em> ≤ \(\pi \).</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region, in terms of <em>θ</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The area of the shaded region is 12 cm<sup>2</sup>. Find the value of<em> θ</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find area of segment <em><strong>(M1)</strong></em></p>
<p><em>eg </em> area of sector – area of triangle, \(\frac{1}{2}{r^2}\left( {\theta - {\text{sin}}\theta } \right)\)</p>
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\(\frac{1}{4}{\left( 4 \right)^2}\theta - \frac{1}{2}{\left( 4 \right)^2}{\text{sin}}\theta ,\,\,\frac{1}{2} \times 16\left[ {\theta - {\text{sin}}\theta } \right]\)</p>
<p>area = 80 – 8 sin<em>θ</em>, 8(<em>θ – </em>sin<em>θ</em>) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>setting <strong>their</strong> area expression equal to 12 <em><strong>(M1)</strong></em></p>
<p><em>eg </em> 12 = 8(<em>θ – </em>sin<em>θ</em>) </p>
<p>2.26717</p>
<p><em>θ </em>= 2.27 (do not accept an answer in degrees) <em><strong>A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = 12\,\,{\text{cos}}\,x - 5\,\,{\text{sin}}\,x,\,\, - \pi \leqslant x \leqslant 2\pi \), be a periodic function with \(f\left( x \right) = f\left( {x + 2\pi } \right)\)</p>
<p>The following diagram shows the graph of \(f\).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">There is a maximum point at A. The minimum value of \(f\) is −13 .</p>
</div>
<div class="specification">
<p>A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.</p>
<p style="text-align: center;"><img 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"></p>
<p>The distance, <em>d</em> centimetres, of the centre of the ball from O at time <em>t</em> seconds, is given by</p>
<p style="padding-left: 90px;">\(d\left( t \right) = f\left( t \right) + 17,\,\,0 \leqslant t \leqslant 5.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of A.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the graph of \(f\), write down the amplitude.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the graph of \(f\), write down the period.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum speed of the ball.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
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<p>Find the first time when the ball’s speed is changing at a rate of 2 cm s<sup>−2</sup>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>−0.394791,13</p>
<p>A(−0.395, 13) <em><strong>A1A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px;">
<p>13 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
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<div class="question" style="padding-left: 20px;">
<p>\({2\pi }\), 6.28 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
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<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> recognizing that amplitude is <em>p</em> or shift is <em>r</em></p>
<p>\(f\left( x \right) = 13\,\,{\text{cos}}\,\left( {x + 0.395} \right)\) (accept <em>p</em> = 13, <em>r</em> = 0.395) <em><strong>A1A1 N3</strong></em></p>
<p><strong>Note:</strong> Accept any value of <em>r</em> of the form \(0.395 + 2\pi k,\,\,k \in \mathbb{Z}\)</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px;">
<p>recognizing need for <em>d </em>′(<em>t</em>) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> −12 sin(<em>t</em>) − 5 cos(<em>t</em>)</p>
<p>correct approach (accept any variable for <em>t</em>) <em><strong>(A1)</strong></em></p>
<p><em>eg </em> −13 sin(<em>t + </em>0.395), sketch of <em>d</em>′, (1.18, −13), <em>t</em> = 4.32</p>
<p>maximum speed = 13 (cms<sup>−1</sup>) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
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<div class="question" style="padding-left: 20px;">
<p>recognizing that acceleration is needed <em><strong>(M1)</strong></em></p>
<p><em>eg </em>a(<em>t</em>), <em>d "</em>(t)</p>
<p>correct equation (accept any variable for <em>t</em>) <em><strong> (A1)</strong></em></p>
<p><em>eg </em>\(a\left( t \right) = - 2,\,\,\left| {\frac{{\text{d}}}{{{\text{d}}t}}\left( {d'\left( t \right)} \right)} \right| = 2,\,\, - 12\,\,{\text{cos}}\,\left( t \right) + 5\,\,{\text{sin}}\,\left( t \right) = - 2\)</p>
<p>valid attempt to solve <strong>their</strong> equation <em><strong>(M1)</strong></em></p>
<p><em>eg </em> sketch, 1.33</p>
<p>1.02154</p>
<p>1.02 <em><strong>A2 N3</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
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<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
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<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
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