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</div><h2>SL Paper 1</h2><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(\sin \theta = \frac{2}{{\sqrt {13} }}\) , where \(\frac{\pi }{2} < \theta < \pi \) .</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;">Find \(\cos \theta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\tan 2\theta \) .</span></p>
<div class="marks">[5]</div>
<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 \(f(x) = \frac{{\cos x}}{{\sin x}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , for \(\sin x \ne 0\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In the following table, \(f'\left( {\frac{\pi }{2}} \right) = p\) and \(f''\left( {\frac{\pi }{2}} \right) = q\) . The table also gives approximate values of \(f'(x)\) and \(f''(x)\) near \(x = \frac{\pi }{2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/batman.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 quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .</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;">Find \(f''(x)\) .</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 value of <em>p</em> and of <em>q</em>.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use information from the table to explain why there is a point of inflexion on the </span><span style="font-family: times new roman,times; font-size: medium;">graph of <em>f</em> where \(x = \frac{\pi }{2}\) .</span></p>
<div class="marks">[2]</div>
<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 \(f(t) = a\cos b(t - c) + d\) , \(t \ge 0\) . Part of the graph of \(y = f(t)\) is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/evening.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t = 3\) , there is a maximum value of 29, at M.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> When \(t = 9\) , there is a minimum value of 15.</span></p>
<p> </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) Find the value of <em>a</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(b = \frac{\pi }{6}\) </span><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;">(iii) Find the value of <em>d</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iv) Write down a value for <em>c</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a(i), (ii), (iii) and (iv).</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 transformation <em>P</em> is given by a horizontal stretch of a scale factor of \(\frac{1}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, followed </span><span style="font-family: times new roman,times; font-size: medium;">by a translation of \(\left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 10}<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let \({M'}\) be the image of M under <em>P</em>. Find the coordinates of \({M'}\) .</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;">The graph of <em>g</em> is the image of the graph of <em>f</em> under <em>P</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(g(t)\) in the form \(g(t) = 7\cos B(t - c) + D\) .</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;"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is the image of the graph of <em>f</em> under <em>P</em>.</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of the transformation that maps the graph of <em>g </em></span><span style="font-family: times new roman,times; font-size: medium;">to the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<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 \(g(x) = 3\sin 2x\) . </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;">Write down the period of <em>g</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the diagram below, sketch the curve of <em>g</em>, for \(0 \le x \le 2\pi \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cake.png" alt></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;">Write down the number of solutions to the equation \(g(x) = 2\) , for \(0 \le x \le 2\pi \) .</span></p>
<div class="marks">[2]</div>
<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;">Let \(f(x) = {(\sin x + \cos x)^2}\) .</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 \(f(x)\) can be expressed as \(1 + \sin 2x\) .</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;">The graph of <em>f</em> is shown below for \(0 \le x \le 2\pi \) .</span></p>
<p><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/pirate.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = 1 + \cos x\) . On the same set of axes, sketch the graph of <em>g</em> for </span><span style="font-family: times new roman,times; font-size: medium;">\(0 \le x \le 2\pi \) .</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;">The graph of <em>g</em> can be obtained from the graph of <em>f</em> under a horizontal stretch of </span><span style="font-family: times new roman,times; font-size: medium;">scale factor <em>p</em> followed by a translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>k\\<br>0<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>p</em> and a possible value of <em>k</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \sqrt 3 {{\rm{e}}^{2x}}\sin x + {{\rm{e}}^{2x}}\cos x\) , for \(0 \le x \le \pi \) . Solve </span><span style="font-family: times new roman,times; font-size: medium;">the </span><span style="font-family: times new roman,times; font-size: medium;">equation \(f(x) = 0\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 3\sin \left( {\frac{\pi }{2}x} \right)\), for \(0 \leqslant x \leqslant 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the amplitude of \(f\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the period of \(f\).</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">On the following grid sketch the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_05.59.27.png" alt="M16/5/MATME/SP1/ENG/TZ1/03.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Given that \(\sin x = \frac{3}{4}\), </span>where \(x\) is an obtuse angle,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">find the value of \(\cos x;\)</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>find the value of \(\cos 2x.\)</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>Note: In this question, distance is in metres and time is in seconds.</strong></p>
<p>Two particles \({P_1}\) and \({P_2}\) start moving from a point A at the same time, along different straight lines.</p>
<p>After \(t\) seconds, the position of \({P_1}\) is given by <strong><em>r</em></strong> = \(\left( {\begin{array}{*{20}{c}} 4 \\ { - 1} \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)\).</p>
</div>
<div class="specification">
<p>Two seconds after leaving A, \({P_1}\) is at point B.</p>
</div>
<div class="specification">
<p>Two seconds after leaving A, \({P_2}\) is at point C, where \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 4 \end{array}} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of A.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\overrightarrow {{\text{AB}}} \);</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 \(\left| {\overrightarrow {{\text{AB}}} } \right|\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\cos {\rm{B\hat AC}}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the distance between \({P_1}\) and \({P_2}\) two seconds after they leave A.</p>
<div class="marks">[4]</div>
<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 \(f(x) = 6 + 6\sin x\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/abba.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the curve of <em>f</em> , the <em>x</em>-axis, and the <em>y</em>-axis.</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;">Solve for \(0 \le x < 2\pi \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(6 + 6\sin x = 6\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(6 + 6\sin x = 0\) .</span></p>
<div class="marks">[5]</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 exact value of the <em>x</em>-intercept of <em>f</em> , for \(0 \le x < 2\pi \) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The area of the shaded region is <em>k</em> . Find the value of <em>k</em> , giving your answer in </span><span style="font-family: times new roman,times; font-size: medium;">terms of \(\pi \) .</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;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of this transformation.</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;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_p^{p + \frac{{3\pi }}{2}} {g(x){\rm{d}}x} = k\) and \(0 \le p < 2\pi \) , write down the two values of <em>p</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The following diagram shows triangle \(ABC\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_04.52.14.png" alt></p>
<p>Let \(\overrightarrow {{\text{AB}}} \bullet \overrightarrow {{\text{AC}}} = - 5\sqrt 3 \) and \(\left| {\overrightarrow {{\text{AB}}} } \right|\left| {\overrightarrow {{\text{AC}}} } \right| = 10\). Find the area of triangle \(ABC\).</p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \(\cos 2x - 3\cos x - 3 - {\cos ^2}x = {\sin ^2}x\) , for \(0 \le x \le 2\pi \) .</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f:x \mapsto {\sin ^3}x\) .</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;">(i) Write down the range of the function <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Consider \(f(x) = 1\) , \(0 \le x \le 2\pi \) . Write down the number of solutions to </span><span style="font-family: times new roman,times; font-size: medium;">this equation. Justify your answer.</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;">Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where \(a{\text{, }}p{\text{, }}q \in \mathbb{Z}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">. Find the volume generated when the </span><span style="font-family: times new roman,times; font-size: medium;">curve of <em>g</em> is revolved through \(2\pi \) about the <em>x</em>-axis.</span></p>
<div class="marks">[7]</div>
<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 part of the graph of a function \(f\) .</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/lap.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph has a maximum at A(\(1\), \(5\)) and a minimum at B(\(3\), \( -1\)) .</span></p>
<p><span style="font-size: medium;"><span style="font-family: times new roman,times;">The function \(f\)</span> <span style="font-family: times new roman,times;">can be written in the form \(f(x) = p\sin (qx) + r\) . Find the value of<br></span></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;">(a) \(p\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \(q\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) \(r\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(p\)</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;">\(q\)</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;">\(r\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a circle with centre \(O\) and a radius of \(10\) cm. Points \(A\), \(B\) and \(C\) lie on the circle.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_05.32.02.png" alt></p>
<p class="p1">Angle \(AOB\) is \(1.2\) radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the length of \({\text{arc ACB}}\).</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 the perimeter of the shaded region.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable \(A\), in terms of an angle \(\theta \).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\cos \theta = \frac{3}{4}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\tan \theta > 0\), find \(\tan \theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between \(x = \theta \) and \(x = \frac{\pi }{4}\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(4 - \cos 2\theta + 5\sin \theta = 2{\sin ^2}\theta + 5\sin \theta + 3\) .</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;"><strong>Hence</strong>, solve the equation \(4 - \cos 2\theta + 5\sin \theta = 0\) for \(0 \le \theta \le 2\pi \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve \({\log _2}(2\sin x) + {\log _2}(\cos x) = - 1\), for \(2\pi < x < \frac{{5\pi }}{2}\).</p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The straight line with equation \(y = \frac{3}{4}x\) </span><span style="font-family: times new roman,times; font-size: medium;">makes an acute angle \(\theta \) with the <em>x</em>-axis.</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;">Write down the value of \(\tan \theta \) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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) \(\sin 2\theta \) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(\cos 2\theta \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(\sin \theta = \frac{{\sqrt 5 }}{3}\), where \(\theta \) is acute.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\cos \theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\cos 2\theta \).</p>
<div class="marks">[2]</div>
<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;">Let \(f(x) = {\sin ^3}x + {\cos ^3}x\tan x,\frac{\pi }{2} < x < \pi \) .</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;">Show that \(f(x) = \sin x\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(\sin x = \frac{2}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">. Show that \(f(2x) = - \frac{{4\sqrt 5 }}{9}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[5]</div>
<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 the graph of \(f(x) = a\sin (b(x - c)) + d\) , for \(2 \le x \le 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/deanna.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum point at P(4, 12) and a minimum point at Q(8, −4) .</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;">Use the graph to write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>a</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>c</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>d</em> .</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;">Show that \(b = \frac{\pi }{4}\) .</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 \(f'(x)\) .</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;">At a point R, the gradient is \( - 2\pi \) . Find the <em>x</em>-coordinate of R.</span></p>
<div class="marks">[6]</div>
<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 expression \(6\sin x\cos x\) can be expressed in the form \(a\sin bx\) .</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>a</em> and of <em>b</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><span style="font-family: times new roman,times; font-size: medium;">Hence or otherwise, solve the equation \(6\sin x\cos x = \frac{3}{2}\) , for \(\frac{\pi }{4} \le x \le \frac{\pi }{2}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Let \(\sin {100^ \circ } = m\)</span><span style="font-family: TimesNewRomanPSMT;">. Find an expression for </span></span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><span style="font-family: times new roman,times; font-size: medium;">\(\cos {100^ \circ }\)</span> in terms of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">m</span></em><span style="font-family: TimesNewRomanPS-ItalicMT;">.</span></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;"><span style="font-family: TimesNewRomanPSMT;">Let \(\sin {100^ \circ } = m\) </span><span style="font-family: TimesNewRomanPSMT;">. Find an expression for </span></span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT;">\(\tan {100^ \circ }\)</span></span> in terms of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">m</span></em><span style="font-family: TimesNewRomanPS-ItalicMT;">.</span></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;">Let \(\sin {100^ \circ } = m\). Find an expression for </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(\sin {200^ \circ }\)</span> in terms of <em>m</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\int_\pi ^a {\cos 2x{\text{d}}x} = \frac{1}{2}{\text{, where }}\pi < a < 2\pi \). Find the value of \(a\).</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram represents a large Ferris wheel, with a diameter of 100 metres.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ferris.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let P be a point on the wheel. The wheel starts with P at the lowest point, </span><span style="font-family: times new roman,times; font-size: medium;">at ground level. The wheel rotates at a constant rate, in an anticlockwise </span><span style="font-family: times new roman,times; font-size: medium;">(counter-clockwise) direction. One revolution takes 20 minutes.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(h(t)\) metres be the height of P above ground level after <em>t</em> minutes. Some values </span><span style="font-family: times new roman,times; font-size: medium;">of \(h(t)\) are given in the table below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/garage.png" alt></span></p>
<p> </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 height of P above ground level after</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 10 minutes;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) 15 minutes.</span></p>
<p align="LEFT"> </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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(h(8) = 90.5\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(h(21)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(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;"><strong>Sketch</strong> the graph of <em>h</em> , for \(0 \le t \le 40\) .</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;">Given that <em>h</em> can be expressed in the form \(h(t) = a\cos bt + c\) , find <em>a</em> , <em>b</em> and <em>c</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f(x) = 15 - {x^2}\), for \(x \in \mathbb{R}\). The following diagram shows part of the graph of \(f\) and the rectangle OABC, where A is on the negative \(x\)-axis, B is on the graph of \(f\), and C is on the \(y\)-axis.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_13.13.04.png" alt="N17/5/MATME/SP1/ENG/TZ0/06"></p>
<p>Find the \(x\)-coordinate of A that gives the maximum area of OABC.</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \sin \left( {x + \frac{\pi }{4}} \right) + k\). The graph of <em>f </em>passes through the point \(\left( {\frac{\pi }{4},{\text{ }}6} \right)\).</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 value of \(k\).</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 minimum value of \(f(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(g(x) = \sin x\). The graph of <em>g </em>is translated to the graph of \(f\) by the vector </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)\)</span><span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">.</span></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;">Write down the value of \(p\) and of \(q\).</span></p>
<div class="marks">[2]</div>
<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-size: medium; font-family: 'times new roman', times;">The following diagram shows a right-angled triangle, \(\rm{ABC}\), where \(\sin \rm{A} = \frac{5}{{13}}\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_1_2.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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\cos A = \frac{{12}}{{13}}\).</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 \(\cos 2A\).</span></p>
<div class="marks">[3]</div>
<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;">Let \(f(x) = \cos 2x\) and \(g(x) = 2{x^2} - 1\) .</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 \(f\left( {\frac{\pi }{2}} \right)\) .</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 \((g \circ f)\left( {\frac{\pi }{2}} \right)\) .</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;">Given that \((g \circ f)(x)\) can be written as \(\cos (kx)\) , find the value of <em>k</em>, \(k \in \mathbb{Z}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a triangle <span class="s1">ABC </span>and a sector <span class="s1">BDC </span>of a circle with centre <span class="s1">B </span>and radius <span class="s1">6 cm</span>. The points <span class="s1">A </span>, <span class="s1">B </span>and <span class="s1">D </span>are on the same line.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_09.25.23.png" alt="M16/5/MATME/SP1/ENG/TZ2/05"></p>
<p class="p1" style="text-align: center;"><span class="s1">\({\text{AB}} = 2\sqrt 3 {\text{ cm, BC}} = 6{\text{ cm, area of triangle ABC}} = 3\sqrt 3{\text{ c}}{{\text{m}}^{\text{2}}}{\rm{, A\hat BC}}\) </span>is obtuse.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\rm{A\hat BC}}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the exact area of the sector <span class="s1">BDC</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(\overrightarrow {{\text{OA}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ 0 \\ 4 \end{array}} \right)\) and \(\overrightarrow {{\text{OB}}} = \left( {\begin{array}{*{20}{c}} 4 \\ 1 \\ 3 \end{array}} \right)\).</p>
</div>
<div class="specification">
<p class="p1">The point <span class="s1">C </span>is such that \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ 1 \\ { - 1} \end{array}} \right)\).</p>
</div>
<div class="specification">
<p class="p1">The following diagram shows triangle <span class="s1">ABC</span>. Let <span class="s1">D </span>be a point on <span class="s1">[BC]</span>, with acute angle \({\text{ADC}} = \theta \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_05.46.25.png" alt="N16/5/MATME/SP1/ENG/TZ0/08.c.d.e"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Find \(\overrightarrow {{\text{AB}}} \).</p>
<p>(ii) Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the coordinates of <span class="s1">C </span><span class="s2">are \(( - 2,{\text{ }}1,{\text{ }}3)\).</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 class="p1">Write down an expression in terms of \(\theta \) <span class="s1">for</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>angle <span class="s2">ADB</span><span class="s1">;</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>area of triangle <span class="s2">ABD</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 class="p1"><span class="s1">Given that \(\frac{{{\text{area }}\Delta {\text{ABD}}}}{{{\text{area }}\Delta {\text{ACD}}}} = 3\)</span>, show that \(\frac{{{\text{BD}}}}{{{\text{BC}}}} = \frac{3}{4}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence or otherwise, find the coordinates of point <span class="s1">D</span>.</p>
<div class="marks">[4]</div>
<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 diagram below shows part of the graph of \(f(x) = a\cos (b(x - c)) - 1\) , where \(a > 0\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/huw.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point \({\rm{P}}\left( {\frac{\pi }{4},2} \right)\) is a maximum point and the point \({\rm{Q}}\left( {\frac{{3\pi }}{4}, - 4} \right)\) is a minimum point.</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 value of <em>a</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that the period of <em>f</em> is \(\pi \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find the value of <em>b</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(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;">Given that \(0 < c < \pi \) , write down the value of <em>c</em> .</span></p>
<div class="marks">[1]</div>
<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;"> Let \(p = \sin 40^\circ \) and \(q = \cos 110^\circ \) . Give your answers to the following in terms of <em>p</em> and/or <em>q .</em></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;">Write down an expression for </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(\sin 140^\circ \) ; </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) \(\cos 70^\circ \) .</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 an expression for \(\cos 140^\circ \) .</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 an expression for \(\tan 140^\circ \) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows triangle ABC, with \({\text{AB}} = 3{\text{ cm}}\), \({\text{BC}} = 8{\text{ cm}}\), and \({\rm{A\hat BC = }}\frac{\pi }{3}\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.17.57.png" alt="N17/5/MATME/SP1/ENG/TZ0/04"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{AC}} = 7{\text{ cm}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The shape in the following diagram is formed by adding a semicircle with diameter [AC] to the triangle.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_10.50.00.png" alt="N17/5/MATME/SP1/ENG/TZ0/04.b"></p>
<p>Find the exact perimeter of this shape.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows triangle PQR.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.36.55.png" alt="M17/5/MATME/SP1/ENG/TZ1/03"></p>
<p>Find PR.</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^{ - 3x}}\) and \(g(x) = \sin \left( {x - \frac{\pi }{3}} \right)\) .</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;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(g'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)\) . Find the exact value of \(h'\left( {\frac{\pi }{3}} \right)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(\cos A = \frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">and \(0 \le A \le \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, find \(\cos 2A\) .</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;">Given that \(\sin B = \frac{2}{3}\) and \(\frac{\pi }{2} \le B \le \pi \) , find \(\cos B\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 3\sin (\pi x)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the amplitude of \(f\).</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 period of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following grid, sketch the graph of \(y = f(x)\), for \(0 \le x \le 3\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_07.47.02.png" alt></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The first two terms of an infinite geometric sequence are <em>u</em><sub>1</sub> = 18 and <em>u</em><sub>2</sub> = 12sin<sup>2</sup> <em>θ</em> , where 0 < <em>θ</em> < 2\(\pi \) , and <em>θ</em> ≠ \(\pi \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <em>r</em> in terms of <em>θ</em>.</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 possible values of <em>r</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the sum of the infinite sequence is \(\frac{{54}}{{2 + {\text{cos}}\,\left( {2\theta } \right)}}\).</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 values of <em>θ</em> which give the greatest value of the sum.</p>
<div class="marks">[6]</div>
<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 the graph of \(f(x) = a\cos (bx)\) , for \(0 \le x \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q3.jpg" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a minimum point at P(2, − 3) and a maximum point at Q(4, 3) .</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) Write down the value of <em>a</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of <em>b</em> .</span></p>
<div class="marks">[3]</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 gradient of the curve at P.</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 equation of the normal to the curve at P.</span></p>
<div class="marks">[2]</div>
<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 semicircle centre O, diameter [AB], with radius 2.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let P be a point on the circumference, with \({\rm{P}}\widehat {\rm{O}}{\rm{B}} = \theta \) radians.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/three_mins.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>S</em> be the total area of the two segments shaded in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flo.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 area of the triangle OPB, in terms 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;">Explain why the area of triangle OPA is the same as the area triangle OPB.</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;">Show that \(S = 2(\pi - 2\sin \theta )\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\theta \) when <em>S</em> is a local minimum, justifying that it is a minimum.</span></p>
<div class="marks">[8]</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;">Find a value of \(\theta \) for which <em>S</em> has its greatest value.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Let \(f(x) = 6x\sqrt {1 - {x^2}} \)</span>, for \( - 1 \leqslant x \leqslant 1\)<span class="s1">, and \(g(x) = \cos (x)\)</span>, for \(0 \leqslant x \leqslant \pi \)<span class="s1">.</span></p>
<p class="p2">Let \(h(x) = (f \circ g)(x)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the range of \(h\).</p>
<div class="marks">[2]</div>
<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 two concentric circles with centre O.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/222.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The radius of the smaller circle is 8 cm and the radius of the larger circle is 10 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Points A, B and C are on the circumference of the larger circle such that \({\rm{A}}\widehat {\rm{O}}{\rm{B}}\) is \(\frac{\pi }{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">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 the length of the arc ACB .</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>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The vertices of the triangle PQR are defined by the position vectors</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\overrightarrow {{\rm{OP}}} = \left( {\begin{array}{*{20}{c}}<br>4\\<br>{ - 3}\\<br>1<br>\end{array}} \right)\) , \(\overrightarrow {{\rm{OQ}}} = \left( {\begin{array}{*{20}{c}}<br>3\\<br>{ - 1}\\<br>2<br>\end{array}} \right)\) and \(\overrightarrow {{\rm{OR}}} = \left( {\begin{array}{*{20}{c}}<br>6\\<br>{ - 1}\\<br>5<br>\end{array}} \right)\) .</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</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(\overrightarrow {{\rm{PQ}}} \) </span><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;">(ii) \(\overrightarrow {{\rm{PR}}} \) </span><span style="font-family: times new roman,times; font-size: medium;">.</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;"> Show that \(\cos {\rm{R}}\widehat {\rm{P}}{\rm{Q}} = \frac{1}{2}\) .</span></p>
<div class="marks">[7]</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 \({\rm{sinR}}\widehat {\rm{P}}{\rm{Q}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find the area of triangle PQR, giving your answer in the form \(a\sqrt 3 \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \(2\cos x = \sin 2x\) , for \(0 \le x \le 3\pi \) .</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos x + \sqrt 3 \sin x\) , \(0 \le x \le 2\pi \) . The following diagram shows the graph of \(f\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/movie.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The \(y\)-intercept is at (\(0\), \(1\)) , there is a minimum point at A (\(p\), \(q\)) and a maximum </span><span style="font-family: times new roman,times; font-size: medium;">point at B.</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 \(f'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Hence</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) show that \(q = - 2\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) verify that A is a minimum point.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b(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 maximum value of \(f(x)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function \(f(x)\) can be written in the form \(r\cos (x - a)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> and of <em>a</em> .</span></p>
<div class="marks">[2]</div>
<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;">A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bike.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point P(<em>x</em> , <em>y</em>) is a vertex of the rectangle and also lies on the circle. The angle </span><span style="font-family: times new roman,times; font-size: medium;">between (OP) and the <em>x</em>-axis is \(\theta \) radians, where \(0 \le \theta \le \frac{\pi }{2}\) .</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;">Write down an expression in terms of \(\theta \) for</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(x\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(y\) .</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;">Let the area of the rectangle be <em>A</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(A = 18\sin 2\theta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) </span><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;">(ii) Hence, find the exact value of \(\theta \) which maximizes the area of </span><span style="font-family: times new roman,times; font-size: medium;">the rectangle.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Use the second derivative to justify that this value of \(\theta \) does give </span><span style="font-family: times new roman,times; font-size: medium;">a maximum.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{{6x}}{{\cos x}}\) . Find \(h'(0)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows a circle with centre O and radius<em> r</em> cm.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The points A and B lie on the circumference of the circle, and \({\text{A}}\mathop {\text{O}}\limits^ \wedge {\text{B}}\) = <em>θ</em>. The area of the shaded sector AOB is 12 cm<sup>2</sup> and the length of arc AB is 6 cm.</p>
<p>Find the value of <em>r</em>.</p>
</div>
<br><hr><br><div class="question">
<p>Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.<br>This is shown in the following diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The vectors <em><strong>p</strong></em> , <em><strong>q</strong></em> and <em><strong>r</strong></em> are shown on the diagram.</p>
<p>Find <em><strong>p</strong></em>•(<em><strong>p</strong></em> + <em><strong>q</strong></em> + <em><strong>r</strong></em>).</p>
</div>
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