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</div><h2>HL Paper 2</h2><div class="specification">
<p>A function is defined by \(f(x) = A\sin (Bx) + C,{\text{ }} - \pi \le x \le \pi \), where \(A,{\text{ }}B,{\text{ }}C \in \mathbb{Z}\). The following diagram represents the graph of \(y = f(x)\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-29_om_10.14.17.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(A\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(B\);</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>\(C\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \(f(x) = 3\) for \(0 \le x \le \pi \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function \(f\) defined by \(f(x) = 3x\arccos (x)\) where \( - 1 \leqslant x \leqslant 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Sketch the graph of \(f\) </span>indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the range of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the inequality \(\left| {3x\arccos (x)} \right| > 1\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Given \(\Delta \)</span><span style="font-family: times new roman,times; font-size: medium;">ABC, with lengths shown in the diagram below, find the length of the line segment [CD].</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider triangle ABC with \({\rm{B}}\hat {\rm{A}}{\rm{C}} = 37.8^\circ \) , AB = 8.75 and BC = 6 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find AC.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows a semi-circle of diameter 20 cm, centre O and two points A and B such that \({\rm{A\hat OB}} = \theta \), where \(\theta \) is in radians.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-17_om_06.17.13.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the shaded area can be expressed as \(50\theta - 50\sin \theta \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\theta \) for which the shaded area is equal to half that of the unshaded area, giving your answer correct to four significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In triangle \({\text{PQR, PR}} = 12{\text{ cm, QR}} = p{\text{ cm, PQ}} = r{\text{ cm}}\) and \({\rm{Q\hat PR}} = 30^\circ \).</p>
</div>
<div class="specification">
<p>Consider the possible triangles with \({\text{QR}} = 8{\text{ cm}}\).</p>
</div>
<div class="specification">
<p>Consider the case where \(p\), the length of QR is not fixed at 8 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the cosine rule to show that \({r^2} - 12\sqrt 3 r + 144 - {p^2} = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the two corresponding values of PQ.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area of the smaller triangle.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the range of values of \(p\) for which it is possible to form two triangles.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find AM.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}\) in radians.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the smaller circle lies on the circumference of the bigger circle. O is the centre of the bigger circle and the two circles intersect at points A and B.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-15_om_11.00.51.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\rm{B\hat OC}}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the area of the shaded region.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Triangle ABC has AB = 5 cm, BC = 6 cm and area 10 \({\text{c}}{{\text{m}}^2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(\sin \hat B\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong>, find the two possible values of AC, giving your answers correct to two decimal places.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In a triangle ABC, \(\hat A = 35^\circ \), BC = 4 cm and AC = 6.5 cm. Find the possible values of \(\hat B\) and the corresponding values of AB.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\arctan \frac{1}{2} - \arctan \frac{1}{3} = \arctan a,{\text{ }}a \in {\mathbb{Q}^ + }\), find the value of <em>a</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, solve the equation \(\arcsin x = \arctan a\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the triangle \({\text{PQR}}\) where \({\rm{Q\hat PR = 30^\circ }}\), \({\text{PQ}} = (x + 2){\text{ cm}}\) and \({\text{PR}} = {(5 - x)^2}{\text{ cm}}\), where \( - 2 < x < 5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the area, \(A\;{\text{c}}{{\text{m}}^2}\), of the triangle is given by \(A = \frac{1}{4}({x^3} - 8{x^2} + 5x + 50)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State \(\frac{{{\text{d}}A}}{{{\text{d}}x}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Verify that \(\frac{{{\text{d}}A}}{{{\text{d}}x}} = 0\) when \(x = \frac{1}{3}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}A}}{{{\text{d}}{x^2}}}\) and hence justify that \(x = \frac{1}{3}\) gives the maximum area of triangle \(PQR\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State the maximum area of triangle \(PQR\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find \(QR\) when the area of triangle \(PQR\) is a maximum.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f(x) = 3\sin x + 4\cos x\) is defined for \(0 < x < 2\pi \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the coordinates of the minimum point on the graph of <em>f </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The points \({\text{P}}(p,{\text{ }}3)\) and \({\text{Q}}(q,{\text{ }}3){\text{, }}q > p\), lie on the graph of \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find <em>p </em>and <em>q </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point of intersection of the normals to the graph at the points P and Q.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A circle of radius 4 cm , centre O , is cut by a chord [AB] of length 6 cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
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" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\rm{A\hat OB}}\), expressing your answer in radians correct to four significant figures.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the area of the shaded region.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the triangle ABC where \({\rm{B\hat AC}} = 70^\circ \), AB = 8 cm and AC = 7 cm. The point D on the side BC is such that \(\frac{{{\text{BD}}}}{{{\text{DC}}}} = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the length of AD.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The depth, <em>h</em>(<em>t</em>) metres, of water at the entrance to a harbour at <em>t</em> hours after midnight on a particular day is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[h(t) = 8 + 4\sin \left( {\frac{{\pi t}}{6}} \right),{\text{ }}0 \leqslant t \leqslant 24.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the maximum depth and the minimum depth of the water.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the values of <em>t</em> for which \(h(t) \geqslant 8\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Farmer Bill owns a rectangular field, 10 m by 4 m. Bill attaches a rope to a wooden post at one corner of his field, and attaches the other end to his goat Gruff.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to graze. Give your answer correct to the nearest integer.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Bill replaces Gruff’s rope with another, this time of length \(a,{\text{ }}4 < a < 10\), so that Gruff can now graze exactly one half of Bill’s field.</p>
<p>Show that \(a\) satisfies the equation</p>
<p>\[{a^2}\arcsin \left( {\frac{4}{a}} \right) + 4\sqrt {{a^2} - 16} = 40.\]</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In triangle \({\text{ABC}}\), \({\text{AB}} = 5{\text{ cm}}\), \({\text{BC}} = 12{\text{ cm}}\) and \({\rm{A\hat BC}} = 100^\circ \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the area of the triangle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(AC\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.<br>“<em>Seaview</em>” \((S)\) is at an angle of depression of 25°.<br>“<em>Nauti Buoy</em>” \((N)\) is at an angle of depression of 35°.<br>The following three dimensional diagram shows Barry and the two yachts at S and N.<br>X lies at the foot of the cliff and angle \({\text{SXN}} = \) 70°.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-08_om_11.45.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/05"></p>
<p>Find, to 3 significant figures, the distance between the two yachts.</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An electricity station is on the edge of a straight coastline. A lighthouse is located in the sea 200 m from the electricity station. The angle between the coastline and the line joining the lighthouse with the electricity station is 60°. A cable needs to be laid connecting the lighthouse to the electricity station. It is decided to lay the cable in a straight line to the coast and then along the coast to the electricity station. The length of cable laid along the coastline is <em>x</em> metres. This information is illustrated in the diagram below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 24px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The cost of laying the cable along the sea bed is US$80 per metre, and the cost of laying it on land is US$20 per metre.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in terms of <em>x</em>, an expression for the cost of laying the cable.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>x</em>, to the nearest metre, such that this cost is minimized.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The vertices of an equilateral triangle, with perimeter <em>P</em> and area <em>A</em> , lie on a circle with radius <em>r</em> . Find an expression for \(\frac{P}{A}\) in the form \(\frac{k}{r}\), where \(k \in {\mathbb{Z}^ + }\).</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}\).</p>
</div>
<div class="specification">
<p>Let \(u = \tan x\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for \(f’(x)\) in terms of \(x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of \(y = f’(x)\) for \(0 \leqslant x < \frac{\pi }{2}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(x\)-coordinate(s) of the point(s) of inflexion of the graph of \(y = f(x)\), labelling these clearly on the graph of \(y = f’(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \(\sin x\) in terms of \(\mu \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \(\sin 2x\) in terms of \(u\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \(f(x) = 0\) can be expressed as \({u^3} - 7{u^2} + 15u - 9 = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation \(f(x) = 0\), giving your answers in the form \(\arctan k\) where \(k \in \mathbb{Z}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The diagram shows two circles with centres at the points <span class="s1">A </span>and <span class="s1">B </span>and radii \(2r\) and \(r\), respectively. The point <span class="s1">B </span>lies on the circle with centre <span class="s1">A</span>. The circles intersect at the points <span class="s1">C </span>and <span class="s1">D</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.29.37.png" alt="N16/5/MATHL/HP2/ENG/TZ0/09"></p>
<p class="p1">Let \(\alpha \) be the measure of the angle <span class="s1">CAD </span>and \(\theta \) be the measure of the angle <span class="s1">CBD </span>in radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for the shaded area in terms of \(\alpha \), \(\theta \) and \(r\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\alpha = 4\arcsin \frac{1}{4}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the value of \(r\) given that the shaded area is equal to 4.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The shaded region <em>S </em>is enclosed between the curve \(y = x + 2\cos x\), for \(0 \leqslant x \leqslant 2\pi \), and the line \(y = x\), as shown in the diagram below.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-12_om_06.15.17.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the points where the line meets the curve.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region \(S\) is rotated by \(2\pi \) about the \(x\)-axis to generate a solid.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down an integral that represents the volume \(V\) of the solid.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the volume \(V\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In a triangle \({\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}\) and \({\rm{B\hat AC}} = \frac{\pi }{9}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the cosine rule to find the two possible values for <span class="s1">AC</span><span class="s2">.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the difference between the areas of the two possible triangles <span class="s1">ABC</span><span class="s2">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(z = r(\cos \alpha + {\text{i}}\sin \alpha )\), where \(\alpha \) is measured in degrees, be the solution of \({z^5} - 1 = 0\) which has the smallest positive argument.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Use the binomial theorem to expand \({(\cos \theta + {\text{i}}\sin \theta )^5}\).</p>
<p>(ii) Hence use De Moivre’s theorem to prove</p>
<p>\[\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta .\]</p>
<p>(iii) State a similar expression for \(\cos 5\theta \) in terms of \(\cos \theta \) and \(\sin \theta \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(r\) and the value of \(\alpha \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using (a) (ii) and your answer from (b) show that \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Hence express \(\sin 72^\circ \) </span>in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="text-align: center;"><img src="images/12b.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In triangle ABC, BC = <em>a</em> , AC = <em>b</em> , AB = <em>c</em> and [BD] is perpendicular to [AC].</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 26px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \({\text{CD}} = b - c\cos A\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong>, by using Pythagoras’ Theorem in the triangle BCD, prove the cosine rule for the triangle ABC.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) If \({\rm{A\hat BC}} = 60^\circ \) , use the cosine rule to show that \(c = \frac{1}{2}a \pm \sqrt {{b^2} - \frac{3}{4}{a^2}} \) .</span></p>
<div class="marks">[12]</div>
<div class="question_part_label">Part A.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The above three dimensional diagram shows the points P and Q which are respectively west and south-west of the base R of a vertical flagpole RS on horizontal ground. The angles of elevation of the top S of the flagpole from P and Q are respectively 25° and 40° , and PQ = 20 m .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the height of the flagpole.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">Part B.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)\) where \(0 < x < \frac{\pi }{2}\).</p>
<p>Express \(f\left( x \right)\) in terms of sin \(x\) and cos \(x\).</p>
</div>
<br><hr><br><div class="question">
<p class="p1">ABCD is a quadrilateral where \({\text{AB}} = 6.5,{\text{ BC}} = 9.1,{\text{ CD}} = 10.4,{\text{ DA}} = 7.8\) and \({\rm{C\hat DA}} = 90^\circ \). Find \({\rm{A\hat BC}}\)<span class="s1">, giving your answer correct to the nearest degree.</span></p>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is \(\theta \) radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of \(0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water \(V{\text{ }}({{\text{m}}^3})\) in the trough in terms of \(\theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate \(\frac{{{\text{d}}\theta }}{{{\text{d}}t}}\) when \(\theta = \frac{\pi }{3}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Triangle \(ABC\) has area \({\text{21 c}}{{\text{m}}^{\text{2}}}\). The sides \(AB\) and \(AC\) have lengths \(6\) cm and \(11\) cm respectively. Find the two possible lengths of the side \(BC\).</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the planes \({\pi _1}:x - 2y - 3z = 2{\text{ and }}{\pi _2}:2x - y - z = k\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the angle between the planes \({\pi _1}\)and \({\pi _2}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The planes \({\pi _1}\) and \({\pi _2}\) intersect in the line \({L_1}\) . Show that the vector equation of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({L_1}\) is \(r = \left( {\begin{array}{*{20}{c}}<br>0\\<br>{2 - 3k}\\<br>{2k - 2}<br>\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}<br>1\\<br>5\\<br>{ - 3}<br>\end{array}} \right)\)</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) has Cartesian equation \(5 - x = y + 3 = 2 - 2z\) . The lines \({L_1}\) and \({L_2}\) intersect at a point X. Find the coordinates of X.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine a Cartesian equation of the plane \({\pi _3}\) containing both lines \({L_1}\) and \({L_2}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let Y be a point on \({L_1}\) and Z be a point on \({L_2}\) such that XY is perpendicular to YZ and the area of the triangle XYZ is 3. Find the perimeter of the triangle XYZ.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In triangle \(ABC\),</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(3\sin B + 4\cos C = 6\) and</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(4\sin C + 3\cos B = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\sin (B + C) = \frac{1}{2}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Robert conjectures that \({\rm{C\hat AB}}\) can have two possible values.</p>
<p class="p1">Show that Robert’s conjecture is incorrect by proving that \({\rm{C\hat AB}}\) has only one possible value.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of \(k\) that satisfy the inequality \({k^2} - k - 12 < 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that \(\cos B < \frac{1}{4}\), find the range of possible values for AB.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A triangle <span class="s1">\(ABC\) </span>has \(\hat A = 50^\circ \), <span class="s1">\({\text{AB}} = 7{\text{ cm}}\) </span>and <span class="s1">\({\text{BC}} = 6{\text{ cm}}\)</span>. Find the area of the triangle given that it is smaller than \(10{\text{ c}}{{\text{m}}^2}\).</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A rectangle is drawn around a sector of a circle as shown. If the angle of the sector is 1 radian and the area of the sector is \(7{\text{ c}}{{\text{m}}^2}\), find the dimensions of the rectangle, giving your answers to the nearest millimetre.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 22px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows two concentric circles with centre O and radii 2 cm and 4 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P and Q lie on the larger circle and \({\rm{P}}\hat {\text{O}}{\text{Q}} = x\) , where \(0 < x < \frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Show that the area of the shaded region is \(8\sin x - 2x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the maximum area of the shaded region.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">In the right circular cone below, O is the centre of the base which has radius 6 cm. The points B and C are on the circumference of the base of the cone. The height AO of the cone is 8 cm and the angle \({\rm{B\hat OC}}\) is 60°.</span> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the size of the angle \({\rm{B\hat AC}}\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Points A , B and T lie on a line on an indoor soccer field. The goal, [AB] , is 2 metres wide. A player situated at point P kicks a ball at the goal. [PT] is perpendicular to (AB) and is 6 metres from a parallel line through the centre of [AB] . Let PT <span class="s1">be \(x\) metros and let \(\alpha = {\rm{A\hat PB}}\) measured in degrees. Assume that the ball travels along the floor.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-03_om_11.38.31.png" alt="M16/5/MATHL/HP2/ENG/TZ2/11"></span></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The maximum for \(\tan \alpha \) </span>gives the maximum for \(\alpha \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(\alpha \) when \(x = 10\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\tan \alpha = \frac{{2x}}{{{x^2} + 35}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha )\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise find the value of \(\alpha \) <span class="s1">such that \(\frac{{\text{d}}}{{{\text{d}}x}}(\tan \alpha ) = 0\).</span></p>
<p class="p2"><span class="s1">(iii) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}}}{{{\text{d}}{x^2}}}(\tan \alpha )\) </span>and hence show that the value of \(\alpha \) <span class="s1">never exceeds 10°.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of values of \(x\) for which \(\alpha \geqslant 7^\circ \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the equation \(3{\cos ^2}x - 8\cos x + 4 = 0\), where \(0 \leqslant x \leqslant 180^\circ \), expressing your answer(s) to the nearest degree.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the exact values of \(\sec x\) satisfying the equation \(3{\sec ^4}x - 8{\sec ^2}x + 4 = 0\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The points P and Q lie on a circle, with centre O and radius 8 cm, such that \({\rm{P\hat OQ}} = 59^\circ \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 20px/normal Times; text-align: center; margin: 0px;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of the shaded segment of the circle contained between the arc PQ and the chord [PQ].</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph below shows \(y = a\cos (bx) + c\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>a</em>, the value of <em>b</em> and the value of <em>c</em>.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A system of equations is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\cos x + \cos y = 1.2\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\sin x + \sin y = 1.4{\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) For each equation express <em>y</em> in terms of <em>x</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong> solve the system for \(0 < x < \pi ,{\text{ }}0 < y < \pi \) .</span></p>
</div>
<br><hr><br><div class="question">
<p>This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius \({\text{OB}} = 9{\text{ cm}}\) and four equal sectors of a smaller circle of radius \({\text{OA}} = 3{\text{ cm}}\).<br>The angle \({\text{BOC}} = \) 20°.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-08_om_11.16.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/03"></p>
<p>Find the area of the pendant.</p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A ship, S, is 10 km north of a motorboat, M, at 12.00pm. The ship is travelling northeast with a constant velocity of \(20{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\). The motorboat wishes to intercept the ship and it moves with a constant velocity of \(30{\text{ km}}\,{\text{h}}{{\text{r}}^{ - 1}}\) in a direction \(\theta \) degrees east of north. In order for the interception to take place, determine</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the value of \(\theta \).<br></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the time at which the interception occurs, correct to the nearest minute.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A straight street of width 20 metres is bounded on its parallel sides by two vertical walls, one of height 13 metres, the other of height 8 metres. The intensity of light at point P at ground level on the street is proportional to the angle \(\theta \) where \(\theta = {\rm{A\hat PB}}\), as shown in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \(\theta \) in terms of <em>x</em>, where <em>x</em> is the distance of P from the base of the wall of height 8 m.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the value of \(\theta \) when <em>x</em> = 0.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the value of \(\theta \) when <em>x</em> = 20.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(\theta \), for \(0 \leqslant x \leqslant 20\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{{5(744 - 64x - {x^2})}}{{({x^2} + 64)({x^2} - 40x + 569)}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the result in part (d), or otherwise, determine the value of <em>x</em> corresponding to the maximum light intensity at P. Give your answer to four significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point P moves across the street with speed \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\). Determine the rate of change of \(\theta \) with respect to time when P is at the midpoint of the street.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The interior of a circle of radius 2 cm is divided into an infinite number of sectors. The areas of these sectors form a geometric sequence with common ratio <em>k</em>. The angle of the first sector is \(\theta \) radians.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(\theta = 2\pi (1 - k)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The perimeter of the third sector is half the perimeter of the first sector.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>k</em> and of \(\theta \).</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">The diagram below shows a fenced triangular enclosure in the middle of a large grassy field. The points A and C are 3 m <span class="s1">apart. A goat \(G\) </span>is tied by a 5 m length of rope at point A on the outside edge of the enclosure.</p>
<p class="p1">Given that the corner of the enclosure at C <span class="s1">forms an angle of \(\theta \) </span>radians and the area of field that can be reached by the goat is \({\text{44 }}{{\text{m}}^{\text{2}}}\)<span class="s1">, find the value of \(\theta \).</span></p>
<p class="p1"><span class="s1"><img src="images/Schermafbeelding_2017-01-26_om_08.36.38.png" alt="M16/5/MATHL/HP2/ENG/TZ1/04"></span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The radius of the circle with centre C is 7 cm and the radius of the circle with centre D is 5 cm. If the length of the chord [AB] is 9 cm, find the area of the shaded region enclosed by the two arcs AB.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Compactness is a measure of how compact an enclosed region is.</p>
<p class="p1">The compactness, <em>\(C\) </em>, of an enclosed region can be defined by \(C = \frac{{4A}}{{\pi {d^2}}}\), where <em>\(A\) </em>is the area of the region and <em>\(d\) </em>is the maximum distance between any two points in the region.</p>
<p class="p1">For a circular region, \(C = 1\).</p>
<p class="p1">Consider a regular polygon of <em>\(n\) </em>sides constructed such that its vertices lie on the circumference of a circle of diameter <em>\(x\) </em>units.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(n > 2\) and even, show that \(C = \frac{n}{{2\pi }}\sin \frac{{2\pi }}{n}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).</p>
<p class="p1">Find the regular polygon with the least number of sides for which the compactness is more than \(0.99\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).</p>
<p class="p1">Comment briefly on whether <em>C </em>is a good measure of compactness.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Two non-intersecting circles C<sub>1</sub> , containing points M and S , and C<sub>2</sub> , containing points N and R, have centres P and Q where PQ \( = 50\) . The line segments [MN] and [SR] are common tangents to the circles. The size of the reflex angle MPS is \( \alpha\), the size of the obtuse angle NQR is \( \beta\) , and the size of the angle MPQ is \( \theta\) . The arc length MS is \({l_1}\) and the arc length NR is \({l_2}\) . This information is represented in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The radius of C<sub>1</sub> is \(x\) , where \(x \geqslant 10\) and the radius of C<sub>2</sub> is \(10\).</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Explain why \(x < 40\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Show that cosθ = x −10 </span><span style="font-family: times new roman,times; font-size: medium;">50</span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) (i) Find an expression for MN in terms of \(x\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the value of \(x\) that maximises MN.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) Find an expression in terms of \(x\) for</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) \( \alpha\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \( \beta\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) The length of the perimeter is given by \({l_1} + {l_2} + {\text{MN}} + {\text{SR}}\).</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Find an expression, \(b (x)\) , for the length of the perimeter in terms of \(x\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the maximum value of the length of the perimeter.</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Find the value of \(x\) that gives a perimeter of length \(200\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider a triangle ABC with \({\rm{B\hat AC}} = 45.7^\circ \) , AB = 9.63 cm and BC = 7.5 cm .<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By drawing a diagram, show why there are two triangles consistent with this information.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the possible values of AC .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;">Points A, B and C are on the circumference of a circle, centre O and radius \(r\) . </span><span style="font-family: times new roman,times; font-size: medium;">A trapezium OABC is formed such that AB is parallel to OC, and the angle \({\rm{A}}\hat {\text{O}}{\text{C}}\) </span><span style="font-family: times new roman,times; font-size: medium;">is \(\theta\) , \(\frac{\pi }{2} \leqslant \theta \leqslant \pi \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that angle \({\rm{B\hat OC}}\) is \(\pi - \theta \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that the area, <em>T</em>, of the trapezium can be expressed as</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[T = \frac{1}{2}{r^2}\sin \theta - \frac{1}{2}{r^2}\sin 2\theta .\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Show that when the area is maximum, the value of \(\theta \) satisfies</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\cos \theta = 2\cos 2\theta .\]</span></p>
<p style="margin: 0px 0px 0px 30px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) <strong>Hence</strong> determine the maximum area of the trapezium when <em>r</em> = 1.</span></p>
<p style="margin: 0px 0px 0px 60px; font: 26px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (Note: It is not required to prove that it is a maximum.)</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows the plan of an art gallery <em>a </em>metres wide. [AB] represents a doorway, leading to an exit corridor <em>b </em>metres wide. In order to remove a painting from the art gallery, CD (denoted by <em>L </em>) is measured for various values of \(\alpha \) , as represented in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">If </span><span style="font: 12.5px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\alpha \)</span> </span><span style="font-family: 'times new roman', times; font-size: medium;">is the angle between [CD] and the wall, show that \(L = \frac{a }{{\sin \alpha }} + \frac{b}{{\cos \alpha }}{\text{, }}0 < \alpha < \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: arial, helvetica, sans-serif;"><span style="font-family: 'times new roman', times; font-size: medium;">If <em>a </em>= 5 and <em>b </em>= 1, find the maximum length of a painting that can be removed through this doorway.</span><br></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\frac{{{\text{d}}L}}{{{\text{d}}\alpha }}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find, in terms of <em>k </em>, the maximum length of a painting that can be removed from the gallery through this doorway.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>a </em>= 3<em>k </em>and <em>b </em>= <em>k </em>.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the minimum value of <em>k </em>if a painting 8 metres long is to be removed through this doorway.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two discs, one of radius 8 cm and one of radius 5 cm, are placed such that they touch each other. A piece of string is wrapped around the discs. This is shown in the diagram below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the length of string needed to go around the discs.</span></p>
</div>
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