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</div><h2>HL Paper 1</h2><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that the points (0, 0) and (\(\sqrt {2\pi } \) , \( - \sqrt {2\pi } \) ) on the curve \({{\text{e}}^{\left( {x + y} \right)}} = \cos \left( {xy} \right)\) have a</span> <span style="font-family: times new roman,times; font-size: medium;">common tangent.</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;">Let \(f(x) = \sqrt {\frac{x}{{1 - x}}} ,{\text{ }}0 < x < 1\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f'(x) = \frac{1}{2}{x^{ - \frac{1}{2}}}{(1 - x)^{ - \frac{3}{2}}}\) and deduce that <em>f </em>is an increasing function.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the curve \(y = f(x)\) has one point of inflexion, and find its coordinates.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p> <span style="font-family: 'times new roman', times; font-size: medium;">Use the substitution \(x = {\sin ^2}\theta \) to show that \(\int {f(x){\text{d}}x} = \arcsin \sqrt x - \sqrt {x - {x^2}} + c\) .</span></p>
<p> </p>
<div class="marks">[11]</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The first set of axes below shows the graph of \(y = {\text{ }}f(x)\) for \( - 4 \leqslant x \leqslant 4\).</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_2014-09-11_om_12.03.04.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;">Let \(g(x) = \int_{ - 4}^x {f(t){\text{d}}t} \) for \( - 4 \leqslant x \leqslant 4\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State the value of <em>x </em>at which \(g(x)\) is a minimum.</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) On the second set of axes, sketch the graph of \(y = g(x)\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = {{\text{e}}^{{x^2} - 2x - 1.5}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(f'(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) You are given that \(y = \frac{{f(x)}}{{x - 1}}\) has a local minimum at <em>x</em> = <em>a</em>, <em>a</em> > 1. Find the</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">value of <em>a</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of \(y = \frac{{{{(\ln x)}^2}}}{x},{\text{ }}x > 0\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_06.37.32.png" alt="M16/5/MATHL/HP1/ENG/TZ1/13"></p>
</div>
<div class="specification">
<p class="p1">The region \(R\) is enclosed by the curve, the \(x\)-axis and the line \(x = e\).</p>
</div>
<div class="specification">
<p class="p1">Let \({I_n} = \int_1^{\text{e}} {\frac{{{{(\ln x)}^n}}}{{{x^2}}}{\text{d}}x,{\text{ }}n \in \mathbb{N}} \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that the curve passes through the point \((a,{\text{ }}0)\)<span class="s1">, state the value of \(a\).</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 class="p1">Use the substitution \(u = \ln x\) to find the area of the region \(R\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \({I_0}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Prove that \({I_n} = \frac{1}{{\text{e}}} + n{I_{n - 1}},{\text{ }}n \in {\mathbb{Z}^ + }\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Hence find the value of \({I_1}\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the volume of the solid formed when the region \(R\) <span class="s1">is rotated through \(2\pi \) </span>about the \(x\)-axis.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions \(f,\,\,g,\) defined for \(x \in \mathbb{R}\), given by \(f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x\) and \(g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f'\left( x \right)\).</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 \(g'\left( x \right)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find \(\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} \).</p>
<div class="marks">[4]</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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A drinking glass is modelled by rotating the graph of \(y = {{\text{e}}^x}\) about the <em>y</em>-axis, for \(1 \leqslant y \leqslant 5\) . Find the volume of the glass.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A curve is defined by \(xy = {y^2} + 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that there is no point where the tangent to the curve is horizontal.</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">Find the coordinates of the points where the tangent to the curve is vertical.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function defined by \(f(x) = x\sqrt {1 - {x^2}} \) <span class="s1">on the domain \( - 1 \le x \le 1\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(f\) is an odd function.</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 \(f'(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the \(x\)-coordinates of any local maximum or minimum points.</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 class="p1">Find the range of \(f\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch the graph of \(y = f(x)\) indicating clearly the coordinates of the \(x\)-intercepts and any local maximum or minimum points.</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 class="p1">Find the area of the region enclosed by the graph of \(y = f(x)\) and the \(x\)-axis for \(x \ge 0\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\int_{ - 1}^1 {\left| {x\sqrt {1 - {x^2}} } \right|{\text{d}}x > \left| {\int_{ - 1}^1 {x\sqrt {1 - {x^2}} {\text{d}}x} } \right|} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(y(x) = x{e^{3x}},{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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">Prove by induction that \(\frac{{{{\text{d}}^n}y}}{{{\text{d}}{x^n}}} = n{3^{n - 1}}{{\text{e}}^{3x}} + x{3^n}{{\text{e}}^{3x}}\) for \(n \in {\mathbb{Z}^ + }\).</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 class="p1">Find the coordinates of any local maximum and minimum points on the graph of \(y(x)\).</p>
<p class="p1">Justify whether any such point is a maximum or a minimum.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of any points of inflexion on the graph of \(y(x)\). Justify whether any such point is a point of inflexion.</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 sketch the graph of \(y(x)\), indicating clearly the points found in parts (c) and (d) and any intercepts with the axes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Using integration by parts find \(\int {x\sin x{\text{d}}x} \).</p>
</div>
<br><hr><br><div class="question">
<p class="p1">By using the substitution \(u = {{\text{e}}^x} + 3\), find \(\int {\frac{{{{\text{e}}^x}}}{{{{\text{e}}^{2x}} + 6{{\text{e}}^x} + 13}}{\text{d}}x} \).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function defined by \(f(x) = {x^3} - 3{x^2} + 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the values of \(x\) for which \(f(x)\) is a decreasing function.</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">There is a point of inflexion, \(P\)<span class="s1">, on the curve \(y = f(x)\)</span>.</p>
<p class="p1">Find the coordinates of \(P\)<span class="s1">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\sin \left( {\theta + \frac{\pi }{2}} \right) = \cos \theta \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Consider \(f(x) = \sin (ax)\) where \(a\) is a constant. Prove by mathematical induction that \({f^{(n)}}(x) = {a^n}\sin \left( {ax + \frac{{n\pi }}{2}} \right)\) where \(n \in {\mathbb{Z}^ + }\) and \({f^{(n)}}(x)\) represents the \({{\text{n}}^{{\text{th}}}}\) derivative of \(f(x)\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<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;">Find the value of the integral \(\int_0^{\sqrt 2 } {\sqrt {4 - {x^2}} {\text{d}}x} \) .</span></p>
<div class="marks">[7]</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of the integral \(\int_0^{0.5} {\arcsin x {\text{d}}x} \) .</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the substitution \(t = \tan \theta \) , find the value of the integral</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;">\[\int_0^{\frac{\pi }{4}} {\frac{{{\text{d}}\theta }}{{3{{\cos }^2}\theta + {{\sin }^2}\theta }}} {\text{ }}.\]</span></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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of the function \(f(x) = \frac{{x + 1}}{{{x^2} + 1}}\) is shown below.</span></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_2014-09-15_om_08.53.43.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The point (1, 1) is a point of inflexion. There are two other points of inflexion.</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 Helvetica;"><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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the \(x\)-coordinates of the points where the gradient of the graph of \(f\) is zero.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f''(x)\) expressing your answer in the form \(\frac{{p(x)}}{{{{({x^2} + 1)}^3}}}\), where \(p(x)\) is a polynomial of degree 3.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-coordinates of the other two points of inflexion.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 style="background-color: #f7f7f7;">Find the area of the shaded region. Express your answer in the form \(\frac{\pi }{a} - \ln \sqrt b \), where </span>\(a\) <span style="background-color: #f7f7f7;">and </span>\(b\) <span style="background-color: #f7f7f7;">are integers.</span></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the complex number \(z = \cos \theta + {\text{i}}\sin \theta \).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The region <em>S</em> is bounded by the curve \(y = \sin x{\cos ^2}x\) and the <em>x</em>-axis between \(x = 0\) and \(x = \frac{\pi }{2}\).</span></p>
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<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;">Use De Moivre’s theorem to show that \({z^n} + {z^{ - n}} = 2\cos n\theta ,{\text{ }}n \in {\mathbb{Z}^ + }\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Expand \({\left( {z + {z^{ - 1}}} \right)^4}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence show that \({\cos ^4}\theta = p\cos 4\theta + q\cos 2\theta + r\), where \(p,{\text{ }}q\) and \(r\) are constants to </span><span style="font-family: 'times new roman', times; font-size: medium;">be determined.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({\cos ^6}\theta = \frac{1}{{32}}\cos 6\theta + \frac{3}{{16}}\cos 4\theta + \frac{{15}}{{32}}\cos 2\theta + \frac{5}{{16}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the value of \(\int_0^{\frac{\pi }{2}} {{{\cos }^6}\theta {\text{d}}\theta } \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
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<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;"><em>S</em> is rotated through \(2\pi \) radians about the <em>x</em>-axis. Find the value of the volume generated.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
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<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 an expression for the constant term in the expansion of \({\left( {z + {z^{ - 1}}} \right)^{2k}}\), \(k \in {\mathbb{Z}^ + }\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence determine an expression for \(\int_0^{\frac{\pi }{2}} {{{\cos }^{2k}}\theta {\text{d}}\theta } \) in terms of <em>k</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
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<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 graphs of \(f(x) = - {x^2} + 2\) and \(g(x) = {x^3} - {x^2} - bx + 2,{\text{ }}b > 0\), intersect and create two closed regions. Show that these two regions have equal areas.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
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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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the part of the curve \(4{x^2} + {y^2} = 4\) shown in the diagram below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find an expression for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) in terms of <em>x</em> and <em>y</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the gradient of the tangent at the point \(\left( {\frac{2}{{\sqrt 5 }},\frac{2}{{\sqrt 5 }}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) A bowl is formed by rotating this curve through \(2\pi \) radians about the <em>x</em>-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the volume of this bowl.</span></p>
</div>
<br><hr><br><div class="specification">
<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 curve C with equation \(y = f(x)\) satisfies the differential equation</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{{\ln y}}(x + 2),{\text{ }}y > 1,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">and <em>y</em> = e when <em>x</em> = 2.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the tangent to C at the point (2, e).</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f(x)\).</span></p>
<div class="marks">[11]</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: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the largest possible domain of <em>f</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the equation \(f(x) = f'(x)\) has no solution.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<address>Find the area enclosed by the curve \(y = \arctan x\) , the x-axis and the line \(x = \sqrt 3 \) .</address>
</div>
<br><hr><br><div class="specification">
<p class="p1">A curve has equation \(3x - 2{y^2}{{\text{e}}^{x - 1}} = 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find an expression for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) </span>in terms of \(x\) and \(y\).</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 equations of the tangents to this curve at the points where the curve intersects <span class="s1">the line \(x = 1\).</span></p>
<div class="marks">[4]</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;">A curve has equation \(\arctan {x^2} + \arctan {y^2} = \frac{\pi }{4}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) in terms of <em>x </em>and <em>y</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the gradient of the curve at the point where \(x = \frac{1}{{\sqrt 2 }}\) and \(y < 0\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({x^3}y = a\sin nx\)<em> </em>. Using implicit differentiation, show that</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;">\[{x^3}\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} + 6{x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + ({n^2}{x^2} + 6)xy = 0\] .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(y = {{\text{e}}^x}\sin x\).</p>
</div>
<div class="specification">
<p class="p1">Consider the function \(f\)<span class="Apple-converted-space"> </span>defined by \(f(x) = {{\text{e}}^x}\sin x,{\text{ }}0 \leqslant x \leqslant \pi \).</p>
</div>
<div class="specification">
<p class="p1">The curvature at any point \((x,{\text{ }}y)\) on a graph is defined as \(\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^x}\cos x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Show that the function \(f\) </span>has a local maximum value when \(x = \frac{{3\pi }}{4}\).</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">Find the \(x\)</span>-coordinate of the point of inflexion of the graph of <span class="s1">\(f\).</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 class="p1"><span class="s1">Sketch the graph of \(f\)</span>, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.</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 class="p1">Find the area of the region enclosed by the graph of \(f\) <span class="s1">and the </span>\(x\)<span class="s1">-axis.</span></p>
<p class="p2">The curvature at any point \((x,{\text{ }}y)\) on a graph is defined as \(\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of the curvature of the graph of \(f\) <span class="s1">at the local maximum point.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value \(\kappa \) for \(x = \frac{\pi }{2}\) and comment on its meaning with respect to the shape of the graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows a circular lake with centre O, diameter AB and radius 2 km.</span></p>
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<p style="font: normal normal normal 29px/normal Helvetica; text-align: center; margin: 0px;"><img 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alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Jorg needs to get from A to B as quickly as possible. He considers rowing to point P and then walking to point B. He can row at \(3{\text{ km}}\,{{\text{h}}^{ - 1}}\) and walk at \(6{\text{ km}}\,{{\text{h}}^{ - 1}}\). Let \({\rm{P\hat AB}} = \theta \) radians, and <em>t</em> be the time in hours taken by Jorg to travel from A to B.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(t = \frac{2}{3}(2\cos \theta + \theta )\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\theta \) for which \(\frac{{{\text{d}}t}}{{{\text{d}}\theta }} = 0\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">What route should Jorg take to travel from A to B in the least amount of time?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Give reasons for your answer.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the definition of a derivative as \(f'(x) = \mathop {\lim }\limits_{h \to 0} \left( {\frac{{f(x + h) - f(x)}}{h}} \right)\) , show that the derivative of \(\frac{1}{{2x + 1}}{\text{ is }}\frac{{ - 2}}{{{{(2x + 1)}^2}}}\).</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;">Prove by induction that the \({n^{{\text{th}}}}\) derivative of \({(2x + 1)^{ - 1}}\) is \({( - 1)^n}\frac{{{2^n}n!}}{{{{(2x + 1)}^{n + 1}}}}\).</span></p>
<div class="marks">[9]</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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the exact value of \(\int_1^{\text{e}} {{x^2}\ln x{\text{d}}x} \) .</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ \begin{array}{r}1 - 2x,\\{\textstyle{3 \over 4}}{(x - 2)^2} - 3,\end{array} \right.\begin{array}{*{20}{c}}{x \le 2}\\{x > 2}\end{array}\]</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;">Determine whether or not \(f\)is continuous.</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;">The graph of the function \(g\) is obtained by applying the following transformations to the graph of \(f\):</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;">a reflection in the \(y\)–axis followed by a translation by the vector \(\left( \begin{array}{l}2\\0\end{array} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(g(x)\).</span></p>
<div class="marks">[4]</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined, for \( - \frac{\pi }{2} \leqslant x \leqslant \frac{\pi }{2}\) , by \(f(x) = 2\cos x + x\sin x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether <em>f</em> is even, odd or neither even nor odd.</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f''(0) = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">John states that, because \(f''(0) = 0\) , the graph of <em>f</em> has a point of inflexion at the point (0, 2) . Explain briefly whether John’s statement is correct or not.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A window is made in the shape of a rectangle with a semicircle of radius \(r\) metres on top, as shown in the diagram. The perimeter of the window is a constant P metres.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-08_om_17.46.34.png" alt="M17/5/MATHL/HP1/ENG/TZ2/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the window in terms of P and \(r\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the width of the window in terms of P when the area is a maximum, justifying that this is a maximum.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that in this case the height of the rectangle is equal to the radius of the semicircle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A tranquilizer is injected into a muscle from which it enters the bloodstream.</p>
<p class="p1">The concentration \(C\) in \({\text{mg}}{{\text{l}}^{ - 1}}\), of tranquilizer in the bloodstream can be modelled by the function \(C(t) = \frac{{2t}}{{3 + {t^2}}},{\text{ }}t \ge 0\) where \(t\) is the number of minutes after the injection.</p>
<p class="p1">Find the maximum concentration of tranquilizer in the bloodstream.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\cot \alpha = \tan \left( {\frac{\pi }{2} - \alpha } \right)\) for \(0 < \alpha < \frac{\pi }{2}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find \(\int_{\tan \alpha }^{\cot \alpha } {\frac{1}{{1 + {x^2}}}{\text{d}}x,{\text{ }}0 < \alpha < \frac{\pi }{2}} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Given that \(\int_{ - 2}^2 {f\left( x \right){\text{d}}x = 10} \) and \(\int_0^2 {f\left( x \right){\text{d}}x = 12} \), find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>\(\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x} \).</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>\(\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1"><span class="s1">By using the substitution \(t = \tan x\)</span>, find \(\int {\frac{{{\text{d}}x}}{{1 + {{\sin }^2}x}}} \)<span class="s1">.</span></p>
<p class="p2">Express your answer in the form \(m\arctan (n\tan x) + c\), where \(m\), \(n\) are constants to be determined.</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;">Find the equation of the normal to the curve \(5x{y^2} - 2{x^2} = 18\) at the point (1, 2) .</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;">Consider the function \(f(x) = \frac{{\ln x}}{x},{\text{ }}x > 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The sketch below shows the graph of \(y = {\text{ }}f(x)\) and its tangent at a point A.</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_2014-09-11_om_14.26.30.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 \(f'(x) = \frac{{1 - \ln x}}{{{x^2}}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of B, at which the curve reaches its maximum value.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of C, the point of inflexion on the curve.</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(y = {\text{ }}f(x)\) crosses the \(x\)-axis at the point A.</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 the equation of the tangent to the graph of \(f\) at the point A.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(y = {\text{ }}f(x)\) crosses the \(x\)-axis at the point A.</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 the area enclosed by the curve \(y = f(x)\), the tangent at A, and the line \(x = {\text{e}}\).</span></p>
<div class="marks">[7]</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed between the curves \(y = \sqrt x {{\text{e}}^x}\) and \(y = {\text{e}}\sqrt x \) is rotated through \(2\pi \) about the <em>x</em>-axis. Find the volume of the solid obtained.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve <em>C</em> is given by \(y = \frac{{x\cos x}}{{x + \cos x}}\), for \(x \geqslant 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{{\cos }^2}x - {x^2}\sin x}}{{{{(x + \cos x)}^2}}},{\text{ }}x \geqslant 0\).</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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the tangent to <em>C</em> at the point \(\left( {\frac{\pi }{2},0} \right)\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\int {(1 + {{\tan }^2}x){\text{d}}x} \).</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 \(\int {{{\sin }^2}x{\text{d}}x} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">By using the substitution \(u = 1 + \sqrt x \), find \(\int {\frac{{\sqrt x }}{{1 + \sqrt x }}{\text{d}}x} \).</p>
</div>
<br><hr><br><div class="specification">
<p>Let \(y = {\text{si}}{{\text{n}}^2}\theta ,\,\,0 \leqslant \theta \leqslant \pi \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{{\text{d}}y}}{{{\text{d}}\theta }}\)</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>Hence find the values of <em>θ</em> for which \(\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2y\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle moves along a straight line. Its displacement, \(s\) metres, at time \(t\) seconds is given by \(s = t + \cos 2t,{\text{ }}t \geqslant 0\). The first two times when the particle is at rest are denoted by \({t_1}\) and \({t_2}\), where \({t_1} < {t_2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({t_1}\) and \({t_2}\).</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>Find the displacement of the particle when \(t = {t_1}\)</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the substitution \(x = \tan \theta \) show that \(\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x = } \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta } \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of \(\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Use the substitution \(u = \ln x\) to find the value of \(\int_{\text{e}}^{{{\text{e}}^2}} {\frac{{{\text{d}}x}}{{x\ln x}}} \).</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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the identity \(\cos 2\theta = 2{\cos ^2}\theta - 1\) to prove that \(\cos \frac{1}{2}x = \sqrt {\frac{{1 + \cos x}}{2}} ,{\text{ }}0 \leqslant x \leqslant \pi \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find a similar expression for \(\sin \frac{1}{2}x,{\text{ }}0 \leqslant x \leqslant \pi \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the value of \(\int_0^{\frac{\pi }{2}} {\left( {\sqrt {1 + \cos x} + \sqrt {1 - \cos x} } \right){\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<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;">Paint is poured into a tray where it forms a circular pool with a uniform thickness of 0.5 cm. If the paint is poured at a constant rate of \(4{\text{ c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\), find the rate of increase of the radius of the circle when the radius is 20 cm.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider two functions \(f\) and \(g\) and their derivatives \(f'\) and \(g'\). The following table shows the values for the two functions and their derivatives at \(x = 1\), \(2\) and \(3\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-07_om_15.01.17.png" alt></p>
<p class="p1">Given that \(p(x) = f(x)g(x)\) and \(h(x) = g \circ f(x)\), find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(p'(3)\);</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">\(h'(2)\).</p>
<div class="marks">[4]</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;">The region bounded by the curve \(y = \frac{{\ln (x)}}{x}\) and the lines <em>x</em> = 1, <em>x</em> = <em>e</em>, <em>y</em> = 0 is rotated through \(2\pi \) radians about the <em>x</em>-axis.</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 volume of the solid generated.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the curve \(y = \frac{1}{{1 - x}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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">Determine the equation of the normal to the curve at the point \(x = 3\) in the form \(ax + by + c = 0\) where \(a,{\text{ }}b,{\text{ }}c \in \mathbb{Z}\).</p>
<div class="marks">[4]</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f\) is given by \(f(x) = x{{\text{e}}^{ - x}}{\text{ }}(x \geqslant 0)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find an expression for \(f'(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence determine the coordinates of the point A, where \(f'(x) = 0\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \(f''(x)\) and hence show the point A is a maximum.</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 coordinates of B, the point of inflexion.</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: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of the function \(g\) is obtained from the graph of \(f\) by stretching it in the <em>x</em>-direction by a scale factor 2.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Write down an expression for \(g(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) State the coordinates of the maximum C of \(g\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Determine the <em>x</em>-coordinates of D and E, the two points where \(f(x) = g(x)\).</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><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graphs of \(y = f(x)\) and \(y = g(x)\) on the same axes, showing clearly the points A, B, C, D and E.</span></p>
<div class="marks">[4]</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find an exact value for the area of the region bounded by the curve \(y = g(x)\), the <em>x</em>-axis and the line \(x = 1\).</span></p>
<div class="marks">[3]</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;">If \(f(x) = x - 3{x^{\frac{2}{3}}},{\text{ }}x > 0\) ,</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) find the <em>x</em>-coordinate of the point P where \(f'(x) = 0\) ;</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) determine whether P is a maximum or minimum point.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The normal to the curve \(x{{\text{e}}^{ - y}} + {{\text{e}}^y} = 1 + x\), at the point (<em>c</em>, \(\ln c\)), has a <em>y</em>-intercept \({c^2} + 1\).</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;">Determine the value of <em>c</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve <em>C</em> is given implicitly by the equation \(\frac{{{x^2}}}{y} - 2x = \ln y\) for \(y > 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Express \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) in terms of <em>x</em> and <em>y</em>.</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) at the point on <em>C</em> where <em>y</em> = 1 and \(x > 0\).</span></p>
<div class="marks">[2]</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows the graph of the function defined by \(y = x{(\ln x)^2}{\text{ for }}x > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><br><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function has a local maximum at the point A and a local minimum at the point B.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the points A and B.</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 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;">Given that the graph of the function has exactly one point of inflexion, find its coordinates.</span></p>
<div class="marks">[3]</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following functions:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(h(x) = \arctan (x),{\text{ }}x \in \mathbb{R}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(g(x) = \frac{1}{x}\), \(x\in \mathbb{R}\)</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">, \({\text{ }}x \ne 0\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = h(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for the composite function \(h \circ g(x)\) and state its domain.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f(x) = h(x) + h \circ g(x)\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) find \(f'(x)\) in simplified form;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) show that \(f(x) = \frac{\pi }{2}\) for \(x > 0\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Nigel states that \(f\) is an odd function and Tom argues that \(f\) is an even function.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State who is correct and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence find the value of \(f(x)\) for \(x < 0\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = {{\text{e}}^x}\sin x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f''(x) = 2{{\text{e}}^x}\sin \left( {x + \frac{\pi }{2}} \right)\) .</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Obtain a similar expression for \({f^{(4)}}(x)\) .</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Suggest an expression for \({f^{(2n)}}(x)\), \(n \in {\mathbb{Z}^ + }\), and prove your conjecture using mathematical induction.</span></p>
<div class="marks">[8]</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = x{{\text{e}}^{2x}}\) .</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;">It can be shown that \({f^{(n)}}(x) = ({2^n}x + n{2^{n - 1}}){{\text{e}}^{2x}}\) for all \(n \in {\mathbb{Z}^ + }\), where \({f^{(n)}}(x)\) represents the \({n^{{\text{th}}}}\) derivative of \(f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) By considering \({f^{(n)}}(x){\text{ for }}n = 1{\text{ and }}n = 2\) , show that there is one minimum point P on the graph of <em>f</em> , and find the coordinates of P.</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 <em>f</em> has a point of inflexion Q at <em>x</em> = −1.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Determine the intervals on the domain of <em>f</em> where <em>f</em> is</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;">(i) concave up;</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) concave down.</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;">(d) Sketch <em>f</em> , clearly showing any intercepts, asymptotes and the points P and Q.</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;">(e) Use mathematical induction to prove that \({f^{(n)}}(x) = ({2^n}x + n{2^{n - 1}}){{\text{e}}^{2x}}{\text{ for all }}n \in {\mathbb{Z}^ + },{\text{ where }}{f^{(n)}}{\text{ represents the }}{n^{{\text{th}}}}{\text{ derivative of }}f(x)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \({f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether \({f_n}\) is an odd or even function, justifying your answer.</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>By using mathematical induction, prove that</p>
<p style="text-align: center;">\({f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}\) where \(m \in \mathbb{Z}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find an expression for the derivative of \({f_n}(x)\) with respect to \(x\).</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>Show that, for \(n > 1\), the equation of the tangent to the curve \(y = {f_n}(x)\) at \(x = \frac{\pi }{4}\) is \(4x - 2y - \pi = 0\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the functions \(f(x) = \tan x,{\text{ }}0 \le \ x < \frac{\pi }{2}\) and \(g(x) = \frac{{x + 1}}{{x - 1}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \(g \circ f(x)\), stating its domain.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence show that \(g \circ f(x) = \frac{{\sin x + \cos x}}{{\sin x - \cos x}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(y = g \circ f(x)\)<span class="s1">, find an exact value for \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) </span>at the point on the graph of \(y = g \circ f(x)\) where \(x = \frac{\pi }{6}\), expressing your answer in the form \(a + b\sqrt 3 ,{\text{ }}a,{\text{ }}b \in \mathbb{Z}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area bounded by the graph of \(y = g \circ f(x)\), the \(x\)-axis and the lines \(x = 0\) and \(x = \frac{\pi }{6}\) is \(\ln \left( {1 + \sqrt 3 } \right)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find \(\int {\arcsin x\,{\text{d}}x} \)</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\int_0^{\frac{\pi }{6}} {x\sin 2x{\text{d}}x = \frac{{\sqrt 3 }}{8} - \frac{\pi }{{24}}} \).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A function \(f\) is defined by \(f(x) = \frac{{3x - 2}}{{2x - 1}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne \frac{1}{2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({f^{ - 1}}(x)\).</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 class="p1">Given that \(f(x)\) can be written in the form \(f(x) = A + \frac{B}{{2x - 1}}\), find the values of the constants \(A\) and \(B\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, write down \(\int {\frac{{3x - 2}}{{2x - 1}}} {\text{d}}x\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The function \(f\) is defined as \(f(x) = a{x^2} + bx + c\) where \(a,{\text{ }}b,{\text{ }}c \in \mathbb{R}\).</p>
<p class="p1">Hayley conjectures that \(\frac{{f({x_2}) - f({x_1})}}{{{x_2} - {x_1}}} = \frac{{f'({x_2}) + f'({x_1})}}{2},{\text{ }}x1 \ne x2\).</p>
<p class="p1">Show that Hayley’s conjecture is correct.</p>
</div>
<br><hr><br><div class="question">
<p class="p1">Find the \(x\)-coordinates of all the points on the curve \(y = 2{x^4} + 6{x^3} + \frac{7}{2}{x^2} - 5x + \frac{3}{2}\) <span class="s1">at which</span></p>
<p class="p2">the tangent to the curve is parallel to the tangent at \(( - 1,{\text{ }}6)\).</p>
</div>
<br><hr><br><div class="question">
<p>Consider the curve \(y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}\).</p>
<p>Find the <em>x</em>-coordinates of the points on the curve where the gradient is zero.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use the substitution \(x = a\sec \theta \) to show that \(\int_{a\sqrt 2 }^{2a} {\frac{{{\text{d}}x}}{{{x^3}\sqrt {{x^2} - {a^2}} }} = \frac{1}{{24{a^3}}}\left( {3\sqrt 3 + \pi - 6} \right)} \).</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate \(\int_{\frac{\pi }{4}}^{\frac{\pi }{3}} {\frac{{{{\sec }^2}x}}{{\sqrt[3]{{\tan x}}}}{\text{d}}x} \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {{{\tan }^3}x{\text{d}}x} \) .</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\int_0^1 {t\ln (t + 1){\text{d}}t} \).</span></p>
</div>
<br><hr><br><div class="specification">
<p>Let \(y = {\text{arccos}}\left( {\frac{x}{2}} \right)\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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 \(\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x} \).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \({x^2} + 3x + 2\) in the form \({(x + h)^2} + k\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Factorize \({x^2} + 3x + 2\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(f(x)\), indicating on it the equations of the asymptotes, the coordinates of the \(y\)-intercept and the local maximum.</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>Show that \(\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of \(p\) if \(\int_0^1 {f(x){\text{d}}x = \ln (p)} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f\left( {\left| x \right|} \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area of the region enclosed between the graph of \(y = f\left( {\left| x \right|} \right)\), the \(x\)-axis and the lines with equations \(x = - 1\) and \(x = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p>A particle moves in a straight line such that at time \(t\) seconds \((t \geqslant 0)\), its velocity \(v\), in \({\text{m}}{{\text{s}}^{ - 1}}\), is given by \(v = 10t{{\text{e}}^{ - 2t}}\). Find the exact distance travelled by the particle in the first half-second.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(\frac{3}{{x + 1}} + \frac{2}{{x + 3}} = \frac{{5x + 11}}{{{x^2} + 4x + 3}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence find the value of <em>k</em> such that \(\int_0^2 {\frac{{5x + 11}}{{{x^2} + 4x + 3}}{\text{d}}x = \ln k} \) .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution \(u = {x^{\frac{1}{2}}}\) to find \(\int {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} \).</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>Hence find the value of \(\frac{1}{2}\int\limits_1^9 {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} \), expressing your answer in the form arctan \(q\), where \(q \in \mathbb{Q}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find all values of <em>x</em> for \(0.1 \leqslant x \leqslant 1\) such that \(\sin (\pi {x^{ - 1}}) = 0\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_{\frac{1}{{n + 1}}}^{\frac{1}{n}} {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}}){\text{d}}x} \), showing that it takes different integer values when <em>n</em> is even and when <em>n</em> is odd.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Evaluate \(\int_{0.1}^1 {\left| {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}})} \right|{\text{d}}x} \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>The folium of Descartes is a curve defined by the equation \({x^3} + {y^3} - 3xy = 0\), shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_18.23.15.png" alt="N17/5/MATHL/HP1/ENG/TZ0/07"></p>
<p>Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the \(y\)-axis.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A body is moving in a straight line. When it is \(s\) metres from a fixed point O on the line its velocity, \(v\), is given by \(v = - \frac{1}{{{s^2}}},{\text{ }}s > 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the acceleration of the body when it is 50 cm from O.</span></p>
</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 curve \(y = x{{\text{e}}^x}\) and the line \(y = kx,{\text{ }}k \in \mathbb{R}\).</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) Let <em>k</em> = 0.</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;">(i) Show that the curve and the line intersect once.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the angle between the tangent to the curve and the line at the point of intersection.</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) Let <em>k</em> =1. Show that the line is a tangent to the curve.</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;">(c) (i) Find the values of <em>k</em> for which the curve \(y = x{{\text{e}}^x}\) and the line \(y = kx\) meet in two distinct points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down the coordinates of the points of intersection.</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;">(iii) Write down an integral representing the area of the region <em>A</em> enclosed by the curve and the line.</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;">(iv) <strong>Hence</strong>, given that \(0 < k < 1\), show that \(A < 1\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following graph shows the relation \(x = 3\cos 2y + 4,{\text{ }}0 \leqslant y \leqslant \pi \).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-27_om_09.08.34.png" alt="M16/5/MATHL/HP1/ENG/TZ2/11"></p>
<p class="p1"><span class="s1">The curve is rotated 360° </span>about the \(y\)-axis to form a volume of revolution.</p>
</div>
<div class="specification">
<p class="p1">A container with this shape is made with a solid base of diameter 14 cm . The container is filled with water at a rate of \({\text{2 c}}{{\text{m}}^{\text{3}}}\,{\text{mi}}{{\text{n}}^{ - 1}}\)<span class="s1">. At time \(t\) minutes, the water depth is \(h{\text{ cm, }}0 \leqslant h \leqslant \pi \) and the volume of water in the container is \(V{\text{ c}}{{\text{m}}^{\text{3}}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the value of the volume generated.</p>
<div class="marks">[8]</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>Given that \(\frac{{{\text{d}}V}}{{{\text{d}}h}} = \pi {(3\cos 2h + 4)^2}\), find an expression for \(\frac{{{\text{d}}h}}{{{\text{d}}t}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(\frac{{{\text{d}}h}}{{{\text{d}}t}}\) <span class="s1">when \(h = \frac{\pi }{4}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(\frac{{{{\text{d}}^2}h}}{{{\text{d}}{t^2}}}\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Find the values of \(h\) <span class="s1">for which \(\frac{{{{\text{d}}^2}h}}{{{\text{d}}{t^2}}} = 0\).</span></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>By making specific reference to the shape of the container, interpret \(\frac{{{\text{d}}h}}{{{\text{d}}t}}\) <span class="s2">at the values of \(h\) found in part (c)(ii).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve <em>C</em> has equation \(y = \frac{1}{8}(9 + 8{x^2} - {x^4})\) .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the points on <em>C</em> at which \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The tangent to <em>C</em> at the point P(1, 2) cuts the <em>x</em>-axis at the point T. Determine the coordinates of T.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The normal to <em>C</em> at the point P cuts the <em>y</em>-axis at the point N. Find the area of triangle PTN.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined 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;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x - 1,}&{x \leqslant 2} \\ <br> {a{x^2} + bx - 5,}&{2 < x < 3} <br>\end{array}} \right.\]<br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where a , \(b \in \mathbb{R}\) .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>f</em> and its derivative, \(f'\) , are continuous for all values in the domain of <em>f</em> , find the values of <em>a</em> and <em>b</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<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;">Show that <em>f</em> is a one-to-one function.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Obtain expressions for the inverse function \({f^{ - 1}}\) and state their domains.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
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<p class="p1">A curve is given by the equation \(y = \sin (\pi \cos x)\).</p>
<p class="p2">Find the coordinates of all the points on the curve for which \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0,{\text{ }}0 \leqslant x \leqslant \pi \).</p>
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<br><hr><br><div class="specification">
<p>It is given that \({\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x\) where \(r,\,x \in {\mathbb{R}^ + }\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p>Express \(y\) in terms of \(x\). Give your answer in the form \(y = p{x^q}\), where <em>p</em> , <em>q</em> are constants.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region <em>R</em>, is bounded by the graph of the function found in part (b), the <em>x</em>-axis, and the lines \(x = 1\) and \(x = \alpha \) where \(\alpha > 1\). The area of <em>R</em> is \(\sqrt 2 \).</p>
<p>Find the value of \(\alpha \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(y = \frac{1}{{1 - x}}\), use mathematical induction to prove that \(\frac{{{{\text{d}}^n}y}}{{{\text{d}}{x^n}}} = \frac{{n!}}{{{{(1 - x)}^{n + 1}}}},{\text{ }}n \in {\mathbb{Z}^ + }\).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">A curve is defined by the equation \(8y\ln x - 2{x^2} + 4{y^2} = 7\). Find the equation of the tangent to the curve at the point where <em>x</em> = 1 and \(y > 0\).</span></p>
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<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;">Consider the curve with equation \({x^2} + xy + {y^2} = 3\).</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;">(a) Find in terms of <em>k</em>, the gradient of the curve at the point (−1, <em>k</em>).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that the tangent to the curve is parallel to the <em>x</em>-axis at this point, find the value of <em>k</em>.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the curve defined by the equation \({x^2} + \sin y - xy = 0\) .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the gradient of the tangent to the curve at the point \((\pi ,{\text{ }}\pi )\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, show that \(\tan \theta = \frac{1}{{1 + 2\pi }}\), where \(\theta \) is the acute angle between the tangent to the curve at \((\pi ,{\text{ }}\pi )\) and the line <em>y </em>= <em>x </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p>Consider the function \(f\) defined by \(f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}\) where \(a\) is a positive constant.</p>
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<div class="specification">
<p>The function \(g\) is defined by \(g(x) = x\sqrt {f(x)} \) for \(\left| x \right| > a\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any \(x\) and \(y\) intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p>\(y = f(x)\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any \(x\) and \(y\) intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p>\(y = \frac{1}{{f(x)}}\);</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any \(x\) and \(y\) intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p>\(y = \left| {\frac{1}{{f(x)}}} \right|\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int {f(x)\cos x{\text{d}}x} \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By finding \(g'(x)\) explain why \(g\) is an increasing function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p class="p1">In triangle \({\text{ABC, BC}} = \sqrt 3 {\text{ cm}}\), \({\rm{A\hat BC}} = \theta \) and \({\rm{B\hat CA}} = \frac{\pi }{3}\).</p>
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<p>Show that length \({\text{AB}} = \frac{3}{{\sqrt 3 \cos \theta + \sin \theta }}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(AB\) has a minimum value, determine the value of \(\theta \) <span class="s1">for which this occurs.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the functions <em>f</em> and <em>g</em> defined by \(f(x) = {2^{\frac{1}{x}}}\) and \(g(x) = 4 - {2^{\frac{1}{x}}}\) , \(x \ne 0\) .</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;">(a) Find the coordinates of <em>P</em>, the point of intersection of the graphs of <em>f</em> and <em>g</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) Find the equation of the tangent to the graph of <em>f</em> at the point P.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve <em>C </em>has equation \(2{x^2} + {y^2} = 18\). Determine the coordinates of the four points on <em>C </em>at which the normal passes through the point (1, 0) .</span></p>
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<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;">Find the area between the curves \(y = 2 + x - {x^2}{\text{ and }}y = 2 - 3x + {x^2}\) .</span></p>
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<br><hr><br><div class="question">
<p class="p1">Show that \(\int_1^2 {{x^3}\ln x{\text{d}}x = 4\ln 2 - \frac{{15}}{{16}}} \).</p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the exact value of \(\int_1^2 {\left( {{{(x - 2)}^2} + \frac{1}{x} + \sin \pi x} \right){\text{dx}}} \).</span></p>
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<br><hr><br><div class="question">
<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;">André wants to get from point A located in the sea to point Y located on a straight stretch of beach. P is the point on the beach nearest to A such that AP = 2 km and PY = 2 km. He does this by swimming in a straight line to a point Q located on the beach and then running to Y.</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;"> </span></p>
<p style="font: normal normal normal 19px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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alt></p>
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<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;">When André swims he covers 1 km in \(5\sqrt 5 \) minutes. When he runs he covers 1 km in 5 minutes.</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;">(a) If PQ = <em>x</em> km, \(0 \leqslant x \leqslant 2\) , find an expression for the time <em>T</em> minutes taken by André to reach point Y.</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;">(b) Show that \(\frac{{{\text{d}}T}}{{{\text{d}}x}} = \frac{{5\sqrt 5 x}}{{\sqrt {{x^2} + 4} }} - 5\).</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;">(c) (i) Solve \(\frac{{{\text{d}}T}}{{{\text{d}}x}} = 0\).</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;">(ii) Use the value of <em>x</em> found in <strong>part (c) (i)</strong> to determine the time, <em>T</em> minutes, taken for André to reach point Y.</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;">(iii) Show that \(\frac{{{{\text{d}}^2}T}}{{{\text{d}}{x^2}}} = \frac{{20\sqrt 5 }}{{{{({x^2} + 4)}^{\frac{3}{2}}}}}\) and <strong>hence</strong> show that the time found in <strong>part (c) (ii)</strong> is a minimum.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that \(\alpha > 1\), use the substitution \(u = \frac{1}{x}\) to show that</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;">\[\int_1^\alpha {\frac{1}{{1 + {x^2}}}{\text{d}}x = \int_{\frac{1}{\alpha }}^1 {\frac{1}{{1 + {u^2}}}{\text{d}}x} } .\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) <strong>Hence</strong> show that \(\arctan \alpha + \arctan \frac{1}{\alpha } = \frac{\pi }{2}\).</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The quadratic function \(f(x) = p + qx - {x^2}\) has a maximum value of 5 when <em>x </em>= 3.</span></p>
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<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;">Find the value of <em>p</em> and the value of <em>q</em> .</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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of <em>f</em>(<em>x</em>) is translated 3 units in the positive direction parallel to the <em>x</em>-axis. Determine the equation of the new graph.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle P moves in a straight line with displacement relative to origin given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[s = 2\sin (\pi t) + \sin (2\pi t),{\text{ }}t \geqslant 0,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where <em>t</em> is the time in seconds and the displacement is measured in centimetres.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the period of the function <em>s</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find expressions for the velocity, <em>v</em>, and the acceleration, <em>a</em>, of P.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Determine all the solutions of the equation <em>v</em> = 0 for \(0 \leqslant t \leqslant 4\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Consider the function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = A\sin (ax) + B\sin (bx),{\text{ }}A,{\text{ }}a,{\text{ }}B,{\text{ }}b,{\text{ }}x \in \mathbb{R}.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use mathematical induction to prove that the\({(2n)^{{\text{th}}}}\) derivative of <em>f</em> is given by \(({f^{(2n)}}(x) = {( - 1)^n}\left( {A{a^{2n}}\sin (ax) + B{b^{2n}}\sin (bx)} \right)\), for all \(n \in {\mathbb{Z}^ + }\).</span></p>
<div class="marks">[8]</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph below shows the two curves \(y = \frac{1}{x}\) and \(y = \frac{k}{x}\), where \(k > 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of region <em>A </em>in terms of <em>k </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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of region <em>B </em>in terms of <em>k </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 ratio of the area of region <em>A </em>to the area of region <em>B </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of \(y = f\left( x \right)\) has a local maximum at A. Find the coordinates of A.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there is exactly one point of inflexion, B, on the graph of \(y = f\left( x \right)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The coordinates of B can be expressed in the form B\(\left( {{2^a},\,b \times {2^{ - 3a}}} \right)\) where <em>a</em>, <em>b</em>\( \in \mathbb{Q}\). Find the value of <em>a</em> and the value of <em>b</em>.</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>Sketch the graph of \(y = f\left( x \right)\) showing clearly the position of the points A and B.</p>
<div class="marks">[4]</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A normal to the graph of \(y = \arctan (x - 1)\) , for \(x > 0\), has equation \(y = - 2x + c\) , where \(x \in \mathbb{R}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>c</em>.</span></p>
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<br><hr><br><div class="specification">
<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 function <em>f</em> is defined by \(f(x) = \frac{{2x - 1}}{{x + 2}}\), with domain \(D = \{ x: - 1 \leqslant x \leqslant 8\} \).</span></p>
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<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;">Express \(f(x)\) in the form \(A + \frac{B}{{x + 2}}\), where \(A\) and \(B \in \mathbb{Z}\).</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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence show that \(f'(x) > 0\) on <em>D</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">State the range of <em>f</em>.</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find an expression for \({f^{ - 1}}(x)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Sketch the graph of \(y = f(x)\), showing the points of intersection with both axes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) On the same diagram, sketch the graph of \(y = f'(x)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) On a different diagram, sketch the graph of \(y = f(|x|)\) where \(x \in D\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find all solutions of the equation \(f(|x|) = - \frac{1}{4}\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</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;">At 12:00 a boat is 20 km due south of a freighter. The boat is travelling due east at \(20{\text{ km}}\,{{\text{h}}^{ - 1}}\), and the freighter is travelling due south at \(40{\text{ km}}\,{{\text{h}}^{ - 1}}\).</span></p>
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<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;">Determine the time at which the two ships are closest to one another, and justify your answer.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If the visibility at sea is 9 km, determine whether or not the captains of the two ships can ever see each other’s ship.</span></p>
<div class="marks">[3]</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;">The function <em>f</em> is defined by \(f(x) = \frac{1}{{4{x^2} - 4x + 5}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Express \(4{x^2} - 4x + 5\) in the form \(a{(x - h)^2} + k\) where <em>a</em>, <em>h</em>, \(k \in \mathbb{Q}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(y = {x^2}\) is transformed onto the graph of \(y = 4{x^2} - 4x + 5\). Describe a sequence of transformations that does this, making the order of transformations clear.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = f(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the range of <em>f</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By using a suitable substitution show that \(\int {f(x){\text{d}}x = \frac{1}{4}\int {\frac{1}{{{u^2} + 1}}{\text{d}}u} } \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \(\int_1^{3.5} {\frac{1}{{4{x^2} - 4x + 5}}{\text{d}}x = \frac{\pi }{{16}}} \).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">f.</div>
</div>
<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;">A curve has equation \({x^3}{y^2} + {x^3} - {y^3} + 9y = 0\). Find the coordinates of the three points on the curve where \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A packaging company makes boxes for chocolates. An example of a box is shown below. This box is closed and the top and bottom of the box are identical regular hexagons of side <em>x</em> cm.</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;"> </span></p>
<p style="font: normal normal normal 19px/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: 19.0px 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: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that the area of each hexagon is \(\frac{{3\sqrt 3 {x^2}}}{2}{\text{c}}{{\text{m}}^2}\) .</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;">(b) Given that the volume of the box is \({\text{90 c}}{{\text{m}}^2}\) , show that when \(x = \sqrt[3]{{20}}\) the total surface area of the box is a minimum, justifying that this value gives a minimum.</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 function <em>f </em>is defined on the domain \(\left[ {0,\,\frac{{3\pi }}{2}} \right]\) by \(f(x) = {e^{ - x}}\cos x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the two zeros of <em>f </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of <em>f </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The region bounded by the graph, the <em>x</em>-axis and the <em>y</em>-axis is denoted by <em>A </em>and the region bounded by the graph and the <em>x</em>-axis is denoted by <em>B </em>. Show that the ratio of the area of <em>A </em>to the area of <em>B </em>is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\frac{{{e^\pi }\left( {{e^{\frac{\pi }{2}}} + 1} \right)}}{{{e^\pi } + 1}}.\]</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<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 function <em>f</em> is defined on the domain \(x \geqslant 0\) by \(f(x) = {{\text{e}}^x} - {x^{\text{e}}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find an expression for \(f'(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that the equation \(f'(x) = 0\) has two roots, state their values.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of <em>f</em> , showing clearly the coordinates of the maximum and minimum.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence show that \({{\text{e}}^\pi } > {\pi ^{\text{e}}}\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f\) is defined as \(f(x) = {{\text{e}}^{3x + 1}},{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \({f^{ - 1}}(x)\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State the domain of \({f^{ - 1}}\).</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">The function \(g\) is defined as \(g(x) = \ln x,{\text{ }}x \in {\mathbb{R}^ + }\).</p>
<p class="p1">The graph of \(y = g(x)\) and the graph of \(y = {f^{ - 1}}(x)\) intersect at the point \(P\).</p>
<p class="p1">Find the coordinates of \(P\).</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 class="p1">The graph of \(y = g(x)\) intersects the \(x\)-axis at the point \(Q\).</p>
<p class="p1">Show that the equation of the tangent \(T\) to the graph of \(y = g(x)\) at the point \(Q\) is \(y = x - 1\).</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 class="p1">A region \(R\) is bounded by the graphs of \(y = g(x)\), the tangent \(T\) and the line \(x = {\text{e}}\).</p>
<p class="p1">Find the area of the region \(R\).</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">A region \(R\) is bounded by the graphs of \(y = g(x)\), the tangent \(T\) and the line \(x = {\text{e}}\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(g(x) \le x - 1,{\text{ }}x \in {\mathbb{R}^ + }\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>By replacing \(x\) with \(\frac{1}{x}\) in part (e)(i), show that \(\frac{{x - 1}}{x} \le g(x),{\text{ }}x \in {\mathbb{R}^ + }\).</p>
<div class="marks">[6]</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A function is defined as \(f(x) = k\sqrt x \), with \(k > 0\) and \(x \geqslant 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Sketch the graph of \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that <em>f</em> is a one-to-one function.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the inverse function, \({f^{ - 1}}(x)\) and state its domain.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) If the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) intersect at the point (4, 4) find the value of <em>k</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Consider the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) using the value of <em>k</em> found in part (d).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the area enclosed by the two graphs.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The line <em>x</em> = <em>c</em> cuts the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) at the points P and Q respectively. Given that the tangent to \(y = f(x)\) at point P is parallel to the tangent to \(y = {f^{ - 1}}(x)\) at point Q find 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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Let \(a > 0\) . Draw the graph of \(y = \left| {x - \frac{a}{2}} \right|\) for \( - a \leqslant x \leqslant a\) on the grid below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 29px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
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<p style="margin: 0px; font: 29px Helvetica; text-align: justify;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find <em>k</em> such that \(\int_{ - a}^0 {\left| {x - \frac{a}{2}} \right|{\text{d}}x = k\int_0^a {\left| {x - \frac{a}{2}} \right|{\text{d}}x} } \) .</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f(x) = 1 + \sin x,{\text{ }}0 \leqslant x \leqslant \frac{{3\pi }}{2}\),</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">sketch the graph of \(f\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 31px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">show that \({\left( {f(x)} \right)^2} = \frac{3}{2} + 2\sin x - \frac{1}{2}\cos 2x\);</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the volume of the solid formed when the graph of <em>f</em> is rotated through \(2\pi \) radians about the <em>x</em>-axis.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = \frac{{\ln x}}{x}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , \(0 < x < {{\text{e}}^2}\) .</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;">(i) Solve the equation \(f'(x) = 0\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence show the graph of \(f\) has a local maximum.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the range of the function \(f\) .</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;">Show that there is a point of inflexion on the graph and determine its coordinates.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(y = f(x)\) , indicating clearly the asymptote, <em>x</em>-intercept and </span><span style="font-family: times new roman,times; font-size: medium;">the local maximum.</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;">Now consider the functions \(g(x) = \frac{{\ln \left| x \right|}}{x}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(h(x) = \frac{{\ln \left| x \right|}}{{\left| x \right|}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , where \(0 < x < {{\text{e}}^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of \(y = g(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the range of \(g\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Find the values of \(x\) such that \(h(x) > g(x)\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A tangent to the graph of \(y = \ln x\) passes through the origin.</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) Sketch the graphs of \(y = \ln x\) and the tangent on the same set of axes, and hence find the equation of the tangent.</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) Use your sketch to explain why \(\ln x \leqslant \frac{x}{{\text{e}}}\) for \(x > 0\) .</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;">(c) Show that \({x^{\text{e}}} \leqslant {{\text{e}}^x}\) for \(x > 0\) .</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;">(d) Determine which is larger, \({\pi ^{\text{e}}}\) or \({{\text{e}}^\pi }\) .</span></p>
</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 diagram below shows a sketch of the gradient function \(f'(x)\) of the curve \(f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 27px/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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">On the graph below, sketch the curve \(y = f(x)\) given that \(f(0) = 0\) . Clearly indicate on the graph any maximum, minimum or inflexion points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 27px/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="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 \(f(x) = \frac{{{x^2} - 5x + 4}}{{{x^2} + 5x + 4}}\).</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 the equations of all asymptotes of the graph of <em>f</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) Find the coordinates of the points where the graph of <em>f</em> meets the <em>x</em> and <em>y</em> axes.</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) Find the coordinates of</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;">(i) the maximum point and justify your answer;</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) the minimum point and justify your answer.</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;">(d) Sketch the graph of <em>f</em>, clearly showing all the features found above.</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;">(e) <strong>Hence</strong>, write down the number of points of inflexion of the graph of <em>f</em>.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>f</em> be a function defined by \(f(x) = x - \arctan x\) , \(x \in \mathbb{R}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(f(1)\) and \(f\left( { - \sqrt 3 } \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that \(f( - x) = - f(x)\) , for \(x \in \mathbb{R}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that \(x - \frac{\pi }{2} < f(x) + \frac{\pi }{2}\) , for \(x \in \mathbb{R}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find expressions for \(f'(x)\) and \(f''(x)\) . Hence describe the behaviour of the graph of <em>f</em> at the origin and justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Sketch a graph of <em>f</em> , showing clearly the asymptotes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Justify that the inverse of <em>f</em> is defined for all \(x \in \mathbb{R}\) and sketch its graph.</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: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the graphs of \(y = \sin x\) and \(y = \sin 2x\) , on the same set of axes, for \(0 \leqslant x \leqslant \frac{\pi }{2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the x-coordinates of the points of intersection of the graphs in the domain \(0 \leqslant x \leqslant \frac{\pi }{2}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the area enclosed by the graphs.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\int_0^1 {\sqrt {\frac{x}{{4 - x}}} }{{\text{d}}x} \) using the substitution \(x = 4{\sin ^2}\theta \) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The increasing function <em>f</em> satisfies \(f(0) = 0\) and \(f(a) = b\) , where \(a > 0\) and \(b > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) By reference to a sketch, show that \(\int_0^a {f(x){\text{d}}x = ab - \int_0^b {{f^{ - 1}}(x){\text{d}}x} } \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) <strong>Hence</strong> find the value of \(\int_0^2 {\arcsin \left( {\frac{x}{4}} \right){\text{d}}x} \) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<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 graph of \(y = f(x)\) is shown below, where A is a local maximum point and D is a local minimum point.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">On the axes below, sketch the graph of \(y = \frac{1}{{f(x)}}\) , clearly showing the coordinates of the images of the points A, B and D, labelling them \({{\text{A}'}}\), \({{\text{B}'}}\), and \({{\text{D}'}}\) respectively, and the equations of any vertical asymptotes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </p>
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<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: 'times new roman', times; font-size: medium;">On the axes below, sketch the graph of the derivative \(y = f'(x)\) , clearly showing the coordinates of the images of the points A and D, labelling them </span><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{A}}}''\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> and </span><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{D}}}''\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> respectively.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"> </p>
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<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
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<![endif]--> <!--StartFragment-->The graphs of \(y = \left| {x + 1} \right|\) and \(y = \left| {x - 3} \right|\) are shown below.</span></p>
<p><img src="data:image/png;base64,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UnhideWhenUsed="false" Name="Medium List 2"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1"/>
<w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2"/>
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UnhideWhenUsed="false" Name="Medium Grid 3"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List"/>
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UnhideWhenUsed="false" Name="Colorful Grid"/>
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UnhideWhenUsed="false" Name="Light Shading Accent 1"/>
<w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 1"/>
<w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 1"/>
<w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 1"/>
<w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 1"/>
<w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 1"/>
<w:LsdException Locked="false" UnhideWhenUsed="false" Name="Revision"/>
<w:LsdException Locked="false" Priority="34" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="List Paragraph"/>
<w:LsdException Locked="false" Priority="29" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Quote"/>
<w:LsdException Locked="false" Priority="30" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Quote"/>
<w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 1"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 1"/>
<w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 1"/>
<w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 1"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 1"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 1"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 1"/>
<w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 1"/>
<w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 2"/>
<w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 2"/>
<w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 2"/>
<w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 2"/>
<w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 2"/>
<w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 2"/>
<w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 2"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 2"/>
<w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 2"/>
<w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 2"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 2"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 2"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 2"/>
<w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 2"/>
<w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 3"/>
<w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 3"/>
<w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 3"/>
<w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 3"/>
<w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 3"/>
<w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 3"/>
<w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 3"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 3"/>
<w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 3"/>
<w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 3"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 3"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 3"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 3"/>
<w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 3"/>
<w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 4"/>
<w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 4"/>
<w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 4"/>
<w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 4"/>
<w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 4"/>
<w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 4"/>
<w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 4"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 1 Accent 4"/>
<w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 4"/>
<w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 4"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 4"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 4"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 4"/>
<w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 4"/>
<w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 5"/>
<w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 5"/>
<w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 5"/>
<w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 5"/>
<w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 5"/>
<w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 5"/>
<w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 5"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
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<w:LsdException Locked="false" Priority="68" SemiHidden="false"
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<w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 5"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 5"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 5"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 5"/>
<w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 5"/>
<w:LsdException Locked="false" Priority="60" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Shading Accent 6"/>
<w:LsdException Locked="false" Priority="61" SemiHidden="false"
UnhideWhenUsed="false" Name="Light List Accent 6"/>
<w:LsdException Locked="false" Priority="62" SemiHidden="false"
UnhideWhenUsed="false" Name="Light Grid Accent 6"/>
<w:LsdException Locked="false" Priority="63" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 1 Accent 6"/>
<w:LsdException Locked="false" Priority="64" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Shading 2 Accent 6"/>
<w:LsdException Locked="false" Priority="65" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 1 Accent 6"/>
<w:LsdException Locked="false" Priority="66" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium List 2 Accent 6"/>
<w:LsdException Locked="false" Priority="67" SemiHidden="false"
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<w:LsdException Locked="false" Priority="68" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 2 Accent 6"/>
<w:LsdException Locked="false" Priority="69" SemiHidden="false"
UnhideWhenUsed="false" Name="Medium Grid 3 Accent 6"/>
<w:LsdException Locked="false" Priority="70" SemiHidden="false"
UnhideWhenUsed="false" Name="Dark List Accent 6"/>
<w:LsdException Locked="false" Priority="71" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Shading Accent 6"/>
<w:LsdException Locked="false" Priority="72" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful List Accent 6"/>
<w:LsdException Locked="false" Priority="73" SemiHidden="false"
UnhideWhenUsed="false" Name="Colorful Grid Accent 6"/>
<w:LsdException Locked="false" Priority="19" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis"/>
<w:LsdException Locked="false" Priority="21" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis"/>
<w:LsdException Locked="false" Priority="31" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference"/>
<w:LsdException Locked="false" Priority="32" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Intense Reference"/>
<w:LsdException Locked="false" Priority="33" SemiHidden="false"
UnhideWhenUsed="false" QFormat="true" Name="Book Title"/>
<w:LsdException Locked="false" Priority="37" Name="Bibliography"/>
<w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/>
</w:LatentStyles>
</xml><![endif]--> <!--[if gte mso 10]>
<style>
/* Style Definitions */
table.MsoNormalTable
{mso-style-name:Standaardtabel;
mso-tstyle-rowband-size:0;
mso-tstyle-colband-size:0;
mso-style-noshow:yes;
mso-style-priority:99;
mso-style-parent:"";
mso-padding-alt:0cm 5.4pt 0cm 5.4pt;
mso-para-margin:0cm;
mso-para-margin-bottom:.0001pt;
mso-pagination:widow-orphan;
font-size:12.0pt;
font-family:Cambria;
mso-ascii-font-family:Cambria;
mso-ascii-theme-font:minor-latin;
mso-hansi-font-family:Cambria;
mso-hansi-theme-font:minor-latin;
mso-ansi-language:NL;}
</style>
<![endif]--> <!--StartFragment--><span style="font-size: 12.0pt; font-family: 'TimesNewRomanPSMT','serif'; mso-fareast-font-family: 'ＭＳ 明朝'; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: TimesNewRomanPSMT; mso-ansi-language: EN-US; mso-fareast-language: NL; mso-bidi-language: AR-SA;">Let <em>f </em>(<em>x</em>) = \(\left| {\,x + 1\,} \right| - \left| {\,x - 3\,} \right|\).</span><!--EndFragment--></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Draw the graph of <em>y </em>= <em>f </em>(<em>x</em>) on the blank grid below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkgAAAGrCAIAAAC0RP9JAAAgAElEQVR4nO3dcWyc9Z3n8edP/hjpZMTd1mpztm4uUqRDx0kUVUbVqb5TcrComygVQg3IMt3NJrl2lQ0SAUdJYJfASUxVGh8poVcv4Gtp2GZw48E+3xKjbLZNSzJSjXDUOqWu4hyW4tTB1mDHMM7z3B/jGMeZfGOb3+Pf9zu/90vzR0y874zc3+PPzBPDRtmGRh48ePDgwaNmHlG2oTEBAKAmMGwAgJrCsAEAagrDBgCoKQwbAKCmMGwAgJrCsAEAagrDBgCoKQwbAKCmMGwAgJrCsAEAagrDBgCoKQwbAKCmMGwAgJrCsAHLNXNp6Bdv/bw4FidJkiRXPuj/x/6h0qznJwXgGoYNWJbJs92Hn9/enIke7hyeSZJkdqjjvsy6bYUPfT8xAHMYNmDZ4pEjD0VfPfCriSRJkuTjsy/t2HfyT56fE4BrGDZg+WZ+daAx25IfSZIkufLe4cc6hz6NfT8nAHMYNmD5ysVcds3Gzt/HSWnw8FOHB0u+nxCAzzBswPLFfzzyUGM29+vLA68+/ca5T5MkSZJDLx760p990e/zApAwbMCKjORbsmu2tv39c70fluMkSaanp2/P3HF75o7x8XHfzw0IHcMGrECpmPtaZv0LZybnfsr/zTffbPpK07/+V/+m87VOv88MAMMGLF881v/U/vz5K5WPpqenm77S9Oabb37pz77Y9JWm6elpv88OCBzDBixTPDH4yvMvD3w0/3OQ//R//6myZ//u3zZUFs7n0wOCx7ABSxRfGfpx673bnn9+33PHL5QX/MbmTZsrdyCzDY2Ve5K+niKAhGEDliy+MvSz/U+++NbQxMJ/Z216evrQi4cqPzOSbWicnp5+7rn/wY+QAB4xbIAzXEqABgwbsHSzpfOn+7q6egbGqv6HRriUAA0YNmAp4vLoiYOt92/c91r/2bHyTT6JSwnQgGEDbmm2NPhq69p7d77xu5L4n4TkUgI0YNiAW4jH3n7ynjX37D9x+Vb/oWMuJUADhg2QTRRz90V1O/IffnrLT+VSAjRg2CTdhd7KY+GvKx8u+t2lf1i1Rlxv/KfP7KyLorX3N/+39V/78r+/7bb/sL3j3bFyXLWWbWhc9KjZLwtxF/EE6WDYJKmePOKrH19Bv1zMZaP65mdPjJXjJP5ooP3BTHT3jsLI6v9UpN0vO/HVjweOYZPYPdbEHfVnxwp/k4mu/f8UTZL4YmF7XZTZVhir9tkMG3El8cAxbICkXMxlFwxbcvW3HRvqo5b8aLVP5lICNGDYJHZfrxF31h/v21mXuSt3ZqbyYfz7zo2Ndx44NVXtc3nHRlxJPHAMm8TusSburj85cHBTpum7xak4qdyKrN/SMVR11xg24lrigWPYJHaPNXGX/fKF4/s3Ne9sP3L0lQOt33q2/4KX//KI3S878dWPB45hk9g91sRd92dLI+8Xi+culaV/SZthI64kHjiGDXCGSwnQgGGT2H29RtxLn3dsxJXEA8ewSewea+Je+gwbcSXxwDFsErvHmriXPsNGXEk8cAybxO6xJu6lz7ARVxIPHMMGOMOlBGjAsAHOcCkBGjBsErs3Ioh76XMrkriSeOAYNondY03cS59hI64kHjiGTWL3WBP30mfYiCuJB45hA5zhUgI0YNgAZ7iUAA0YNondGxHEvfS5FUlcSTxwDJvE7rEm7qXPsBFXEg8cwyaxe6yJe+kzbMSVxAPHsAHOcCkBGjBsgDNcSoAGDJvE7o0I4l763IokriQeOIZNYvdYE/fSZ9iIK4kHjmGT2D3WxL30GTbiSuKBY9gkdo81cS99ho24knjgGDbAGS4lQAOGTWL39RpxL33esRFXEg8cwybpLvRWHgt/PX8cV/Zh1Rpx73Enf1a2oXHRw/qXhXiq8QTpYNgkqZ484qsfT7vPOzbiSuKBY9gkdo81cS99ho24knjgGDbAGS4lQAOGTWL39RpxL33esRFXEg8cwyaxe6yJe+kzbMSVxAPHsEnsHmviXvoMG3El8cAxbBK7x5q4lz7DRlxJPHAMG+AMlxKgAcMmsft6jbiXPu/YiCuJB45hk9g91sS99Bk24krigWPYJHaPNXEvfYaNuJJ44Bg2id1jTdxLn2EjriQeOIYNcIZLCdCAYQOc4VICNGDYJHZvRBD30udWJHEl8cAxbBK7x5q4lz7DRlxJPHAMm8TusSbupc+wEVcSDxzDBjjDpQRowLABznApARowbBK7NyKIe+lzK5K4knjgGDaJ3WNN3EufYSOuJB44hk1i91gT99Jn2IgriQeOYQOc4VICNGDYAGe4lAANGDaJ3RsRxL30uRVJXEk8cAybxO6xJp5KP54Yeqfn5PDHN/t9ho24knjgGDaJ3WNNPIX+7OSZF9Znsi35kZt9BsNGXEk8cAybxO6xJu68H18+8VTzusaIYSNuIB44hg1Ygni0v21H7uihR/wNG4AlYtgkdl+vEXfaL1/s27elvVgazbfwjo24hXjgGDZJd6G38lj46/njuLIPq9aIe48Lf1b5fH7HN3/4Yr6nu3Nvc1RfGbaqn5xtaFz0sP5lIZ5qPEE6GDZJqieP+OrHV9IvD+e/vavzg6kkSRLesRE3Eg8cwyaxe6yJO+pP/eHId9bvfv1UsVgsFot9uQeiNQ/keorvjZTiKp/NsBFXEg8cwwYIRvIt2ehG2VyxXOWzuZQADRg2id3Xa8RT6XMrkriReOAYNondY008lT7DRtxIPHAMm8TusSaeSp9hI24kHjiGTWL3WBP30mfYiCuJB45hA5zhUgI0YNgkdl+vEffS5x0bcSXxwDFsErvHmriXPsNGXEk8cAybxO6xJu6lz7ARVxIPHMMmsXusiXvpM2zElcQDx7ABznApARowbIAzdi8l3j2gljBsErs3Ioh76XMrkriSeOAYNondY03cS59hI64kHjiGTWL3WBP30mfYiCuJB45hA5zhUgI0YNgAZ7iUAA0YNondGxHEvfS5FUlcSTxwDJvE7rEm7qXPsBFXEg8cwyaxe6yJe+kzbMSVxAPHsAHOcCkBGjBsgDNcSoAGDJvE7o0I4l763IokriQeOIZNYvdYE/fSZ9iIK4kHjmGT2D3WxL30GTbiSuKBY9gkdo81cS99u8OWKrsHxm48cAwb4AyXEqABwyax+3qNuJe+3XdsxGssHjiGTdJd6K08Fv56/jiu7MOqNeLe407+rGxD46KHlWdee/+DmognSAfDJkn15BFf/Xjafd6xEVcSDxzDJrF7rIl76TNsxJXEA8ewAc5wKQEaMGwSu6/XiHvp846NuJJ44Bg2id1jTdxLn2EjriQeOIZNYvdYE/fSZ9iIK4kHjmGT2D3WxL30GTbiSuKBY9gAZ7iUAA0YNond12vEvfR5x0ZcSTxwDJvE7rEm7qXPsBFXEg8cwyaxe6yJe+kzbMSVxAPHsEnsHmviXvoMG3El8cAxbIAzXEqABgwb4AyXEqABwyaxeyOCuJc+tyKJK4kHjmGT2D3WxL30GTbiSuKBY9gkdo81cS99ho24knjgGDbAGS4lQAOGDXCGSwnQgGGT2L0RQdxLn1uRxJXEA8ewSewea+Je+gwbcSXxwDFsErvHmriXPsNGXEk8cAwb4AyXEqABwwY4w6UEaMCwSezeiCDupc+tSOJK4oFj2CR2jzVxp/3Z0vnTfV35rr7T50uzwucxbMSVxAPHsEnsHmvi7vozo8efac5Ec9Zu7RiciG/yqQwbcSXxwDFsErvHmrirfnzxrV2PtvcPT8bJzOip7z9YF2U2HB78pPq0MWzElcQDx7ABgquT73S8OjR17cPxk/uaomhbfrRc9bO5lAANGDaJ3ddrxNPpzwx3PhzVt/VPXq3627xjI64kHjiGTdJd6K08Fv56/jiu7MOqNeLe40v7s44c+Ma6ptzpqZv8WdmGxkUPNc98hX+W3f9BTcQTpINhk6R68oivfvzz9ePy+Tda733m5OWb/mAk79iIK4kHjmGT2D3WxN33y8P5HdvaBz662Y9EJgwbcTXxwDFswBLEE++/vOfp4xeq/9DINVxKgAYMm8Tu6zXiLvvxxGDH47vyw9dWbeZPo+NVF87uO7ZU2T0wduOBY9gkdo81cWf9eGKwY+u65t0/OprP5/P5/NEjP3hsS/tvPqn2uXaHjXiNxQPHsEnsHmvijvqloc5ta6NFNr909uOqn82wEVcSDxzDJrF7rIl76TNsxJXEA8ewAc5wKQEaMGwSu6/XiHvp233Hliq7B8ZuPHAMm8TusSbupW932IjXWDxwDJvE7rEm7qXPsBFXEg8cwyaxe6yJe+kzbMSVxAPHsAHOcCkBGjBsgDNcSoAGDJvE7o0I4l763IokriQeOIZNYvdYE/fSZ9iIK4kHjmGT2D3WxL30GTbiSuKBY9gAZ7iUAA0YNsAZLiVAA4ZNYvdGBHEvfW5FElcSDxzDJrF7rIl76TNsxJXEA8ewSewea+Je+gwbcSXxwDFsgDNcSoAGDBvgDJcSoAHDJrF7I4K4lz63IokriQeOYZPYPdbEvfQZNuJK4oFj2CR2jzVxL32GjbiSeOAYNondY03cS59hI64kHjiGDXCGSwnQgGGT2H29RtxLn3dsxJXEA8ewSboLvZXHwl/PH8eVfVi1Rtx73MmflW1oXPSw/mUhnmo8QToYNkmqJ4/46sfT7vOOjbiSeOAYNondY03cS59hI64kHjiGDXCGSwnQgGGT2H29RtxLn3dsxJXEA8ewSewea+Je+gwbcSXxwDFsErvHmriXPsNGXEk8cAybxO6xJu6lz7ARVxIPHMMGOMOlBGjAsEnsvl4j7qXPOzbiSuKBY9gkdo81cS99ho24knjgGDaJ3WNN3EufYSOuJB44hk1i91gT99Jn2IgriQeOYQOc4VICNGDYAGe4lAANGDaJ3RsRxL30uRVJXEk8cAybxO6xJu6lz7ARVxIPHMMmsXusiXvpM2zElcQDx7ABznApARowbIAzXEqABgybxO6NCOJe+tyKJK4kHjiGTWL3WBP30mfYiCuJB45hk9g91sS99Bk24krigWPYAGe4lAANGDbAGS4lQAOGTWL3RgRxL31uRRJXEg8cwyZJ9StDfPXjafeJE1cSDxzDJrF7rIl76RMnriQeOIZNYvdYE/fSJ07cSXx6ejq9PzcEDJukJq+ZkONp94kT//zx8fHx2zN3PLH7iXND59L702sbwyapvWsm8HjafeLEncTPDZ17YvcTlXn75S9+md5zqFUMmyTb0MiDR3eht/JY9GHlnyzrQx48lv5o+OKaL9zxhdszd2zetJl3b8uSZdgEdn/Yl7iXfqqXkt0vO/GVxefft23etHl8fDy9Z1J7GDZJDV8zYcbT7jNsxJ3Ef/mLX87fiuS92gowbIAzXEr4/C5cuHB75o5DLx5i0laMYVui2dL5031dXT0DY7Hvp7Js8cTQOz0nhz/2/TyWrvLVznf1nT5fmvX9ZBaIJ4f/pSff9fbA2EzV37d5KWn9ai+RveM976bfVbjx+DkxbJLuQm+SxOXREwdb79+477X+s2Nlx/G0LIjPTp55YX0m25IfSSHuXnfh2OjxZ5oz0Zy1WzsGJxy+mFj5k7/y259u39R68K1T//za367fXvVZGbwVOZPqV7sizQMze+R7f+32eC+U2jOPy6Mntv7XLy/pu0r50tCv+wtv/J+By7NJEpfHBnreKJwZrf7SChUMm6S78FZp8NXWtffufON3JdeX++oMW3z5xFPN6xojO8P246d3PdrePzwZJzOjp77/YF2U2XB48BNnX/2VPvnS4EsPZdYfPvtpnCTli4Wddfc8c/Ly4vc35oYtvvhW5avdXTiWxle7Ir0DE18+8c3/+CW3x3uhdJ75bOW7ytef+OESvqvEV/7467eO/WDnPWuacu9Ojr79bOvGr61dv++fx1J4YrWDYZMc+/FzT96z5p79Jy6ncP9xNYYtHu1v25E7eugRM8N29cizf/vq0NS1D8dP7muKom35UWdvlVf45Cf7d9dH2Vyx8jzi4c6N0bpH8+cXnQtrw3Z18p2Oyle7u9Cbxle7Iq0DE4/2t+14tO2/uz3eC6XyYmLs7cp3ldeXEZ8Z7nw4WveN1sd/NGDxdvGqY9gEE8XcfVHdjvyHn/p+JitTvti3b0t7sTSab0ntyk/ZzHDnw1F9W//kVa9PI/745FNrFn4NZ351oDGq29m36G9CjF9KSr7aS2T0eK/wu8rs2Ze+GjXtO8nfvS0Jw3Zz430766Jo7f0tW1sfeeDuTOar2zveHStb+dmRuHw+v+ObPxy8EifGrvyFLvXvbmrKnZ669Wemqjya3xZF67YVPrz2D4q5bBRt6Bi6fgKMX0pKvtpLYfZ4r/S7SjzcuTFa23rDTQJUxbDdVLmY+0J0e/OzJ8bKcRJ/NND+YCa6e0dhxNXBSvdW5M//If/tXZ0fTCVJ4vzKX60fe4nL599ovbfKX2U56i9dZdi+liuWrv2DYi4bfXZr8hprtyKvi6fx1Z6POy6Wh+ePd3fn3vSGzfkzLxdz2ai+8l2lu/CzpX5XiS/2P/edlvsb63f3T7p9QjUq9GGLLw305Kt7vf2vbovq5y+Y+GJhe12U2VZw9Ze2n/Oaufkz7+oZ+OPLu7++fvfrp4rFYrFY7Ms9EK15INdTfG/EyY/ApPnMx47Nx8vD+R3b2gc+cvsS9XMM24IbQZVh29g5fP2TMzxsP/+HNL7ac3HHz3zqD0e+M3+8v/d3f+n2eC/k+pnPjhX+JnNthrsLvbf6rhKXL/3h3KXJ0b6D3z15fqjjweiu3JlL77784r+k8bf+tYRhu+k32e7Xn/rCgmFLrv62Y0N91JIfdfRHpzkPp5/8L/XRjW54h6HvmV8btnji/Zf3PH38guMfY1jhk49nirm7Fr4tGO/bWRfd+PLZ6rDFEy9+56E0vtoVrp/5SL4lm97xXiidd2yfDdutvqtMFnPro8b/vP2V90px/Mng4Q2ZzNrWVwf5+ZFbCX3YJON9O+syd+XOzP0LI/HvOzc23nnglIW/gbierb+ESJIknhjseHxXfvjat6mZP42Op/Q9d6mmTh24M9N44FeVw3B1qGNDdOf2wv9bzZ+KTIvCr/ay2Drey/yuEpdG3hu6NP8/zaVzf7hk5q/5fWLYBJMDBzdlmr5bnIqTyq3I+i0dQ/Z2zdiVH08Mdmxd17z7R0crb+OOHvnBY1vaf/OJ56c19UHno3Vzh2FqqGNL3V+8PHhl8bcYe5eS0q/2ctg63jXzXUU3hk3S/fPO4/s3Ne9sP3L0lQOt33q23+W9mtX776ta+uGR0kuP/fnaxfeYNr901tl/MGnlT7584fj+Tet3H/7pS088+NDz/dX+0w/WbkWWhjq3pfrVrkj3qJv64ZEkmTtIzTvbH297zPl3FVQwbJLuQm+SzJZG3i8Wzzm/A1CT/1Vy5fHP3Z8tjbwv/JCCtWEj7is+Wxp5/3sv/C/uK6aEYZPYvGaIe+szbMSVxAPHsAHOcCkBGjBsgDNcSoAGDJvE7o0I4l763IokriQeOIZNYvdYE/fSZ9iIK4kHjmGT2D3WxL30GTbiSuKBY9gkdo81cS99ho24knjgGDbAGS4lQAOGTWL39RpxL33esRFXEg8cwyaxe6yJe+kzbMSVxAPHsEnsHmviXvoMG3El8cAxbBK7x5q4lz7DRlxJPHAMG+AMlxKgAcMmsft6jbiXPu/YiCuJB45hk3QXeiuPhb+eP44r+7Bqjbj3uJM/K9vQuOhh/ctCPNV4gnQwbJJUTx7x1Y+n3ecdG3El8cAxbBK7x5q4lz7DRlxJPHAMG+AMlxKgAcMmsft6jbiXPu/YiCuJB45hk9g91sS99Bk24krigWPYJHaPNXEvfYaNuJJ44Bg2id1jTdxLn2EjriQeOIYNcIZLCdCAYZPYfb1G3Eufd2zElcQDx7BJ7B5r4l76DBtxJfHAMWwSu8eauJc+w0ZcSTxwDJvE7rEm7qXPsBFXEg8cwwY4w6UEaMCwAc5wKQEaMGwSuzciiHvpcyuSuJJ44Bg2id1jTdxLn2EjriQeOIZNYvdYE/fSZ9iIK4kHjmEDnOFSAjRg2ABnuJQADRg2id0bEcS99LkVSVxJPHAMm8TusSbupc+wEVcSDxzDJrF7rIl76TNsxJXEA8ewAc5wKQEaMGzA0s2Wzp/u68p39Z0+X5q98be5lAANGDaJ3RsRxFPoz4wef6Y5E81Zu7VjcCK+/jO4FUlcSTxwDJvE7rEm7rwfX3xr16Pt/cOTcTIzeur7D9ZFmQ2HBz+5btoYNuJK4oFj2CR2jzVx1/2rk+90vDo0de3D8ZP7mqJoW360vPCTGDbiSuKBY9gkdo818ZT7M8OdD0f1bf2TVxf+U4aNuJJ44Bg2YAUu9e9uasqdnrr+n3IpARowbBK7r9eIp9mPy+ffaL33mZOXF/9gJO/YiCuJB45hk3QXeiuPhb+eP44r+7Bqjbj3eHeh99hPDu3fs7dtz958Pp/P59v27G2b+7CrZ2Ds2Pwnl4fzO7b9dfvPbvyzsg2Nix7WvyzEU40nSAfDJkn15BFf/bjcjy8N9OSr6uoZGJv78cd44v2X9zx9/EK5WoF3bMSVxAPHsEnsHmviqfTjicGOx3flh6+t2syfRscXLhzDRlxJPHAMG7A08cRgx9Z1zbt/dLTyNu7okR88tqX9N58s+BQuJUADhk1i9/Uacdf90lDntrXRIptfOvvxwk/iHRtxJfHAMWwSu8eauJc+w0ZcSTxwDJvE7rEm7qXPsBFXEg8cwyaxe6yJe+kzbMSVxAPHsAHOcCkBGjBsEruv14h76fOOjbiSeOAYNondY03cS59hI64kHjiGTWL3WBP30mfYiCuJB45hk9g91sS99Bk24krigWPYAGe4lAANGDbAGS4lQAOGTWL3RgRxL31uRRJXEg8cwyaxe6yJe+kzbMSVxAPHsEnsHmviXvoMG3El8cAxbIAzXEqABgwb4AyXEqABwyaxeyOCuJc+tyKJK4kHjmGT2D3WxL30GTbiSuKBY9gkdo81cS99ho24knjgGDbAGS4lQAOGDXCGSwnQgGGT2L0RQdxLn1uRxJXEA8ewSewea+Je+gwbcSXxwDFsErvHmriXPsNGXEk8cAybxO6xJu6lz7ARVxIPHMMGOMOlBGjAsEnsvl4j7qXPOzbiSuKBY9gk3YXeymPhr+eP48o+rFoj7j3u5M/KNjQuelj/shBPNZ4gHQybJNWTR3z142n3ecdGXEk8cAybxO6xJu6lz7ARVxIPHMMGOMOlBGjAsEnsvl4j7qXPOzbiSuKBY9gkdo81cS99ho24knjgGDaJ3WNN3EufYSOuJB44hk1i91gT99Jn2IgriQeOYQOc4VICNGDYJHZfrxH30ucdG3El8cAxbBK7x5q4lz7DRlxJPHAMm8TusSbupc+wEVcSDxzDJrF7rIl76TNsxJXEA8ewAc5wKQEaMGyAM1xKgAYMm8TujQjiXvrciiSuJB44hk1i91gT99Jn2IgriQeOYZPYPdbEvfQZNuJK4oFj2ABnuJQADRg2wBkuJUADhk1i90YEcS99bkUSVxIPHMMmsXusiXvpM2zElcQDx7BJ7B5r4l76DBtxJfHAMWyAM1xKgAYMG7B88cTQOz0nhz9e9I+5lAANGDaJ3RsRxNPsz06eeWF9JtuSH1n0G9yKJK4kHjiGTWL3WBNPrx9fPvFU87rGiGEjrjceOIZNYvdYE0+rH4/2t+3IHT30CMNGXHE8cAybxO6xJp5Ov3yxb9+W9mJpNN/CsBFXHA8cwwYsUVw+n9/xzR8OXokTH8MGYIkYNond12vE3ffLw/lv7+r8YCpJEi/DZvfLTnz144Fj2CTdhd7KY+Gv54/jyj6sWiPuPd5d6D32k0P79+xt27M3n8/n8/m2PXvb5j7s6hn448u7v/6fvvHE8y8cLBaLxb7cl6M7vvzo333vf75Wij/7s7INjYse1r8sruJO/gda9H++MGL6ywLnGDZJqieP+OrH5X58aaAnX1VXz8Dpoy3Z6EbZXLH8WYF3bMSVxAPHsEnsHmvi6fa5FUlcdzxwDBuwfPzwCKAYwyax+3qNeLp93rER1x0PHMMmsXusiXvpM2zElcQDx7BJ7B5r4l76DBtxJfHAMWwSu8eauJc+w0ZcSTxwDBvgDJcSoAHDBjjDpQRowLBJ7N6IIO6lz61I4krigWPYJHaPNXEvfYaNuJJ44Bg2id1jTdxLn2EjriQeOIYNcIZLCdCAYQOc4VICNGDYJHZvRBD30udWJHEl8cAxbBK7x5q4lz7DRlxJPHAMm8TusSbupc+wEVcSDxzDBjjDpQRowLABznApARowbBK7NyKIe+lzK5K4knjgGDaJ3WNN3EufYSOuJB44hk1i91gT99Jn2IgriQeOYQOc4VICNGDYAGe4lAANGDaJ3RsRxL30uRVJXEk8cAybxO6xJu6lz7ARVxIPHMMmsXusiXvpM2zElcQDx7BJ7B5r4l76DBtxJfHAMWyAM1xKgAYMm8Tu6zXiXvq8YyOuJB44hk3SXeitPBb+ev44ruzDqjXi3uNO/qxsQ+Oih/UvC/FU4wnSwbBJUj15xFc/nnafd2zElcQDx7BJ7B5r4l76DBtxJfHAMWyAM1xKgAYMm8Tu6zXiXvq8YyOuJB44hk1i91gT99Jn2IgriQeOYZPYPdbEvfQZNuJK4oFj2CR2jzVxL32GjbiSeOAYNsAZLiVAA4YNcIZLCdCAYZPYvRFB3EufW5HElcQDx7BJ7B5r4l76DBtxJfHAMWwSu8eauJc+w0ZcSTxwDBvgDJcSoAHDBjjDpQRowLBJ7N6IIO6lz61I4krigWPYJHaPNXEvfYaNuJJ44Bg2id1jTdxLn2EjriQeOIYNcIZLCdCAYQOc4VICNGDYJHZvRBD30udWJHEl8cAxbBK7x5q4lz7DRlxJPHAMm8TusSbupZDjRPkAAAI1SURBVM+wEVcSDxzDBjjDpQRowLABKzBbOn+6r6urZ2AsXvBPuZQADRg2id0bEcRT68fl0RMHW+/fuO+1/rNj5et/j1uRxJXEA8ewSewea+Lp9GdLg6+2rr135xu/K8VVfpthI64kHjiGTWL3WBNPox+Pvf3kPWvu2X/icrVVSxg24mrigWPYJHaPNfEU+hPF3H1R3Y78h5/e7DMYNuJK4oFj2IClGe/bWRdFa+9v2dr6yAN3ZzJf3d7x7lj5uvduXEqABgybxO7rNeLO++ViLhvVNz97YqwcJ/FHA+0PZqK7dxRGVu2nIu1+2YmvfjxwDJuku9BbeSz89fxxXNmHVWvENcSP/eTQ/j172/bszefz+Xy+bc/etrkPu3oGRjv3/8VtUX3zntfm4j9++v5MdNt9T//vBbVsQ+OiRw18WYinF0+QDoYNmBNfGujJV9XVMzD2aTGXjbIt+ZG5z776244N9VFLfnRBgUsJ0IBhA5ZmvG9nXeau3JmZyofx7zs3Nt554NSU32cF4AYMG7BEkwMHN2WavlucipMkiS8Wttdv6Rhi1wB1GDZgycoXju/f1Lyz/cjRVw60fuvZ/gvlW//fAFhtDBuwLLOlkfeLxXOXyjf5l7QB+MawAQBqCsMGAKgpDBsAoKYwbACAmsKwAQBqCsMGAKgpDBsAoKYwbACAmsKwAQBqCsMGAKgpDBsAoKYwbACAmsKwAQBqCsMGAKgpDBsAoKbMDRsPHjx48OBRM4//D2+oj2NcUdNqAAAAAElFTkSuQmCC" alt></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;">Hence state the value 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;">(i) <span lang="NL">\(f'( - 3)\);</span></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;">(ii) <span lang="NL">\(f'(2.7)\);</span><!--EndFragment--></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;">(iii) \(\int_{ - 3}^{ - 2} {f(x)dx} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>