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</div><h2>HL Paper 1</h2><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the function \(f:x \to \sqrt {\frac{\pi }{4} - \arccos x} \).</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;">(a) Find the largest possible domain of <em>f</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;">(b) Determine an expression for the inverse function, \({f^{ - 1}}\), and write down its domain.</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 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;">Write \(\ln ({x^2} - 1) - 2\ln (x + 1) + \ln ({x^2} + x)\) as a single logarithm, in its simplest form.</span></p>
</div>
<br><hr><br><div class="specification">
<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;">Consider the equation \(y{x^2} + (y - 1)x + (y - 1) = 0\).</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: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the set of values of <em>y</em> for which this equation has real roots.</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;">Hence determine the range of the function \(f:x \to \frac{{x + 1}}{{{x^2} + x + 1}}\).</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;">Explain why <em>f</em> has no inverse.</span></p>
<div class="marks">[1]</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;">Let \(f(x) = {x^3} + a{x^2} + bx + c\) , where <em>a </em>, <em>b </em>, \(c \in \mathbb{Z}\) . The diagram shows the graph of <em>y</em> = <em>f</em>(<em>x</em>) .</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the information shown in the diagram, find the values of <em>a </em>, <em>b </em>and <em>c </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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If <em>g</em>(<em>x</em>) = 3<em>f</em>(<em>x </em>− 2) ,</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;">(i) state the coordinates of the points where the graph of <em>g </em>intercepts the <em>x</em>-axis.</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;">(ii) Find the <em>y</em>-intercept of the graph of <em>g </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f\) is defined by \(f(x) = \frac{{3x}}{{x - 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch the graph of \(y = f(x)\), indicating clearly any asymptotes and points of intersection with the \(x\) and \(y\) axes.</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 an expression for \({f^{ - 1}}(x)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find all values of \(x\) for which \(f(x) = {f^{ - 1}}(x)\).</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 class="p1">Solve the inequality \(\left| {f(x)} \right| < \frac{3}{2}\).</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 class="p1">Solve the inequality \(f\left( {\left| x \right|} \right) < \frac{3}{2}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A function is defined by \(h(x) = 2{{\text{e}}^x} - \frac{1}{{{{\text{e}}^x}}},{\text{ }}x \in \mathbb{R}\) . Find an expression for \({h^{ - 1}}(x)\) .</span></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;">The polynomial \(P(x) = {x^3} + a{x^2} + bx + 2\) is divisible by (<em>x</em> +1) and by (<em>x</em> − 2) .</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;">Find the value of <em>a</em> and of <em>b</em>, where \(a,{\text{ }}b \in \mathbb{R}\) .</span></p>
</div>
<br><hr><br><div class="specification">
<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>
</div>
<div class="specification">
<p>The function \(g\) is defined by \(g(x) = x\sqrt {f(x)} \) for \(\left| x \right| > a\).</p>
</div>
<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>
</div>
<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>
</div>
<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>
</div>
<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>
</div>
<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>
</div>
<br><hr><br><div class="specification">
<p class="p1">The functions \(f\) and \(g\) are defined by \(f(x) = 2x + \frac{\pi }{5},{\text{ }}x \in \mathbb{R}\) and \(g(x) = 3\sin x + 4,{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(g \circ f(x) = 3\sin \left( {2x + \frac{\pi }{5}} \right) + 4\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the range of \(g \circ f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(g \circ f\left( {\frac{{3\pi }}{{20}}} \right) = 7\), find the next value of \(x\), greater than \({\frac{{3\pi }}{{20}}}\), for which \(g \circ f(x) = 7\).</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">The graph of \(y = g \circ f(x)\) can be obtained by applying four transformations to the graph of \(y = \sin x\). State what the four transformations represent geometrically and give the order in which they are applied.</p>
<div class="marks">[4]</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="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="specification">
<p class="p1">Let \(p(x) = 2{x^5} + {x^4} - 26{x^3} - 13{x^2} + 72x + 36,{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For the polynomial equation \(p(x) = 0\), state</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>the sum of the roots;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the product of the roots.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A new polynomial is defined by \(q(x) = p(x + 4)\).</p>
<p class="p1">Find the sum of the roots of the equation \(q(x) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</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">The quadratic equation \({x^2} - 2kx + (k - 1) = 0\) has roots \(\alpha \) and \(\beta \) such that \({\alpha ^2} + {\beta ^2} = 4\). Without solving the equation, find the possible values of the real number \(k\).</p>
</div>
<br><hr><br><div class="specification">
<p>A given polynomial function is defined as \(f(x) = {a_0} + {a_1}x + {a_2}{x^2} + \ldots + {a_n}{x^n}\). The roots of the polynomial equation \(f(x) = 0\) are consecutive terms of a geometric sequence with a common ratio of \(\frac{1}{2}\) and first term 2.</p>
<p>Given that \({a_{n - 1}} = - 63\) and \({a_n} = 16\) find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">the degree of the polynomial;</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 value of \({a_0}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the function <em>f</em> , where \(f(x) = \arcsin (\ln x)\).</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;">(a) Find the domain of <em>f</em> .</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) Find \({f^{ - 1}}(x)\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\frac{1}{{\sqrt n + \sqrt {n + 1} }} = \sqrt {n + 1} - \sqrt n \) where \(n \ge 0,{\text{ }}n \in \mathbb{Z}\).</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 show that \(\sqrt 2 - 1 < \frac{1}{{\sqrt 2 }}\).</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>Prove, by mathematical induction, that \(\sum\limits_{r = 1}^{r = n} {\frac{1}{{\sqrt r }} > \sqrt n } \) for \(n \ge 2,{\text{ }}n \in \mathbb{Z}\).</p>
<div class="marks">[9]</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(g(x) = {\log _5}\left| {2{{\log }_3}x} \right|\) . Find the product of the zeros of <em>g</em> .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The functions \(f\) and \(g\) are defined by \(f(x) = a{x^2} + bx + c,{\text{ }}x \in \mathbb{R}\) and \(g(x) = p\sin x + qx + r,{\text{ }}x \in \mathbb{R}\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}p,{\text{ }}q,{\text{ }}r\) are real constants.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(f\) is an even function, show that \(b = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(g\) is an odd function, find the value of \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The function \(h\) is both odd and even, with domain \(\mathbb{R}\).</p>
<p class="p1">Find \(h(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial \(q(x) = 3{x^3} - 11{x^2} + kx + 8\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(q(x)\) has a factor \((x - 4)\), find the value of \(k\).</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>Hence or otherwise, factorize \(q(x)\) as a product of linear factors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial \(P\left( z \right) = {z^5} - 10{z^2} + 15z - 6,{\text{ }}z \in \mathbb{C}\).</p>
</div>
<div class="specification">
<p>The polynomial can be written in the form \(P(z) = {(z - 1)^3}({z^2} + bz + c)\).</p>
</div>
<div class="specification">
<p>Consider the function \(q\left( x \right) = {x^5} - 10{x^2} + 15x - 6,{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the sum and the product of the roots of \(P(z) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \((z - 1)\) is a factor of \(P(z)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(b\) and the value of \(c\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the complex roots of \(P(z) = 0\).</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>Show that the graph of \(y = q(x)\) is concave up for \(x > 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = q(x)\) showing clearly any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The function \(f\) is defined as \(f(x) = \frac{{3x + 2}}{{x + 1}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 1\).</p>
<p class="p1">Sketch the graph of \(y = f(x)\), clearly indicating and stating the equations of any asymptotes and the coordinates of any axes intercepts.</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The cubic equation \({x^3} + p{x^2} + qx + c = 0\)<span class="s1">, has roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \)</span>. By expanding \((x - \alpha )(x - \beta )(x - \gamma )\) show that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) \(p = - (\alpha + \beta + \gamma )\);</p>
<p>(ii) \(q = \alpha \beta + \beta \gamma + \gamma \alpha \);</p>
<p>(iii) \(c = - \alpha \beta \gamma \).</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>It is now given that \(p = - 6\) and \(q = 18\) for parts (b) and (c) below.</p>
<p>(i) In the case that the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form an arithmetic sequence, show that one of the roots is \(2\).</p>
<p>(ii) Hence determine the value of \(c\).</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">In another case the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) <span class="s1">form a geometric sequence. Determine the value of \(c\).</span></p>
<div class="marks">[6]</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;">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>The function \(f\) is defined by \(f(x) = 2{x^3} + 5,{\text{ }} - 2 \leqslant x \leqslant 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of \(f\).</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 an expression for \({f^{ - 1}}(x)\).</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>Write down the domain and range of \({f^{ - 1}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</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 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>
</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;">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>
</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;">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>
</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;">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>
</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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = \frac{{1 - 3x}}{{x - 2}}\), showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_17.42.06.png" alt="N17/5/MATHL/HP1/ENG/TZ0/06.a"></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 or otherwise, solve the inequality \(\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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="specification">
<p class="p1">A rational function is defined by \(f(x) = a + \frac{b}{{x - c}}\) where the parameters \(a,{\text{ }}b,{\text{ }}c \in \mathbb{Z}\) and \(x \in \mathbb{R}\backslash \{ c\} \). The following diagram represents the graph of \(y = f(x)\).</p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_09.42.27.png" alt="N16/5/MATHL/HP1/ENG/TZ0/03"></p>
<p class="p1">Using the information on the graph,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">state the value of \(a\) <span class="s1">and the value of </span>\(c\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">find the value of \(b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The equation \(5{x^3} + 48{x^2} + 100x + 2 = a\) has roots \({r_1}\), \({r_2}\) and \({r_3}\).</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;">Given that \({r_1} + {r_2} + {r_3} + {r_1}{r_2}{r_3} = 0\), find the value of <em>a</em>.</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;">Consider the equation \(9{x^3} - 45{x^2} + 74x - 40 = 0\) .</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;">Write down the numerical value of the sum and of the product of the roots of this equation.</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: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The roots of this equation are three consecutive terms of an arithmetic sequence.</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;">Taking the roots to be \(\alpha {\text{ , }}\alpha \pm \beta \) , solve the equation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let <em>f</em>(<em>x</em>) = <em>x</em><sup>4</sup> + <em>px</em><sup>3</sup> + <em>qx</em> + 5 where <em>p</em>, <em>q</em> are constants.</p>
<p>The remainder when <em>f</em>(<em>x</em>) is divided by (<em>x</em> + 1) is 7, and the remainder when <em>f</em>(<em>x</em>) is divided by (<em>x</em> − 2) is 1. Find the value of <em>p</em> and the value of <em>q</em>.</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;">The cubic polynomial \(3{x^3} + p{x^2} + qx - 2\) has a factor \((x + 2)\) and leaves a remainder 4 when divided by \((x + 1)\). Find the value of <em>p </em>and the value of <em>q</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The quadratic equation \(2{x^2} - 8x + 1 = 0\) has roots \(\alpha \) and \(\beta \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Without solving the equation, find the value of</p>
<p>(i) \(\alpha + \beta \);</p>
<p>(ii) \(\alpha \beta \).</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">Another quadratic equation \({x^2} + px + q = 0,{\text{ }}p,{\text{ }}q \in \mathbb{Z}\) has roots \(\frac{2}{\alpha }\) and \(\frac{2}{\beta }\).</p>
<p class="p1">Find the value of \(p\) and the value of \(q\).</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: 12.0px Times;"><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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \frac{{2{x^2} + 3}}{{75}},{\text{ }}x \geqslant 0\]</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;">\[g(x) = \frac{{\left| {3x - 4} \right|}}{{10}},{\text{ }}x \in \mathbb{R}{\text{ }}.\]</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 range of <em>f </em>and of <em>g </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 an expression for the composite function \(f \circ g(x)\) in the form \(\frac{{a{x^2} + bx + c}}{{3750}}\), where \(a,{\text{ }}b{\text{ and }}c \in \mathbb{Z}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find an expression for the inverse function \({f^{ - 1}}(x)\) .</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;">(ii) State the domain and range of \({f^{ - 1}}\)<span style="font: 7.0px Helvetica;"> </span>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="line-height: normal; font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">The domains of <em>f</em> and <em>g</em> are now restricted to {0, 1, 2, 3, 4} .</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;">By considering the values of <em>f </em>and <em>g </em>on this new domain, determine which of <em>f </em>and <em>g </em>could be used to find a probability distribution for a discrete random variable <em>X </em>, stating your reasons clearly.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Using this probability distribution, calculate the mean of <em>X </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the graphs of \(y = \left| x \right|\) and \(y = - \left| x \right| + b\), where \(b \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graphs on the same set of axes.</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>Given that the graphs enclose a region of area 18 square units, find the value of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the functions given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\[f(x) = 2x + 3\]\[g(x) = \frac{1}{x},x \ne 0\]<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(\left( {g \circ f} \right)\left( x \right)\) and write down the domain of the function.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(\left( {f \circ g} \right)\left( x \right)\) and write down the domain of the function.</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><span style="font-family: times new roman,times; font-size: medium;">Find the coordinates of the point where the graph of \(y = f(x)\) and the graph of \(y = \left( {{g^{ - 1}} \circ f \circ g} \right)(x)\) intersect.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<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>
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<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>
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<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>
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<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>
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<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>
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<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A function <em>f</em> is defined by \(f(x) = \frac{{2x - 3}}{{x - 1}},{\text{ }}x \ne 1\).</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 an expression for \({f^{ - 1}}(x)\).</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;">(b) Solve the equation \(\left| {{f^{ - 1}}(x)} \right| = 1 + {f^{ - 1}}(x)\).</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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>
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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;">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>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider a function <em>f </em>, defined by \(f(x) = \frac{x}{{2 - x}}{\text{ for }}0 \leqslant x \leqslant 1\) .</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \((f \circ f)(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<div class="marks">[3]</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 Helvetica;"><span style="font-family: 'times new roman',times; font-size: medium;">Let \({F_n}(x) = \frac{x}{{{2^n} - ({2^n} - 1)x}}\), where \(0 \leqslant x \leqslant 1\).</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;">Use mathematical induction to show that for any \(n \in {\mathbb{Z}^ + }\)</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;">\[\underbrace {(f \circ f \circ ... \circ f)}_{n{\text{ times}}}(x) = {F_n}(x)\] .</span></p>
<div class="marks">[8]</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({F_{ - n}}(x)\) is an expression for the inverse of \({F_n}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State \({F_n}(0){\text{ and }}{F_n}(1)\) .</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;">(ii) Show that \({F_n}(x) < x\) , given 0 < <em>x </em>< 1, \(n \in {\mathbb{Z}^ + }\) .</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;">(iii) For \(n \in {\mathbb{Z}^ + }\) , let \({A_n}\) be the area of the region enclosed by the graph of \(F_n^{ - 1}\) , the <em>x</em>-axis and the line <em>x </em>= 1. Find the area \({B_n}\) of the region enclosed by \({F_n}\) and \(F_n^{ - 1}\) in terms of \({A_n}\) .<span style="font: 7.0px Helvetica;"><br></span></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the set of values of \(a\) for which the function \(x \mapsto {\log _a}x\) exists, for all \(x \in {\mathbb{R}^ + }\).</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"><span class="s1">Given that \({\log _x}y = 4{\log _y}x\)</span>, find all the possible expressions of \(y\) as a function of \(x\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When the function \(q(x) = {x^3} + k{x^2} - 7x + 3\) is divided by (<em>x</em> + 1) the remainder is seven times the remainder that is found when the function is divided by (<em>x</em> + 2) .</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;">Find the value of <em>k</em> .</span></p>
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<br><hr><br><div class="question">
<p>Solve \({\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}\).</p>
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<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 roots of a quadratic equation \(2{x^2} + 4x - 1 = 0\) are \(\alpha \) and \(\beta \).</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;">Without solving the equation,</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 the value of \({\alpha ^2} + {\beta ^2}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) find a quadratic equation with roots \({\alpha ^2}\) and \({\beta ^2}\).</span></p>
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<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;">When the polynomial \(3{x^3} + ax + b\) is divided by \((x - 2)\), the remainder is 2, and when divided by \((x + 1)\), it is 5. Find the value of <em>a </em>and the value of <em>b</em>.</span></p>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch on the same axes the curve \(y = \left| {\frac{7}{{x - 4}}} \right|\) and the line \(y = x + 2\), clearly indicating any axes intercepts and any asymptotes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the exact solutions to the equation \(x + 2 = \left| {\frac{7}{{x - 4}}} \right|\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows a sketch of the graph of \(y = f(x)\).</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_2014-09-16_om_05.50.26.png" alt></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;">Sketch the graph of \(y = {f^{ - 1}}(x)\) on the same axes.</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;">State the range of \({f^{ - 1}}\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f(x) = \ln (ax + b),{\text{ }}x > 1\), find the value of \(a\) and the value of \(b\).</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the set of values of <em>x</em> for which \(\left| {x - 1} \right| > \left| {2x - 1} \right|\).</span></p>
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<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{4}{{x + 2}},{\text{ }}x \ne - 2{\text{ and }}g(x) = x - 1\).</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;">If \(h = g \circ f\) , find</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) <em>h</em>(<em>x</em>) ;</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) \({h^{ - 1}}(x)\) , where \({h^{ - 1}}\) is the inverse of <em>h</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is given by \(f(x) = \frac{{{3^x} + 1}}{{{3^x} - {3^{ - x}}}}\), for <em>x</em> > 0.</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f(x) > 1\) for all <em>x</em> > 0.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the equation \(f(x) = 4\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">(i) Express each of the complex numbers \({z_1} = \sqrt 3 + {\text{i, }}{z_2} = - \sqrt 3 + {\text{i}}\) and \({z_3} = - 2{\text{i}}\) in modulus-argument form.</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;">(ii) Hence show that the points in the complex plane representing \({z_1}\), \({z_2}\) and \({z_3}\) form the vertices of an equilateral triangle.</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;">(iii) Show that \({\text{z}}_1^{3n} + z_2^{3n} = 2z_3^{3n}\) where \(n \in \mathbb{N}\).</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them in modulus-argument form.</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) If <em>w</em> is the solution to \({z^7} = 1\) with least positive argument, determine the argument of 1 + <em>w</em>. Express your answer in terms of \(\pi \).</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) Show that \({z^2} - 2z\cos \left( {\frac{{2\pi }}{7}} \right) + 1\) is a factor of the polynomial \({z^7} - 1\). State the two other quadratic factors with real coefficients.</span></p>
<div class="marks">[9]</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given the complex numbers \({z_1} = 1 + 3{\text{i}}\) and \({z_2} = - 1 - {\text{i}}\).</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;">Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).</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: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).</span></p>
<div class="marks">[5]</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The same remainder is found when \(2{x^3} + k{x^2} + 6x + 32\) and \({x^4} - 6{x^2} - {k^2}x + 9\) are divided by \(x + 1\) . Find the possible values of <em>k </em>.</span></p>
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<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When \(3{x^5} - ax + b\) is divided by <em>x</em> −1 and <em>x</em> +1 the remainders are equal. Given that a , \(b \in \mathbb{R}\) , find</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) the value of <em>a</em> ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) the set of values of <em>b</em> .</span></p>
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<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Express the quadratic \(3{x^2} - 6x + 5\) in the form \(a{(x + b)^2} + c\), where <em>a</em>, <em>b</em>, <em>c </em>\( \in \mathbb{Z}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Describe a sequence of transformations that transforms the graph of \(y = {x^2}\) to the graph of \(y = 3{x^2} - 6x + 5\).</span></p>
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<br><hr><br><div class="specification">
<p class="p1">The function \(f\) is defined by \(f(x) = \frac{1}{x},{\text{ }}x \ne 0\).</p>
<p class="p1">The graph of the function \(y = g(x)\) is obtained by applying the following transformations to</p>
<p class="p1">the graph of \(y = f(x)\) :</p>
<p class="p1"> \({\text{a translation by the vector }}\left( {\begin{array}{*{20}{c}}{ - 3} \\ 0 \end{array}} \right);\) \({\text{a translation by the vector }}\left( {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right);\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \(g(x)\).</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">State the equations of the asymptotes of the graph of \(g\).</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: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Factorize \({z^3} + 1\) into a linear and quadratic factor.</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: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\gamma \) is one of the cube roots of −1.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that \({\gamma ^2} = \gamma - 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence find the value of \({(1 - \gamma )^6}\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</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="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(A{x^3} + B{x^2} + x + 6\) is exactly divisible by \((x +1)(x − 2)\), find the value of <em>A</em> and the value of <em>B</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: 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="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 \(y = \left| {\cos \left( {\frac{x}{4}} \right)} \right|\) for \(0 \leqslant x \leqslant 8\pi \).</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Solve \(\left| {\cos \left( {\frac{x}{4}} \right)} \right| = \frac{1}{2}\) for \(0 \leqslant x \leqslant 8\pi \).</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: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The functions <em>f</em> and <em>g</em> are defined as:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = {{\text{e}}^{{x^2}}},{\text{ }}x \geqslant 0\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\[g(x) = \frac{1}{{x + 3}},{\text{ }}x \ne - 3.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(h(x){\text{ where }}h(x) = g \circ f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) State the domain of \({h^{ - 1}}(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find \({h^{ - 1}}(x)\) .</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;">When \(f(x) = {x^4} + 3{x^3} + p{x^2} - 2x + q\) is divided by (<em>x</em> − 2) the remainder is 15, and (<em>x</em> + 3) is a factor of <em>f</em>(<em>x</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;">Find the values of <em>p</em> and <em>q</em> .</span></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;">Solve the following equations:</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) \({\log _2}(x - 2) = {\log _4}({x^2} - 6x + 12)\);</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) \({x^{\ln x}} = {{\text{e}}^{{{(\ln x)}^3}}}\).</span></p>
</div>
<br><hr><br><div class="specification">
<p>The function \(f\) is defined by \(f\left( x \right) = \frac{{ax + b}}{{cx + d}}\), for \(x \in \mathbb{R},\,\,x \ne - \frac{d}{c}\).</p>
</div>
<div class="specification">
<p>The function \(g\) is defined by \(g\left( x \right) = \frac{{2x - 3}}{{x - 2}},\,\,x \in \mathbb{R},\,\,x \ne 2\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function \({f^{ - 1}}\), stating its domain.</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>Express \(g\left( x \right)\) in the form \(A + \frac{B}{{x - 2}}\) where A, B are constants.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = g\left( x \right)\). State the equations of any asymptotes and the coordinates of any intercepts with the axes.</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>The function \(h\) is defined by \(h\left( x \right) = \sqrt x \), for \(x\) ≥ 0.</p>
<p>State the domain and range of \(h \circ g\).</p>
<div class="marks">[4]</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: 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="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>
</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 \(f(x)\) 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 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="specification">
<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 random variable <em>X</em> has probability density function <em>f</em> where</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;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {kx(x + 1)(2 - x),}&{0 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise }}{\text{.}}} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of the function. You are not required to find the coordinates of the maximum.</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: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>k</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</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;">Shown below are the graphs of \(y = f(x)\) and \(y = g(x)\).</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;">If \((f \circ g)(x) = 3\), find all possible values of <em>x</em>.</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 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 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><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows the graph of the function \(y = f(x)\) , defined for all \(x \in \mathbb{R}\),</span><br><span style="font-family: times new roman,times; font-size: medium;">where \(b > a > 0\) .</span></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt><br><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(g(x) = \frac{1}{{f(x - a) - b}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the largest possible domain of the function \(g\) .</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="text-align: justify;"><span style="font-family: times new roman,times; font-size: medium;">On the axes below, sketch the graph of \(y = g(x)\) . On the graph, indicate any asymptotes and local maxima or minima, and write down their equations and coordinates</span>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<div class="marks">[6]</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;">Given that \(f(x) = 1 + \sin x,{\text{ }}0 \leqslant x \leqslant \frac{{3\pi }}{2}\),</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: 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>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graphs of \(y = \frac{x}{2} + 1\) and \(y = \left| {x - 2} \right|\) on the following axes.</p>
<p><img 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"></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>Solve the equation \(\frac{x}{2} + 1 = \left| {x - 2} \right|\).</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: 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="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 a polynomial function <em>f </em>of degree 4 is shown below.</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 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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;">Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in \mathbb{R}\) . 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;">(i) \({x^2} - {y^2} = - 5\) ;</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) \(xy = 6\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A.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;">Hence find the two square roots of \( - 5 + 12{\text{i}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">A.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;">For any complex number <em>z </em>, show that \({(z^*)^2} = ({z^2})^*\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">A.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence write down the two square roots of \( - 5 - 12{\text{i}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why, of the four roots of the equation \(f(x) = 0\) , two are real and two are complex.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve passes through the point \(( - 1,\, - 18)\) . Find \(f(x)\) in the form</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;">\(f(x) = (x - a)(x - b)({x^2} + cx + d),{\text{ where }}a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\)<em> </em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">B.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 two complex roots of the equation \(f(x) = 0\) in Cartesian form.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw the four roots on the complex plane (the Argand diagram).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Express each of the four roots of the equation in the form \(r{{\text{e}}^{{\text{i}}\theta }}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">B.e.</div>
</div>
<br><hr><br><div class="question">
<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 graph of \(y = \frac{{a + x}}{{b + cx}}\) is drawn below.</span></p>
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<p style="font: normal normal normal 32px/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: 32.0px 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: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the value of <em>a</em>, the value of <em>b</em> and the value of <em>c</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) Using the values of <em>a</em>, <em>b</em> and <em>c</em> found in part (a), sketch the graph of \(y = \left| {\frac{{b + cx}}{{a + x}}} \right|\) on the axes below, showing clearly all intercepts and asymptotes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 26px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</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;">The graph below shows \(y = f(x)\) , where \(f(x) = x + \ln x\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </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;">(a) On the graph below, sketch the curve \(y = {f^{ - 1}}(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> <br></span></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>
<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;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the coordinates of the point of intersection of the graph of \(y = f(x)\) and the graph of \(y = {f^{ - 1}}(x)\) .</span></p>
</div>
<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="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 diagram shows the graph of <em>y</em> = <em>f</em>(<em>x</em>) . The graph has a horizontal asymptote at <em>y</em> = 2 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = \frac{1}{{f(x)}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Sketch the graph of \(y = x{\text{ }}f(x)\) .</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 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="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;">The diagram below shows a solid with volume <em>V</em> , obtained from a cube with edge \(a > 1\) when a smaller cube with edge \(\frac{1}{a}\) is removed.</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 src="data:image/png;base64,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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;">Let \(x = a - \frac{1}{a}\)</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 <em>V</em> in terms of <em>x</em> .</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) Hence or otherwise, show that the only value of <em>a</em> for which <em>V</em> = 4<em>x</em> is \(a = \frac{{1 + \sqrt 5 }}{2}\) .</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 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,iVBORw0KGgoAAAANSUhEUgAAAZIAAAGMCAIAAAB/PLNGAAAgAElEQVR4nO2dXWwc15Xn+3EeCAQyHM9odzPkpleAHowxBoJh0AgmtjFjOPGy5TGgzCi7K4jOTihF0kqJYzk2NnLi0PIsmlAcwTa9q+VmzdVYtMWmGLZH/oCVmWhj0rB6VjIsAaYRydAXEUvojA2KH8Om6u5DKa1mf1RX1b331Dl1/3/0A6vZ+tXt01W/uveo2Z1RCBIhn58Z3bsjt27VxpHznlJqYXpo49pHRs9fT3pciEvJJD0ARFyWPzv25Jc6dhSvLCu1PHvm77Z8bf+p5aQHhbgUaAuJHO/c8PrMfxg+t6iUVzk/+sT+f5pLekiIU4G2kOiZKWzK3JMvzSrv8lv/bejE55hrIaSBtpDoWZzq71q3+9iF80cG9p8oe0kPB3Et0BYSPZVSPvulrz26Z+/fnVlIeiyIg4G2kBi5UNi09s7H35ip3JxplcvlL37htnK5nOCwEEcCbSHRs/Thf//e0IezK1paY2Njt3TcOjY2ltSgEHcCbSERU7l4bP+BYzOLtffNz89339X9xS/c1n1X9/z8fFJDQxwJtIWEjPcvp1+6f/X9mx7bX+cspdTY2Fj3Xd3/9kud3Xd1Y8KF2A60hYSNNzv9y9cnz802ebuDb6tsZ5fvL/qxIU4F2kJ08+6v372l49ZyuZzt7Jqfn7+l49Z3f/1u0oNC0hxoC9HNyZMn/YVhtrNLKTU2Nvb2W28nPSgkzYG2EGPxtYUgtgNtISGzeOXUWwcH83ueOnR6rvkb46EthCbQFhIilYvHntlw56YXJhv+D7E20BZCE2gLaZfKxXf2PLi2d+TsUpu/PoS2EJpAW0hwFs6P9K3q2Dx8tv1fH0JbCE2gLSQwc5P9t3d05H7yyqGX8vl9g6++X/t3iHWBthCatNfWRPHoRPFo7c+Nm/49kTYDaPz3levJbd+2I/VPc6J4dHx/35czHev6XvzxwHP5H/Xd+6/+8E82/u0Vrzkt29lVd2P7vIzvK2VPc7Qw3t+/1+q+lEZCaUtnB6mE5wf2ZTu7Bl8ctAHnVJPKTKEv438ioFJKXTsz+HAms2FouvmC0epsi1NZWJDtwefn57+zdWuuJ2cD7gfaSgA+UTw6OTlpyVycarJ8pbij46a2vMWpZ7pubtYH2qIkW4L7zvrO1q2jhXHj8GqsawtpFXvm4pPr00P3Z+7Y+eZv/c1KKZ9NaLaFEKTqLOYf44HZlhbZhrl41cS7UNy6bvWuN8ueUqryaXHX7VsnZlo05THboiQbh9c5i/PIoS1dsnFzcauJN/vB0OYHNuUPFg7lt/Q+96vW7ziFtijJZuGN8yzOI4e2DJDNmotlTRavTp8sTV+tBD4I2qIkG4Q3XRtyHjm0ZYZs0FxCa6KgLVqyKXirfhbnkaMlbywudOiDg5a8uEjpwdcFsy2TZCPmEloThdkWLVkfHuwsziOHtgyT9c0ltCYK2qIla8LbzrPYjlxBWzbImuYSWhMFbdGSdeBh1oY8R+4H2rJC1jGX0JooaIuWHBsesp/FcOTVoCVvKw526NGS5x+hPfi6QFsW45q5oC3mSYezFBaJtskxzCW0JgqLRFpyVHhUZ/EZeWOgLevkqOYSWhMFbdGSI8FjzLOYjLxpoC0KciRzCa2JgrZoyeHh8daGHEbeKuhtEcWFPhd6WwyTmn5WbaAtuqTeXNAWt6TSWQqLRGJyGHMJrYnCIpGW3Bau6SzOBYe2qMltzSW0JgraoiUHw/XnWZwLDm0lQA42l9CaKGiLlhwAN7I25Fxw9LaSSSr7XOhtcUha+1m1gbYSS/rMBW0lHhecpbBITJbc1FxCa6KwSKQlN8LNOotzwaGthMmN5hJaEwVt0ZLr4MbnWZwLDm0lT64zl9CaKGiLllwLt7E25Fxw9LZYJB19LvS2Eokj/azaQFtckgJzQVv0cdBZKoy2vv/93ZOTk5Z2z3kiSk/2zbV92w4bcD9YJBLDrQ57tDBuz1mcC95eW9u37ch2dvnmmige9W/VfdfeE2kzgBa8yXxfmrvOD+zzzUX5NE0962xnV91N3MuXYA2j7mu0MJ7ryeV6cqOFcYklVRoJ1ZIffHGwai6z0Rx9UnCrw/bNZWm1aHXkmG2Rkf21Ya4nZ29tyLngYf8n0ZK5OJcmEbIPt9fngraI4TbI1X6WP8+yFM4Fj9CStzfnQhojsUOPljxB3OzB1yXa+7aMm4uz0RMh18JtmAuzLWK4WXKds4TWRB8e+e2mZs3FuTSJkOvgxs0FbRHDDZIb51lCa6IPj/MueYPm4lyaRMiNcLPmgraI4abITdeGQmuiD4/5xz2mzMW5NImQm8INmgvaIoYbIbfqZwmtiT48/rvk0aGnjIgOPVryNoIefGO0/pRa31ycjZ4IOQBuxFyYbRHDNcnBzhJaE3247idAaJqLc2kSIQfD9c0FbRHDdcht51lCa6IPN/DBNTrm4lyaRMht4ZrmgraI4bHJYdaGQmuiDzfzeVuxzcW5NImQw8B1zAVtEcPjkUP2s4TWRB9u7INr0KGnDM8OPVryRoIefNuY/HTTGObibPREyOHh8cyF2RYxPCo5krOE1kQfbvhDmaOai3NpEiFHgscwF7RFDI9EjjrPEloTfbj5z5KPZC7OpUmEHBUe1VzQFjE8PDnG2lBoTfThVr4CI7y5OJcmEXIMeCRzQVvE8JDkeP0soTXRh9v6LHl06CnDpEOPlny8oAcfNRa/cCyMuTgbPRFybHhIc2G2RQxvS9ZxltCa6MPtfk9iW3NxLk0iZB14GHNBW8TwYLLmPEtoTfTh1r/eNdhcnEuTCFkT3tZc0BYxPICsvzYUWhN9OMW3UgeYi3NpEiHrw4PNBW0Rw1uRjfSzhNZEH0709a7o0FMmqQ49WvIhgx68Zui+lRrmokwi5oK2wgTO0g/FIrGaRnNxnogmQjYIb2ouDbhXOf/6nh0j08stH4FFYluyWWcJrYk+nFRbqsFcnEuTCNksvNFc8eFL08Mb12Sy+VKl5UOgrWCy8XmW0Jrow6m1pVaai3NpEiEbh9eZKy589qOh7esf7Ia2YpNtrA2F1kQfTtfbqg36XJTR7nN5C6cP9D5x9NRoX4LaEh30s8wmGW0pmIs2OubyPn8v/81nj/9ucaYAbcUJnGU8CSwSq7FtLiwSa+Oba/u2HdH+mVc+MfDowImypyrJaktizZVSo4Vxe84SWhN9eChtVffh/9y46d8TadO/Z/u2HbV9Lqv7irEZY18J7rotPD+wr9ZcIXb9+iv5b3+9b/+spyaKxZef/Hrmj761b7zlSLKdXXU3cS9f7H013fVoYTzXk8v15EYL4+l4mgb3pTSS5GzLh9ubc9kbue2a2IP75gq9WrxQ2JTN1GdtX/Fy00djtlUbf22Y68nZWxuKq4kpeGK9rdqgz0WZKH2ua+eOTxRu5PDQ7vszqzflD08cP3et6aPR26oG/SyrYaEtBXPRJlaHPuHelqDAWbaT/CKx+rNxc2GRGACPbi605EOlzllShi0LzkhbyrS5oK1guPG/W4S2GudZIoYtDs5LW8qouaCttnCz5nJcW03XhvyHLRHOpbdVG/S5KGPQXC73ttDPogxHbSmYizamzOWstuAs4rBbJFajby4sEsPDfXONHzmiA3dzkRjsLLbDFg3nqy2lbS5oKxLcN5fOdcJBbbWdZ/EctnQ4a20pPXNBW1HhmuZyTVth1oYMh50COHdtKQ1zQVsx4DrmkqutGAnZz0rxoZIgnGlLvi7o0FMmtrncacmjB59sBMy2/MQwF2ZbseHxzCV3thUJHslZfIadJrgYbano5oK2dOAxzOWCtqLOs5gMO2VwSdpSEc0FbWnCo5or9dqKsTbkMOz0wYVpS0UxF7SlD49krnRrK14/K/FhpxIuoyVfF3ToKRPeXCluyaMHzyryZlt+wpgLsy1T8JDmSutsS8dZDF/NFMClakuFMBe0ZRAexlyp1JbmPIvnqykdLlhbqp25oC2z8LbmSp+29NeGbF9N0XDZ2lKB5oK2jMODzZUybRnpZ3F+NeXCRbbk64IOPWUCzJWmljx68Jwjfrblp6m5MNuyBG9lrtTMtgw6i/+rKRGeEm2pZuaCtuzBm5orHdoyO88S8WqKg6dHW6rBXNCWVXijuVKgLeNrQymvpix4qrSlVpoL2rINrzOXdG3Z6GcJejUFwdPQkq8LOvSUqTWX6JY8evCCkkJtKZiLNlVzydXWaGEczhKUtC0Sq7FqLqE1sQf3nSV0kTg/P5/ryVlylsRXkz88lLaq+/B/btz074m0GUAzta/t23bU9rms7ivBp6m/LyPPOj+wzzdX7Y3V82o6ktHCeK4nl+vJjRbGk62hI/u6dOnStx75z9Vqx0tqZ1s+3NKcS3RN7MF9VQma4Vb7WZpnUUDkvpo24JcuXbr3q1/VL3g6e1u1QZ+LLNUOl4hqowdPnKqz9Auefm0pmIsqfm9LhLngLOIYdJZK/SKx+rNZc6WjJsZTbcnbMJfBkTc6y15Z5L6aBuGNztKEu6ItZdRcqamJ2dT+T6Jxc5kaedN5FrRlD950ngVtRYCbMleaamIwdW+AMGsuIyNvtTaEtizBW60NrWsrZUGfy14a37fFqs+FfhZx5ufn//ob37BRcOe0pWAua2n6dlMm5oKziGO14G4tEqvRNFcqa6KfVu+SN2IunZG3PYWwSDQLL5fLwfMs9LZiwnXMldaaaCbgj3v0zRV75GEu+9CWQThBwd3VltIwV4propPgv0nUNFe8kYdcqkBbpuA0BXext1Ub9LkMpu2fUhP3udDPIg5ZwV3XloK5zCXMJ0CQmQvOIg5lwZ1eJFYT1VxMhs0NHvKDa+KZK9LIo55CWCRqwokLDm3dSCRz8Rk2K3j4z9uKYa7wI49x2Ye2dOD0BYe2bia8uVgNmw880scERjVXyJHHW6pAW7HhiRQc2lqRkObiNmwm8KifbhrJXGFGHru9Am3FgydVcLTk64MOfezE+FBmgx169OCJk2DBMdtqkrbm4jnsxOHxPks+pLmCR655CmG2FRWebMGhreYJNhfbYScLj/0VGGHMFTBy/cs+tBUJnnjBoa2WCTAX52EnCNf55p625mo1ciNLFWgrPJxDwaGtoLQyF/NhJwXX/MKxYHM1Hbmp9gq0FRLOpOBoybcJOvTho/89iZE69OjBE4dPwTHbap9Gc4kYNj3cyNe7tjJX3cjNnkKYbbWFsyo4tBUqdeaSMmwjcG/2N78czu/esmvP4C9OXVkMeKSpb6Vuaq7akRu/7ENbwXBuBYe2wqbWXIKGrQtf+nikt3tdrm/n9ofWdWQyax8rXm5pLlPaUs3MVR25jaUKtBUAZ1hwaCtCquaSNWwNuDc3tf/RwrmKUkotz556Idex6iuDHy63eLRBbakGc/kjt9RegbZawXkWHC35aHGsQ+8tfPLRJwveja1KKZ/NZPOlSotHm9WWajAXn5awI2FbcMy2Ise2ufjWZHGqv2vd7mNXW/3euLZUjblGC+P2TiHMthrDueDQVpxs37bDnrm41sSbKw3cvX5oeslr9Qgb2lK/N9d999xn77IPbdVlfn4+15NjW/BQ2qruw/+5cdO/J9JmAE3EvqrmSsHTDLXrIwe++xd/+diBIwEjyXZ21d2MPK/5+fn77rkv29nV37/X9tO0W0Mhh4rvrFxPbrQwbm9fSiOYbcUnW1otcqyJVz4x8N29k1dbTrSUUnZmW9X2Sn//XokzXI6vZmCqBa86y0asawsJiBMdeu+z0z9/eu87F1t14qsxrq26ljCTb4pNcdj24OsCbekm5ebyPjs9tG39D1+bLJVKpVJp6u2Xf9B/5PJS08ea1VbTUwjmshcpzlJYJBohmzUXp5rMTg/3rcmsyKotxU9brBUNaqvxFKqO3Ia5sEgMKLiNoLeVALyRbNBcQmuizGmr6WW/duTGzeW4ttoW3HigrQTgTcmmzCW0JsqQtlotVepGbtZcLmsrZMHNBi15Rkl5n6td9LUVqb2CPpd+BPWzagNtGY7L5tLUVoxTCObSiVBnKSwSbZA1zSW0JkpPW21PoVYjN2IuBxeJsQtuJOhtJQBvS9Yxl9CaKA1thbnsB4xc31yuaUuz4PqBthKAhyHHNpfQmqi42gq5VAkeuaa5nNKWkYJrBi15vnGtzxVDWwbbK+hzhYncflZtoC27ccpcUbVl/BSCuYKTDmcpLBIJyFHNJbQmKqK2op5CIUcez1xMDhWrcEsFjxf0thKARyVHMpfQmqgo2opx2Q8/8hjm4nOoWIJbLXiMQFsJwGOQw5tLaE1UaG3FW6pEGnlUc7E6VIzDCQoeNWjJi0nq+1xhtEXWXolkLqunaLJJTT+rNtAWadJtrrbaIj6F0KFPpbMUFon05LbmEloT1U5bmqdQvJGHNBfPQ0UTnkjBaeDQVgLkYHMJrYkK1Jb+ZT/2yMOYi+2hEhueYMEJ4NBWMuQAcwmtiWqtLSNLFZ2RtzUX50MlBjzxgtuGQ1uJkVuZS2hNVAttmWqvaI482FzMD5VIcCYFtwpHSz7JpKxD36gtVi1hFzr0rApuL5htJUxuNJfQmqgGbZk9hYyMvJW5RBwqbeEMC24JDm0lT64zl9CaqJXaMn7ZNzXypuaScqgEwNkW3AYc2mJBrjWX0JqoGm3ZWKoYHHmjuQQdKk3hzAtuHA5tcSFXzSW0Jur32rLUXjE78jpzyTpU6uAiCm4WjpY8o0jv0Gc7uwS1hNPRoRdUcIPBbIsX2ba5bM+27J1CNkZeNZfEQ0UpNVoYl1VwU3BoixdZKbV92w575rI38vn5eV9bli77lkbumys/sM8GXFkueK4nJ67gRuChtFXdh/9z46Z/T6TNAJrofRnZddVcNE9T/1mPFsZzPblsZ1fdTcTL19+/t9ZcSdUw0r58Z+V6cqOFcYYlDUNTGsFsixe5Cre0WrQx8mp7xci3UreK1Zrbm+FaLfhoYdw4vBrO5yZa8nwjokNf2xK2qi3bEVFt5WoPvi6YbfEi18GNn0tmR153Csmdbdmb4VotuNDTRx8ObfEiN8LNnksGR9542ZeuLcX7OtFYcKGnjz4c2uJFbgo3eC6ZGnnTpUoKtKW4XieaFlzo6aMPh7Z4kVvBTZ1LRkbeqr2SDm0pfteJVgUXevrow9GSFxMmPeOAlrDolnxdmFRboQffLNCWpCR+LgWfQmnSlmJQbQVntQgWibzIbeGa55LOyNueQqlZJFajby6rBRd6+ujDoS1e5DBwnXMp9sjDXPbTpy2V3HUiTMGFnj76cGiLFzkkPPa5FG/kIZcqqdSWSuI6EbLgiR+HScHR25Iass5L+PZKynpbtaHsc6Gf1TbQluAQnEuRTqEUa0tRmQvOChMsEnmRo8KjnkuR4FFPobQuEquJYS6rBedQk0Tg0BYvcgx4pHMpPDzGZT/12lI2rxMxCs6kJvRwaIsXOR48/LkUEh5vqeKCtpSd60S8gvOpCTEcva2UxGDnJXZ7Jd29rdqY7XOhnxU10FZ6YuRc0jmF3NGWMmcuOCtGsEjkRdaEtz2XguGap5Aji8RqwpjLasEZ1oQGDm3xIuvDg8+lALj+Zd81bSm964R+wXnWhAAObfEiG4EHnEut4EaWKg5qS8W9ThgpONua2Iajt5XOROq8mGqvONXbqk3UPhf6WZqBtlKbkOeSwVPIWW2pKOaCs/SDRSIvsll447lUBzd7Crm5SKymqbmsFpx/TSzB22vL6rEoFC5o2HXnUi3c+GVfUFkswRvNhYLbgENbvMg24LXnUhVuY6kiqyyW4K2uEyi4QTi0xYtsCV49l3y4pfaKuLJYgjdeJ/yC3/vVr5bLZYM7ElST8PAwJYK2eJHtwf1zKdvZZa8lLLEsluC114mqsy5dumR2L7JqEhK+d++z3Xd1j42NBRyf0BYvslW4fy799Te+Yem/sYSWxRK8ep2w5CwlsCZh4OVyeWxsrPuubl9eTSdfobSFG2644UZ5+/Ifd/6bP/zXX/zCbbd03Do2NhZZW5z/HzQpuNBh+3B7bxfKuv0GiKb5/vd325hn+RFakzDwcrn8wvMv3NJx68MPPQxtcSeLhkNblOS0wj+e/njv3md9Yb3763ebPgbvkkeMxaq2EEfy8EMPP7778ZMnTwY8JqS2Fq+ceuvgYH7PU4dOz3lGBkcVr3L+9T07RqaXkx5I6Hizv/nlcH73ll17Bn9x6spi0sOpzfLs9NHBPY8/nn9laqbJwCRqi3G1w0Te4a2UCvZJmCZGiEXi+PCxZzbcuemFyWZHqmasT0SXpoc3rslk86WKabKlLH382J+vXZfr27n9oXUdmczax4qXTZZdY+TLs6f/9+a7dxcvL1TOv7r5jiYDk7dIXPp4pLd7Xa6v599326i2sn2EHzlg4/C+Abc08srFY89sWHPfd0L4xJs7cyS/88FVa7aOnJ3zrvzfgb+6o+OBoenlttqqXOzfeOfa3pGzS1YmWZa1NfbR0Pb1D3bL0ZY3N7X/oSf/V0UppZZnT72Q61j1lcEPDV5K44986czQ+jV3/+zkvyilVPn4D+9es+uNKysPCmna8uam9j9aOFdRaqL4uo1qK7tH+Ozgf+mxcXj7sTLyysV39jy4tnfkfxwJDffOjWxcc8fTQy8/O/SPp0rvnfm00m62tXB+pK/jD/5i+OyC9nibx+aL6o2+sLP3iaOnRvsEaWvhk4/+Z+Hvb2xVSvms4bHHHvnymcGvZO7Y+eZvlVJKLV0Y6c10bCnMrBiaOG0tfPLRJwveDbiFaiurh8rpA3++4Wkbh7cfCyNfOD/St6pj8/DZhSjwhemhDZlVvcO/maveFaitucn+2zs6cj955dBL+fy+wVffn6mIaWx5n7+X/+azx3+3OFOw9bpaz+JUf9e63ceuJj0OpVRlptCXydyTL80qpZS6/vmxJ1bf3LwRib2tm2FU7fYReXjH9Il37fhTX1p5jQzS1vKpn/1pZtW6vhffnHp/6o2fbVrTde9T/3BFhLi88omBRwdOlD3/fJPyuq6IN1cauHv90LSd5XnEzJ0ZXJ/JrB88c+OKVynls5m1fcXLtQ+SrC1W1W4XmYd3TJ94F17/yaavrV5xRQnQ1o0LbO9P/fd6XTsz+HAms2Fo2uSC0c4Uevl3kz/dur/0avGosvO6UrylZWl6+JHvjZyda/fwWPDIWbow0lujLX+2VT83kbZIrIEfOWCj2srKsG8c3rOemigW7WnL9MhXTNgnimPtfVL5/J+vzV1+ff/z7/+/0c1r78i/d/XEa6+duaaUyly/cOz5fNO8+g+Ht3fc1Ja3OPVMV8O6QDM6pWk98h99+89uy9SnfmqQ1LADR3741Oz1ieJR5ZVPDHx37+RV45f+uCNfLr/56Kqbr75/CN6cfPmRqi2vvO+Rv7RRbWVl2BcKm7JWD28/pke+fKW4o+Omto6288ncmcH1HX/6tZ0HzyyoxbPD/7Ej+8Cugx/MekoFLxKvTw/dn7mj58ev+JuVUj4rY7Z17dzxiUKh8MST/7VQODy0+/7M6k35wxPHz10ztQPLs63Dp3/+9N53LtqY+MceuXe50Luqem7MnRlcn7l979TKN92I1Jb32emfP/2f+octLbMsDPvG4V0oFJ548kkbh7cf4yP3feL/r85E8Whbn3izFz6Yvnrjdalc/fjjq9XXKLAl710obl23etebZU8pVfm0uOv2rRMzEtb+NZG0+FdKKe+z00Pb1v/wtclSqVQqlabefvkH/UcuLyU9LKW8K8f33HvjYPDOjWy8c+PwdN2w5PW22FY7bEQd3uZ80uZ9W97sB0ObH9iUP1g4lN/S+9yvLLzj1HJEva5qdnq4b83K2f+qLcVPeVwqvNkPhnr/aueLB1/u/5uNz7zT+N9A0rTFutrhIuvwNuaTMH9K/Yur0ydL01dZrVmShQsdtgG4N3vhg3/64MJs01Nb5CLRMlzosG3CF69On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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,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" 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>