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</div><h2>HL Paper 2</h2><div class="specification">
<p class="p1">The graph of \(y = \ln (5x + 10)\) is obtained from the graph of \(y = \ln x\) by a translation of \(a\) units in the direction of the \(x\)-axis followed by a translation of \(b\) units in the direction of the \(y\)-axis.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\) and the value of \(b\).</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 region bounded by the graph of \(y = \ln (5x + 10)\), the \(x\)-axis and the lines \(x = {\text{e}}\) and \(x = 2{\text{e}}\), is rotated through \(2\pi \) radians about the \(x\)-axis. Find the volume generated.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Consider \(p(x) = 3{x^3} + ax + 5a,\;\;\;a \in \mathbb{R}\).</p>
<p class="p1">The polynomial \(p(x)\) leaves a remainder of \( - 7\) when divided by \((x - a)\).</p>
<p class="p1">Show that only one value of \(a\) satisfies the above condition and state its value.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {x{{\sec }^2}x{\text{d}}x} \).</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;">Determine the value of <em>m</em> if \(\int_0^m {x{{\sec }^2}x{\text{d}}x = 0.5} \), where <em>m</em> > 0.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>It is given that \(f(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4\) where \(a\) and \(b\) are positive integers.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \({x^2} - 1\) is a factor of \(f(x)\) find the value of \(a\) and the value of \(b\).</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>Factorize \(f(x)\) into a product of linear factors.</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>Sketch the graph of \(y = f(x)\), labelling the maximum and minimum points and the \(x\) and \(y\) intercepts.</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>Using your graph state the range of values of \(c\) for which \(f(x) = c\) has exactly two distinct real roots.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f(x) = 3\sin x + 4\cos x\) is defined for \(0 < x < 2\pi \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the coordinates of the minimum point on the graph of <em>f </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The points \({\text{P}}(p,{\text{ }}3)\) and \({\text{Q}}(q,{\text{ }}3){\text{, }}q > p\), lie on the graph of \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find <em>p </em>and <em>q </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point, on \(y = f(x)\) , where the gradient of the graph is 3.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the coordinates of the point of intersection of the normals to the graph at the points P and Q.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="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;">Show that the quadratic equation \({x^2} - (5 - k)x - (k + 2) = 0\) has two distinct real roots for all real values of <em>k </em>.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Express \({x^2} + 4x - 2\) <span class="s1">in the form \({(x + a)^2} + b\) </span>where \(a,{\text{ }}b \in \mathbb{Z}\).</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 class="p1"><span class="s1">If \(f(x) = x + 2\) </span>and \((g \circ f)(x) = {x^2} + 4x - 2\) <span class="s1">write down \(g(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following graph represents a function \(y = f(x)\)<span class="s1">, where \( - 3 \le x \le 5\).</span></p>
<p class="p2">The function has a maximum at \((3,{\text{ }}1)\) and a minimum at \(( - 1,{\text{ }} - 1)\).</p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-29_om_14.45.13.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The functions \(u\) and \(v\) are defined as \(u(x) = x - 3,{\text{ }}v(x) = 2x\) where \(x \in \mathbb{R}\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the range of the function \(u \circ f\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State the range of the function \(u \circ v \circ f\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find the largest possible domain of the function \(f \circ v \circ u\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Explain why \(f\) does not have an inverse.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>The domain of \(f\) is restricted to define a function \(g\) so that it has an inverse \({g^{ - 1}}\).</p>
<p class="p1">State the largest possible domain of \(g\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Sketch a graph of \(y = {g^{ - 1}}(x)\), showing clearly the <span class="s1">\(y\)</span>-intercept and stating the coordinates of the endpoints.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the function defined by \(h(x) = \frac{{2x - 5}}{{x + d}}\), \(x \ne - d\) and \(d \in \mathbb{R}\).</p>
<p>(i) Find an expression for the inverse function \({h^{ - 1}}(x)\).</p>
<p>(ii) Find the value of \(d\) such that \(h\) is a self-inverse function.</p>
<p>For this value of \(d\), there is a function \(k\) such that \(h \circ k(x) = \frac{{2x}}{{x + 1}},{\text{ }}x \ne - 1\).</p>
<p>(iii) Find \(k(x)\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> has inverse \({f^{ - 1}}\) and derivative \(f'(x)\) for all \(x \in \mathbb{R}\). For all functions with these properties you are given the result that for \(a \in \mathbb{R}\) with \(b = f(a)\) and \(f'(a) \ne 0\)</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;">\[({f^{ - 1}})'(b) = \frac{1}{{f'(a)}}.\]</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Verify that this is true for \(f(x) = {x^3} + 1\) at <em>x</em> = 2.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(g(x) = x{{\text{e}}^{{x^2}}}\), show that \(g'(x) > 0\) for all values of <em>x</em>.</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;">Using the result given at the start of the question, find the value of the gradient function of \(y = {g^{ - 1}}(x)\) at <em>x</em> = 2.</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) With <em>f</em> and <em>g</em> as defined in parts (a) and (b), solve \(g \circ f(x) = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let \(h(x) = {(g \circ f)^{ - 1}}(x)\). Find \(h'(2)\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = x{(x + 2)^6}\).</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;">Solve the inequality \(f(x) > x\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 \(\int {f(x){\text{d}}x} \).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">One root of the equation \({x^2} + ax + b = 0\) is \(2 + 3{\text{i}}\) where \(a,{\text{ }}b \in \mathbb{R}\). Find the value of \(a\) and the value of \(b\).</span></p>
</div>
<br><hr><br><div class="question">
<p>The polynomial \({x^4} + p{x^3} + q{x^2} + rx + 6\) is exactly divisible by each of \(\left( {x - 1} \right)\), \(\left( {x - 2} \right)\) and \(\left( {x - 3} \right)\).</p>
<p>Find the values of \(p\), \(q\) and \(r\).</p>
</div>
<br><hr><br><div class="question">
<p class="p1">A function \(f\) is defined by \(f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}\). By considering \(f'(x)\) determine whether \(f\) is a one-to-one or a many-to-one function.</p>
</div>
<br><hr><br><div class="specification">
<p>When carpet is manufactured, small faults occur at random. The number of faults in Premium carpets can be modelled by a Poisson distribution with mean 0.5 faults per 20\(\,\)m<sup>2</sup>. Mr Jones chooses Premium carpets to replace the carpets in his office building. The office building has 10 rooms, each with the area of 80\(\,\)m<sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the carpet laid in the first room has fewer than three faults.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly seven rooms will have fewer than three faults in the carpet.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch the graph of \(y = {(x - 5)^2} - 2\left| {x - 5} \right| - 9,{\text{ for }}0 \le x \le 10\).</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">Hence, or otherwise, solve the equation \({(x - 5)^2} - 2\left| {x - 5} \right| - 9 = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The probability density function of a continuous random variable \(X\) is given by</p>
<p>\[f(x) = \left\{ {\begin{array}{*{20}{c}} {0,{\text{ }}x < 0} \\ {\frac{{\sin x}}{4},{\text{ }}0 \le x \le \pi } \\ {a(x - \pi ),{\text{ }}\pi < x \le 2\pi } \\ {0,{\text{ }}2\pi < x} \end{array}.} \right.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch the graph \(y = f(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(X \le \pi )\).</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">Show that \(a = \frac{1}{{{\pi ^2}}}\).</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">Write down the median of \(X\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the mean of \(X\).</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">Calculate the variance of \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}\left( {\frac{\pi }{2} \le X \le \frac{{3\pi }}{2}} \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(\frac{\pi }{2} \le X \le \frac{{3\pi }}{2}\) find the probability that \(\pi \le X \le 2\pi \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The vertical cross-section of a container is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-10_om_11.45.14.png" alt></p>
<p class="p1">The curved sides of the cross-section are given by the equation \(y = 0.25{x^2} - 16\). The horizontal cross-sections are circular. The depth of the container is \(48\) cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If the container is filled with water to a depth of \(h\,{\text{cm}}\), show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of the water is given by \(V = 4\pi \left( {\frac{{{h^2}}}{2} + 16h} \right)\).</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">The container, initially full of water, begins leaking from a small hole at a rate given by \(\frac{{{\text{d}}V}}{{{\text{d}}t}} =<span class="Apple-converted-space"> </span>- \frac{{250\sqrt h }}{{\pi(h + 16)}}\) where <em>\(t\) </em>is measured in seconds.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\frac{{{\text{d}}h}}{{{\text{d}}t}} =<span class="Apple-converted-space"> </span>- \frac{{250\sqrt h }}{{4{\pi ^2}{{(h + 16)}^2}}}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State \(\frac{{{\text{d}}t}}{{{\text{d}}h}}\) and hence show that \(t = \frac{{ - 4{\pi ^2}}}{{250}}\int {\left( {{h^{\frac{3}{2}}} + 32{h^{\frac{1}{2}}} + 256{h^{ - \frac{1}{2}}}} \right){\text{d}}h} \).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find, correct to the nearest minute, the time taken for the container to become empty. (\(60\) seconds = 1 minute)</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Once empty, water is pumped back into the container at a rate of \(8.5\;{\text{c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\). At the same time, water continues leaking from the container at a rate of \(\frac{{250\sqrt h }}{{\pi (h + 16)}}{\text{c}}{{\text{m}}^3}{{\text{s}}^{ - 1}}\).</p>
<p class="p1">Using an appropriate sketch graph, determine the depth at which the water ultimately stabilizes in the container.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The seventh, third and first terms of an arithmetic sequence form the first three terms of a geometric sequence.</p>
<p class="p1">The arithmetic sequence has first term <em>\(a\) </em>and non-zero common difference <em>\(d\)</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(d = \frac{a}{2}\).</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">The seventh term of the arithmetic sequence is \(3\). The sum of the first \(n\) terms in the arithmetic sequence exceeds the sum of the first <em>\(n\) </em>terms in the geometric sequence by at least \(200\).</p>
<p class="p1">Find the least value of <em>\(n\) </em>for which this occurs.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(z = r(\cos \alpha + {\text{i}}\sin \alpha )\), where \(\alpha \) is measured in degrees, be the solution of \({z^5} - 1 = 0\) which has the smallest positive argument.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Use the binomial theorem to expand \({(\cos \theta + {\text{i}}\sin \theta )^5}\).</p>
<p>(ii) Hence use De Moivre’s theorem to prove</p>
<p>\[\sin 5\theta = 5{\cos ^4}\theta \sin \theta - 10{\cos ^2}\theta {\sin ^3}\theta + {\sin ^5}\theta .\]</p>
<p>(iii) State a similar expression for \(\cos 5\theta \) in terms of \(\cos \theta \) and \(\sin \theta \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(r\) and the value of \(\alpha \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using (a) (ii) and your answer from (b) show that \(16{\sin ^4}\alpha - 20{\sin ^2}\alpha + 5 = 0\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Hence express \(\sin 72^\circ \) </span>in the form \(\frac{{\sqrt {a + b\sqrt c } }}{d}\) where \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider \(f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)\)</p>
</div>
<div class="specification">
<p>The function \(f\) is defined by \(f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D\)</p>
</div>
<div class="specification">
<p>The function \(g\) is defined by \(g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain \(D\) for \(f\) to be a 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>Sketch the graph of \(y = f(x)\) showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</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>Explain why \(f\) is an even function.</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>Explain why the inverse function \({f^{ - 1}}\) does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function \({g^{ - 1}}\) and state its domain.</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>Find \(g'(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to \(g'(x) = 0\);</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to \(({g^{ - 1}})'(x) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^4} + 0.2{x^3} - 5.8{x^2} - x + 4,{\text{ }}x \in \mathbb{R}\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The domain of \(f\) </span>is now restricted to \([0,{\text{ }}a]\)<span class="s1">.</span></p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = 2\sin (x - 1) - 3,{\text{ }} - \frac{\pi }{2} + 1 \leqslant x \leqslant \frac{\pi }{2} + 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the solutions of \(f(x) > 0\).</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>For the curve \(y = f(x)\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the coordinates of both local minimum points.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the \(x\)-coordinates of the points of inflexion.</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">Write down the largest value of \(a\) for which \(f\) <span class="s1">has an inverse. Give your answer correct to 3 </span>significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For this value of <span class="s1"><em>a </em></span>sketch the graphs of \(y = f(x)\) and \(y = {f^{ - 1}}(x)\) on the same set of axes, showing clearly the coordinates of the end points of each curve.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \({f^{ - 1}}(x) = 1\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({g^{ - 1}}(x)\), stating the domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \(({f^{ - 1}} \circ g)(x) < 1\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</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;">Let \(f(x) = \left| x \right| - 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The graph of \(y = g(x)\) is drawn below.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-12_om_11.31.12.png" alt></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"> </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) Find the value of \((f \circ g)(1)\).</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) Find the value of \((f \circ g \circ g)(1)\).</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) Sketch the graph of \(y = (f \circ g)(x)\).</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;">(b) (i) Sketch the graph of \(y = f(x)\).</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) State the zeros of <em>f</em>.</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;">(c) (i) Sketch the graph of \(y = (f \circ f)(x)\).</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) State the zeros of \(f \circ f\).</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;">(d) Given that we can denote \(\underbrace {f \circ f \circ f \circ \ldots \circ f}_{n{\text{ times}}}\) as \({f^n}\),</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) find the zeros of \({f^3}\);</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) find the zeros of \({f^4}\);</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) deduce the zeros of \({f^8}\).</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;">(e) The zeros of \({f^{2n}}\) are \({a_1},{\text{ }}{a_2},{\text{ }}{a_3},{\text{ }} \ldots {\text{, }}{a_N}\).</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) State the relation between <em>n </em>and <em>N</em>;</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) Find, and simplify, an expression for \(\sum\limits_{r = 1}^N {\left| {{a_r}} \right|} \) in terms of <em>n</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The function \(f\) is given by \(f(x) = \frac{{3{x^2} + 10}}{{{x^{\text{2}}} - 4}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne 2,{\text{ }}x \ne - 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(f\) is an even 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">Sketch the graph \(y = f(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the range of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</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 \(\left| {0.1{x^2} - 2x + 3} \right| < {\log _{10}}x\) .</span></p>
</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;">Consider the functions \(f(x) = {x^3} + 1\) and \(g(x) = \frac{1}{{{x^3} + 1}}\). The graphs of \(y = f(x)\) and \(y = g(x)\) meet at the point (0, 1) and one other point, P.</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;">Find the coordinates of P.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the size of the acute angle between the tangents to the two graphs at the point P.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Simplify the difference of binomial coefficients</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;">\[\left( {\begin{array}{*{20}{c}}<br> n \\ <br> 3 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> {2n} \\ <br> 2 <br>\end{array}} \right),{\text{ where }}n \geqslant 3.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence, solve the inequality</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;">\[\left( {\begin{array}{*{20}{c}}<br> n \\ <br> 3 <br>\end{array}} \right) - \left( {\begin{array}{*{20}{c}}<br> {2n} \\ <br> 2 <br>\end{array}} \right) > 32n,{\text{ where }}n \geqslant 3.\]</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The function \(f(x) = 4{x^3} + 2ax - 7a\) , \(a \in \mathbb{R}\), leaves a remainder of \(−10\) when divided by \(\left( {x - a} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(a\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that for this value of \(a\) there is a unique real solution to the equation \(f (x) = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the quadratic expression \(2{x^2} + x - 3\) as the product of two linear factors.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, or otherwise, find the coefficient of \(x\) in the expansion of \({\left( {2{x^2} + x - 3} \right)^8}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider \(f(x) = \ln x - {{\text{e}}^{\cos x}},{\text{ }}0 < x \leqslant 10\).</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 \(y = f(x)\), stating the coordinates of any maximum and minimum points and points of intersection with the <em>x</em>-axis.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Solve the inequality \(\ln x \leqslant {{\text{e}}^{\cos x}},{\text{ }}0 < x \leqslant 10\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f </em>is defined as \(f(x) = - 3 + \frac{1}{{x - 2}},{\text{ }}x \ne 2\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Sketch the graph of \(y = f(x)\), clearly indicating any asymptotes and axes intercepts.</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) Write down the equations of any asymptotes and the coordinates of any axes intercepts.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the inverse function \({f^{ - 1}}\), stating its domain.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let \({\text{P}}(X = n)\) be the probability that Kati obtains her third voucher on the \(n{\text{th}}\) <span class="s1">bar opened.</span></p>
<p class="p1">(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)</p>
</div>
<div class="specification">
<p class="p1">It is given that \({\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}\) for \(n \geqslant 3,{\text{ }}n \in \mathbb{N}\).</p>
</div>
<div class="specification">
<p class="p1">Kati’s mother goes to the shop and buys \(x\) chocolate bars. She takes the bars home for Kati to open.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{P}}(X = 3) = 0.001\) and \({\text{P}}(X = 4) = 0.0027\).</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 values of the constants \(a\) <span class="s1">and \(b\).</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>Deduce that \(\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}\) for \(n > 3\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Hence show that \(X\) has two modes \({m_1}\) and \({m_2}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>State the values of \({m_1}\) and \({m_2}\).</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"><span class="s1">Determine the minimum value of \(x\) </span>such that the probability Kati receives at least one free gift is greater than <span class="s2">0.5.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A function \(f\) is defined by \(f(x) = (x + 1)(x-1)(x-5),{\text{ }}x \in \mathbb{R}\).</p>
<p class="p1">Find the values of \(x\) for which \(f(x) < \left| {f(x)} \right|\).</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 function \(g\) is defined by \(g(x) = {x^2} + x - 6,{\text{ }}x \in \mathbb{R}\).</p>
<p class="p1">Find the values of \(x\) for which \(g(x) < \frac{1}{{g(x)}}\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi \).</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve \(y = f(x)\), the \(x\)-axis and the lines \(x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the \(x\)-coordinate of the minimum point on the curve \(y = f(x)\) satisfies the equation \(\tan x = 2x\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of \(x\) for which \(f(x)\) is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\) showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of \(f\) where the normal to the graph is parallel to the line \(y = - x\).</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>This region is now rotated through \(2\pi \) radians about the \(x\)-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Particle <em>A </em>moves such that its velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\), at time <em>t </em>seconds, is given by \(v(t) = \frac{t}{{12 + {t^4}}},{\text{ }}t \geqslant 0\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Particle <em>B </em>moves such that its velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\) is related to its displacement \(s{\text{ m}}\), by the equation \(v(s) = \arcsin \left( {\sqrt s } \right)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(y = v(t)\). Indicate clearly the local maximum and write down its coordinates.</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;">Use the substitution \(u = {t^2}\) to find \(\int {\frac{t}{{12 + {t^4}}}{\text{d}}t} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times;"><span style="font-size: medium; line-height: normal; background-color: #f7f7f7;">Find the exact distance travelled by particle </span>\(A\) <span style="font-size: medium; line-height: normal; background-color: #f7f7f7;">between \(t = 0\) and \(t = 6\) seconds.</span></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;">Give your answer in the form \(k\arctan (b),{\text{ }}k,{\text{ }}b \in \mathbb{R}\).</span></p>
<p> </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>Find the acceleration of particle B when \(s = 0.1{\text{ m}}\).</p>
<div class="marks">[3]</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;">Find the values of \(k\) such that the equation \({x^3} + {x^2} - x + 2 = k\) has three distinct real solutions.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">When \({x^2} + 4x - b\) is divided by \(x - a\) <span class="s1">the remainder is 2</span>.</p>
<p class="p1">Given that \(a,{\text{ }}b \in \mathbb{R}\), find the smallest possible value for \(b\).</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of \(k\) that satisfy the inequality \({k^2} - k - 12 < 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that \(\cos B < \frac{1}{4}\), find the range of possible values for AB.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The equation \({x^2} - 5x - 7 = 0\) has roots \(\alpha \) and \(\beta \). The equation \({x^2} + px + q = 0\) has roots \(\alpha + 1\) and \(\beta + 1\). Find the value of \(p\) and the value of \(q\).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function \(f\) defined by \(f(x) = 3x\arccos (x)\) where \( - 1 \leqslant x \leqslant 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Sketch the graph of \(f\) </span>indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the range of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the inequality \(\left| {3x\arccos (x)} \right| > 1\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A polynomial \(p(x)\) with real coefficients is of degree five. The equation \(p(x) = 0\) has a complex root 2 + i. The graph of \(y = p(x)\) has the <em>x</em>-axis as a tangent at (2, 0) and intersects the coordinate axes at (−1, 0) and (0, 4).</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;">Find \(p(x)\) in factorised form with real coefficients.</span></p>
</div>
<br><hr><br><div class="question">
<p>In the quadratic equation \(7{x^2} - 8x + p = 0,{\text{ }}(p \in \mathbb{Q})\), one root is three times the other root.<br>Find the value of \(p\).</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;">Given that the graph of \(y = {x^3} - 6{x^2} + kx - 4\) has exactly one point at which the gradient is zero, find the value of <em>k </em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vectors <strong><em>a</em></strong> and <strong><em>b</em></strong> are such that <strong><em>a</em></strong> \( = (3\cos \theta + 6)\)<strong><em>i</em></strong> \( + 7\) <strong><em>j</em></strong> and <strong><em>b</em></strong> \( = (\cos \theta - 2)\)<strong><em>i</em></strong> \( + (1 + \sin \theta )\)<strong><em>j</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <strong><em>a</em></strong> and <strong><em>b</em></strong> are perpendicular,</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">show that \(3{\sin ^2}\theta - 7\sin \theta + 2 = 0\);</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">find the smallest possible positive value of \(\theta \).</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: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is of the form \(f(x) = \frac{{x + a}}{{bx + c}}\), \(x \ne - \frac{c}{b}\). Given that the graph of <em>f</em> has asymptotes <em>x</em> = −4 and <em>y</em> = −2 , and that the point \(\left( {\frac{2}{3},{\text{ }}1} \right)\) lies on the graph, find the values of <em>a</em> , <em>b</em> and <em>c</em> .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = \ln (x)\) is transformed into the graph of \(y = \ln \left( {2x + 1} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Describe two transformations that are required to do this.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \(\ln \left( {2x + 1} \right) > 3\cos (x)\), \(x \in [0,10]\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The arithmetic sequence \(\{ {u_n}:n \in {\mathbb{Z}^ + }\} \) has first term \({u_1} = 1.6\) and common difference <em>d</em> = 1.5. The geometric sequence \(\{ {v_n}:n \in {\mathbb{Z}^ + }\} \) has first term \({v_1} = 3\) and common ratio <em>r</em> = 1.2.</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: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \({u_n} - {v_n}\) in terms of <em>n</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the set of values of <em>n</em> for which \({u_n} > {v_n}\).</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;">Determine the greatest value of \({u_n} - {v_n}\). Give your answer correct to four significant figures.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A particle moves in a straight line, its velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\) at time \(t\) seconds is given by \(v = 9t - 3{t^2},{\text{ }}0 \le t \le 5\).</p>
<p class="p1">At time \(t = 0\), the displacement \(s\) of the particle from an origin \(O\) is 3 m.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the displacement of the particle when \(t = 4\).</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">Sketch a displacement/time graph for the particle, \(0 \le t \le 5\), showing clearly where the curve meets the axes and the coordinates of the points where the displacement takes greatest and least values.</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"><span class="s1">For \(t > 5\)</span>, the displacement of the particle is given by \(s = a + b\cos \frac{{2\pi t}}{5}\) <span class="s1">such that \(s\) is continuous for all \(t \ge 0\).</span></p>
<p class="p2">Given further that \(s = 16.5\) when \(t = 7.5\), find the values of \(a\) and \(b\).</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"><span class="s1">For \(t > 5\)</span>, the displacement of the particle is given by \(s = a + b\cos \frac{{2\pi t}}{5}\) <span class="s1">such that \(s\) is continuous for all \(t \ge 0\).</span></p>
<p class="p1">Find the times \({t_1}\) and \({t_2}(0 < {t_1} < {t_2} < 8)\) when the particle returns to its starting point.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that the equation \(3{x^2} + 2kx + k - 1 = 0\) has two distinct real roots for all values of \(k \in \mathbb{R}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>k</em> for which the two roots of the equation are closest together.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Farmer Bill owns a rectangular field, 10 m by 4 m. Bill attaches a rope to a wooden post at one corner of his field, and attaches the other end to his goat Gruff.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that the rope is 5 m long, calculate the percentage of Bill’s field that Gruff is able to graze. Give your answer correct to the nearest integer.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Bill replaces Gruff’s rope with another, this time of length \(a,{\text{ }}4 < a < 10\), so that Gruff can now graze exactly one half of Bill’s field.</p>
<p>Show that \(a\) satisfies the equation</p>
<p>\[{a^2}\arcsin \left( {\frac{4}{a}} \right) + 4\sqrt {{a^2} - 16} = 40.\]</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The functions \(f\) and \(g\) are defined by</p>
<p class="p1">\[f(x) = \frac{{{{\text{e}}^x} + {{\text{e}}^{ - x}}}}{2},{\text{ }}x \in \mathbb{R}\]</p>
<p class="p1">\[g(x) = \frac{{{{\text{e}}^x} - {{\text{e}}^{ - x}}}}{2},{\text{ }}x \in \mathbb{R}\]</p>
</div>
<div class="specification">
<p class="p1">Let \(h(x) = nf(x) + g(x)\) where \(n \in \mathbb{R},{\text{ }}n > 1\).</p>
</div>
<div class="specification">
<p class="p1">Let \(t(x) = \frac{{g(x)}}{{f(x)}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\frac{1}{{4f(x) - 2g(x)}} = \frac{{{{\text{e}}^x}}}{{{{\text{e}}^{2x}} + 3}}\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Use the substitution \(u = {{\text{e}}^x}\) to find \(\int_0^{\ln 3} {\frac{1}{{4f(x) - 2g(x)}}} {\text{d}}x\). Give your answer in the form \(\frac{{\pi \sqrt a }}{b}\) where \(a,{\text{ }}b \in {\mathbb{Z}^ + }\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>By forming a quadratic equation in \({{\text{e}}^x}\)<span class="s1">, solve the equation \(h(x) = k\), where \(k \in {\mathbb{R}^ + }\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise show that the equation \(h(x) = k\) has two real solutions provided that \(k > \sqrt {{n^2} - 1} \) and \(k \in {\mathbb{R}^ + }\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(t'(x) = \frac{{{{[f(x)]}^2} - {{[g(x)]}^2}}}{{{{[f(x)]}^2}}}\) <span class="s1">for \(x \in \mathbb{R}\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence show that \(t'(x) > 0\) for \(x \in \mathbb{R}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let the function \(f\) be defined by \(f(x) = \frac{{2 - {{\text{e}}^x}}}{{2{{\text{e}}^x} - 1}},{\text{ }}x \in D\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine \(D\), the largest possible domain 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 class="p1">Show that the graph of \(f\) has three asymptotes and state their equations.</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 \(f'(x) = - \frac{{3{{\text{e}}^x}}}{{{{(2{{\text{e}}^x} - 1)}^2}}}\).</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">Use your answers from parts (b) and (c) to justify that \(f\) <span class="s1">has an inverse and state its domain.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({f^{ - 1}}(x)\).</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 class="p1"><span class="s1">Consider the region \(R\) </span>enclosed by the graph of \(y = f(x)\) and the axes.</p>
<p class="p1">Find the volume of the solid obtained when \(R\) is rotated through \(2\pi \) about the \(y\)-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
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<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 graphs of \(y = {x^2}{{\text{e}}^{ - x}}\) and \(y = 1 - 2\sin x\) for \(2 \leqslant x \leqslant 7\) intersect at points A and B.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The <em>x</em>-coordinates of A and B are \({x_{\text{A}}}\) and \({x_{\text{B}}}\).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \({x_{\text{A}}}\) and the value of \({x_{\text{B}}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area enclosed between the two graphs for \({x_{\mathbf{A}}} \leqslant x \leqslant {x_{\text{B}}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the solution of the equation</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;">\[\ln {2^{4x - 1}} = \ln {8^{x + 5}} + {\log _2}{16^{1 - 2x}},\]</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;">expressing your answer in terms of \(\ln 2\).</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) Using this value of <em>x</em>, find the value of <em>a</em> for which \({\log _a}x = 2\), giving your answer to three decimal places.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that (<em>x</em> − 2) is a factor of \(f(x) = {x^3} + a{x^2} + bx - 4\) and that division \(f(x)\) by (<em>x</em> − 1) leaves a remainder of −6 , find the value of <em>a</em> and the value of <em>b</em> .</span></p>
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<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;">Given that \(f(x) = \frac{1}{{1 + {{\text{e}}^{ - x}}}}\),</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find \({f^{ - 1}}(x)\), stating its domain;</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the value of <em>x</em> such that \(f(x) = {f^{ - 1}}(x)\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the first term and the common difference.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the smallest value of <em>n </em>such that the sum of the first <em>n </em>terms is greater than 600.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p class="p1">The function \(f\) is defined as \(f(x) = \sqrt {\frac{{1 - x}}{{1 + x}}} ,{\text{ }} - 1 < x \leqslant 1\).</p>
<p class="p1">Find the inverse function, \({f^{ - 1}}\) stating its domain and range.</p>
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<p class="p1">Compactness is a measure of how compact an enclosed region is.</p>
<p class="p1">The compactness, <em>\(C\) </em>, of an enclosed region can be defined by \(C = \frac{{4A}}{{\pi {d^2}}}\), where <em>\(A\) </em>is the area of the region and <em>\(d\) </em>is the maximum distance between any two points in the region.</p>
<p class="p1">For a circular region, \(C = 1\).</p>
<p class="p1">Consider a regular polygon of <em>\(n\) </em>sides constructed such that its vertices lie on the circumference of a circle of diameter <em>\(x\) </em>units.</p>
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<p class="p1">If \(n > 2\) and even, show that \(C = \frac{n}{{2\pi }}\sin \frac{{2\pi }}{n}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p class="p1">If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).</p>
<p class="p1">Find the regular polygon with the least number of sides for which the compactness is more than \(0.99\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If \(n > 1\) and odd, it can be shown that \(C = \frac{{n\sin \frac{{2\pi }}{n}}}{{\pi \left( {1 + \cos \frac{\pi }{n}} \right)}}\).</p>
<p class="p1">Comment briefly on whether <em>C </em>is a good measure of compactness.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows a semi-circle of diameter 20 cm, centre O and two points A and B such that \({\rm{A\hat OB}} = \theta \), where \(\theta \) is in radians.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-17_om_06.17.13.png" alt></span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the shaded area can be expressed as \(50\theta - 50\sin \theta \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(\theta \) for which the shaded area is equal to half that of the unshaded area, giving your answer correct to four significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \ln x\) . The graph of <em>f </em>is transformed into the graph of the function <em>g </em>by a translation of \(\left( {\begin{array}{*{20}{c}}<br> 3 \\ <br> { - 2} <br>\end{array}} \right)\), followed by a reflection in the <em>x</em>-axis. Find an expression for \(g(x)\), giving your answer as a single logarithm.</span></p>
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<p class="p1">Find the acute angle between the planes with equations \(x + y + z = 3\) and \(2x - z = 2\).</p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{{{\text{e}}^{2x}} + 1}}{{{{\text{e}}^x} - 2}}\).</span></p>
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<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The line \({L_2}\) is parallel to \({L_1}\) and tangent to the curve \(y = f(x)\).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equations of the horizontal and vertical asymptotes of the curve \(y = f(x)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \(f'(x)\).</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) Show that the curve has exactly one point where its tangent is horizontal.</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) Find the coordinates of this point.</span></p>
<p> </p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of \({L_1}\), the normal to the curve at the point where it crosses the <em>y</em>-axis.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the line \({L_2}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the curve \(y = \frac{{\cos x}}{{\sqrt {{x^2} + 1} }},{\text{ }} - 4 \leqslant x \leqslant 4\) showing clearly the coordinates of the <em>x-</em>intercepts, any maximum points and any minimum points.</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the gradient of the curve at <em>x </em>= 1 .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of the normal to the curve at <em>x </em>= 1 .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram shows the graphs of a linear function <em>f</em> and a quadratic function <em>g</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;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">On the same axes sketch the graph of \(\frac{f}{g}\). Indicate clearly where the <em>x</em>-intercept and the asymptotes occur.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Express the sum of the first <em>n</em> positive odd integers using sigma notation.</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) Show that the sum stated above is \({n^2}\).</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;">(iii) Deduce the value of the difference between the sum of the first 47 positive odd integers and the sum of the first 14 positive odd integers.</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A number of distinct points are marked on the circumference of a circle, forming a polygon. Diagonals are drawn by joining all pairs of non-adjacent points.</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;">(i) Show on a diagram all diagonals if there are 5 points.</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;">(ii) Show that the number of diagonals is \(\frac{{n(n - 3)}}{2}\) if there are n points, where \(n > 2\).</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;">(iii) Given that there are more than one million diagonals, determine the least number of points for which this is possible.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 random variable \(X \sim B(n,{\text{ }}p)\) has mean 4 and variance 3.</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;">(i) Determine <em>n</em> and <em>p</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;">(ii) Find the probability that in a single experiment the outcome is 1 or 3.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A straight street of width 20 metres is bounded on its parallel sides by two vertical walls, one of height 13 metres, the other of height 8 metres. The intensity of light at point P at ground level on the street is proportional to the angle \(\theta \) where \(\theta = {\rm{A\hat PB}}\), as shown in the diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \(\theta \) in terms of <em>x</em>, where <em>x</em> is the distance of P from the base of the wall of height 8 m.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the value of \(\theta \) when <em>x</em> = 0.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the value of \(\theta \) when <em>x</em> = 20.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(\theta \), for \(0 \leqslant x \leqslant 20\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{{5(744 - 64x - {x^2})}}{{({x^2} + 64)({x^2} - 40x + 569)}}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the result in part (d), or otherwise, determine the value of <em>x</em> corresponding to the maximum light intensity at P. Give your answer to four significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The point P moves across the street with speed \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\). Determine the rate of change of \(\theta \) with respect to time when P is at the midpoint of the street.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle, A, is moving along a straight line. The velocity, \({v_A}{\text{ m}}{{\text{s}}^{ - 1}}\), of A <em>t</em> seconds after its motion begins is given by</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;">\[{v_A} = {t^3} - 5{t^2} + 6t.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \({v_A} = {t^3} - 5{t^2} + 6t\) for \(t \geqslant 0\), with \({v_A}\) on the vertical axis and <em>t</em> on the horizontal. Show on your sketch the local maximum and minimum points, and the intercepts with the <em>t</em>-axis.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the times for which the velocity of the particle is increasing.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the times for which the magnitude of the velocity of the particle is increasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">At <em>t</em> = 0 the particle is at point O on the line.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for the particle’s displacement, \({x_A}{\text{m}}\), from O at time <em>t</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A second particle, B, moving along the same line, has position \({x_B}{\text{ m}}\), velocity \({v_B}{\text{ m}}{{\text{s}}^{ - 1}}\) and acceleration, \({a_B}{\text{ m}}{{\text{s}}^{ - 2}}\), where \({a_B} = - 2{v_B}\) for \(t \geqslant 0\). At \(t = 0,{\text{ }}{x_B} = 20\) and \({v_B} = - 20\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for \({v_B}\) in terms of <em>t</em>.</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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>t</em> when the two particles meet.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the expression \(f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{cot}}\left( {\frac{\pi }{4} - x} \right)\).</p>
</div>
<div class="specification">
<p>The expression \(f\left( x \right)\) can be written as \(g\left( t \right)\) where \(t = {\text{tan}}\,x\).</p>
</div>
<div class="specification">
<p>Let \(\alpha \), <em>β</em> be the roots of \(g\left( t \right) = k\), where 0 < \(k\) < 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f\left( x \right)\) for \( - \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}\).</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>With reference to your graph, explain why \(f\) is a function on the given domain.</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>Explain why \(f\) has no inverse on the given domain.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why \(f\) is not a function for \( - \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}\).</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>Sketch the graph of \(y = g\left( t \right)\) for <em>t</em> ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.</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>Find \(\alpha \) and <em>β</em> in terms of \(k\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\alpha \) + <em>β</em> < −2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the graph of \(y = x + \sin (x - 3),{\text{ }} - \pi \leqslant x \leqslant \pi \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph, clearly labelling the <em>x</em> and <em>y</em> intercepts with 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: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the area of the region bounded by the graph and the <em>x</em> and <em>y</em> axes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Sketch the curve \(y = \left| {\ln x} \right| - \left| {\cos x} \right| - 0.1\) , \(0 < x < 4\) showing clearly the coordinates of the points of intersection with the <em>x</em>-axis and the coordinates of any local maxima and minima.</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;">(b) Find the values of <em>x</em> for which \(\left| {\ln x} \right| > \left| {\cos x} \right| + 0.1\), \(0 < x < 4\) .</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{1 - x}}{{1 + x}}\) </span><span style="font-family: times new roman,times; font-size: medium;">and \(g(x) = \sqrt {x + 1} \), \(x > - 1\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the set of values of \(x\) for which \(f'(x) \leqslant f(x) \leqslant g(x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{4 - {x^2}}}{{4 - \sqrt x }}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) State the largest possible domain for <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the inequality \(f(x) \geqslant 1\).</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(g\) , where \(g(x) = \frac{{3x}}{{5 + {x^2}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Given that the domain of \(g\) is \(x \geqslant a\) , find the least value of \(a\) such that \(g\) has </span><span style="font-family: times new roman,times; font-size: medium;">an inverse function.<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) On the same set of axes, sketch</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) the graph of \(g\) for this value of \(a\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the corresponding inverse, \({g^{ - 1}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Find an expression for \({g^{ - 1}}(x)\) .<br></span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves in a straight line with velocity <em>v </em>metres per second. At any time <em>t </em>seconds, \(0 \leqslant t < \frac{{3\pi }}{4}\), the velocity is given by the differential equation \(\frac{{{\text{d}}v}}{{{\text{d}}t}} + {v^2} + 1 = 0\) .</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;">It is also given that <em>v </em>= 1 when <em>t </em>= 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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for <em>v </em>in terms of <em>t </em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of <em>v </em>against <em>t </em>, clearly showing the coordinates of any intercepts, and the equations of any asymptotes.</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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the time <em>T </em>at which the velocity is zero.</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 distance travelled in the interval [0, <em>T</em>] .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 <em>s </em>, the displacement, in terms of <em>t </em>, given that <em>s </em>= 0 when <em>t </em>= 0 .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, show that \(s = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}\).</span></p>
<div class="marks">[4]</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: 31.0px 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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = {({x^3} + 6{x^2} + 3x - 10)^{\frac{1}{2}}},{\text{ for }}x \in D,\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where \(D \subseteq \mathbb{R}\) is the greatest possible domain of <em>f</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the roots of \(f(x) = 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence specify the set <em>D</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find the coordinates of the local maximum on the graph \(y = f(x)\).</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;">(d) Solve the equation \(f(x) = 3\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Sketch the graph of \(\left| y \right| = f(x),{\text{ for }}x \in D\).</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;">(f) Find the area of the region completely enclosed by the graph of \(\left| y \right| = f(x)\)</span></p>
</div>
<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) Sketch the curve \(f(x) = \left| {1 + 3\sin (2x)} \right|{\text{, for }}0 \leqslant x \leqslant \pi \) . Write down on the graph the values of the <em>x</em> and <em>y</em> intercepts.</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;"> </span></p>
<p style="font: 23px Helvetica; text-align: justify; margin: 0px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdYAAAIYCAIAAABfclpUAAAPlUlEQVR4nO3cz2scZ4LHYf0n3dAnGQQzzMUnG8McjA/ChMUEfMliGyzPHnJYmEUSgh1yWGdAxiabQC5togSGgd0WDksgTmgMgwgamslAWERDFsyo6YMRpmmQ0aGoPbTtyJL1y6rW11Geh8KHVhe8FO2PXr31Vk+UAIRMvO7FQbf94L8Xb15eXB0+f+VpZ/GdqZutXnGCQwM47XYn+Fm3ebVRqzdq9cbkQnswiu7TzuI7jdrV+91nJz1AgNPrtbPgsiweL988+2pzn7QX/viiyABUYI8EP58Lb0vwsHO/2Rnu8W4A3sBeCS4G7YWp2rn59pOyLMtyq/eg+WVv6+TGBfALcKgEF72v/vPBY2sQANXaK8Fl0W1erp253FwryqffP/iLvRAAlTswwf87+NvXbUsQAGOwZ4LLQXt+sj713vt3LEEAjMf+CT5z6aenMwCo2D4JfnTn37/pmwADjM1ej2b0O//T0V+Asdqe4K1++4NL08213ndLn/1FfwHGbXuCn3WbVxuT1+9807X+C3AC9l4LBmDMjpzgzc3Nbrc7jqEA/NIcOcHLrdaF8+c3NzfHMRqAX5SjJXh9fX30VcKffPzJmAYE8MtxtATfmpl5/m3utbrlCIBjOkKCl1utRq3+4e3bowS/e+WK5QiA4zhsgjc2Ni6cPz83O9vtdhu1+udLS6N/xzo4gNPtsAmem51t1Orr6+ujBHe73dF0eH19fazjAzjFDpXghw8fNmr15VarLMuXCR7Ni2/NzIx5hACn1sEJ3tzc3J7alwkuX00zAEd1cIJHCw4v9z9sT3C5bYFijGMEOKUOSPDGxsaO2247Ejx6g23CAG/g4FnwxsbG9s1nOxJclqUpMMCbOfIDyrsTDMCbkWCAGAkGiJFggBgJBoiRYIAYCQaIkWCAGAkGiJFggBgJBoiRYIAYCQaIkWCAGAkGiJFggBgJBoiRYIAYCQaIkWCAGAkGiJFggBgJBoiRYIAYCQaIkWCAGAkGiJFggBgJBoiRYIAYCQaI2ZXg4vHyzXOXm2vFHidIMEBVdiR4q9d6f6p2RoIBTsD2BBfDzkc3fv/7G5MSDHAStiV4uHrn2ked3jfzEgxwIl4m+Gln8V/vdJ6Wg7YEA5yMUYKLYeejG4urw7KUYIATM1GWZTlcvbfwoDeK7uESvPtYbrVOaswAp8REWT7t3P3jcm/r+QuHS/AnH3+83GptP8yLAY5qohy05ydfM6tt1K7e7z7bfYKFCICq7Ho0w1owwEmRYIAYCQaI8TU9ADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQ8zzBRX/l3rWzjVq9cXFhuTvY5wQJBqjKRFmW5fDvy5991y/Ksuj/9e71qcmF9qDY6wQJBqjKRFk++/HRd72XyR205yfPzbef7HWCBANUZedacNFtXr54rzM0CwYYu+0JHnTbzflr/9Hub+1zggQDVOVFggft+cl6o1Zv1KbnW2vDvU8YJfjhw4fdV21ubp7IgAFOj1cWIor+6tLcdKN2dqb1eK+ViFGCdx/mxQBHtWtfcLF2f/rM1Fx7r41pZsEAVdn9aMazbvPqgQk25wU4vt0JftKemz5wIUKCAY5voiweL988d2nui05/qyy3+u0PLl9bWrMpDWD8JspysNa8OTW6qzZ5/U6r098zv2UpwQDV8TU9ADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQM1GWxbD79Z1rZxu1eqM2Pf/Zar/Y7wQJBqjKRNH76t7dr7vDoiy3+quf3Jg8c2lxdbj3CRIMUJWJHx991/tp2vus27zamFxoD/acCUswQFV2rAUXg/bClAQDnIjXJPg3N1u9vZeDJRigKjsS/KQ9d/1O5+k+J4wSfGtmZm52dvsx1lECnErbE1wMOx/d2PdeXCnBANXZluDh6r25L9aG+25JsxABUJ0XCS4ef3n3zwf2t5RggOpMlGVZFr323U/b/a3nrw1/WGquDPY4QYIBqjJRDteW56Ybtfq248zl5tpe82EJBqjKxPLNs6/2t96oXb3ffbbXCRIMUBVf0wMQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAjwQAxEgwQI8EAMRIMECPBADESDBAzSnAx7D5a/q/FG79aaA+K/U+QYICqTJRlWXSb//Te9X+erDcmJRjg5LxciHjWbV6VYICTJMEAMRIMEPOGCb41MzM3O7v9GPdAAU4fCQaIsRABECPBADESDBAjwQAxE2VZloP2/GS9URsdZy431/bJsAQDVMXX9ADESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESDBAjAQDxEgwQIwEA8RIMECMBAPESPDb4vOlpc+XltKjOFVWVlZWVlbSozg9llst//ErJ8Fvi7nZ2bnZ2fQoThWXtFqNWn251UqP4m3U7XZvzcysr6+/wbkS/LbQi8q5pNWS4L2srKxcOH/+za6PBL8t9KJyLmm1JHgfGxsbc7OzjVr93StXjpRHCX5b6EXlXNJqSfCBXk6HP19a2tzcPMwpEvy20IvKuaTVkuDD2Nzc/PD27dF0+Pvvvz/w/ROjj+nhj9/dutWo1X9369ZRT3Tsf1w4f/7C+fPxYZymwyWt9hhlJT6Mn8Uxmgs3avUPb9/efzoswQ6Hw1HxMervhfPnD/y7wUIEQGVGG9Qatfrc7OzGxsaB75dggApsbm5+vrQ0mvw+fPjwkGdJMMBxdbvdd69cOczi7w4SDHAsy63WaPJ7mC0QO0gwwLGsr68ftBH4H8vXft2o1RuT7y/3tsqi/9e716dq9d8sdl6b4K1+54v5i2catbM37q70i1d+JsFjMey2W3+6c216vv0kPZRTYdj9dvH6VK3eqJ25NPdFp7+VHtDP2lZ/9ZMbk6OL2eoOi4PP4HWKXmtmsj419+dv735wf20wenF3goth596lix+0+1tl0WsvvHNpcXW47ccSXL1i7f47V2+8d7ZROyfBFSgef3n302+7g7Isi/7KvWtnGxfvdYTjDRXDvz1YWu0XLy7m1Fx7kB7Tz9aT9ty5Ru3sTOvxy4/jrgQXa/enf/tTCAbt+cmr97vPXv5cgsek6DYvS3AVih9XHvV+mva6sMdS/N+jR/940Yti0F6YmlxoD/w+ezPFoL0wVXulqDsTXHSbl195x5P23G8vN9deXnIJHhOlGJdBe37Sha3Es27zvR1/FnMUowSf2V7UHQl+1m1ebbzyW+5Je+5cY7rZffGCBI+JBI/LoD3/q/eXe5aDj2fYbTcXbix80zcDflNF/5s/zM3928Uz2xdzdiR4Z3Al+MRI8HgUg/YfLpu4Hcto7lZv1OqNiwvLXUvBb6R4vPwvc8u9wfNpbu/vS3e/6hUS/NaQ4LEYrt659pF7cVV4sVFq0p8UR1Ss3Z8+8+K3VzHs3Lu0bW+JhYi3hQSPwdPOtt0/HJ9PaeVedztue4KLtfvTv3Y77gT4cFdtq/fg0yX9rdaOHVMcm01pbwsJrtRWv/3pvXbvxdRhsPbZF49spTo+9zar5tGMt4UEV2fQbS1cGt0+enm8coeDwyp6rZnJ6fnPRs9m9NoLV280f3Bvs0J7PKD8/HnEF5d+Gwkeg9EzM2JRia1e6/2pHf19dScmRzD84f61s6PLOHVtcbljT1rFfE0PQIwEA8RIMECMBAPESDBAjAQDxPw/i78hXcSuejcAAAAASUVORK5CYII=" alt></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) By adding <strong>one</strong> suitable line to your sketch, find the number of solutions to the equation \(\pi f(x) = 4(\pi - x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(f(x) = x + \frac{{8x}}{{{x^2} - 9}}\). Clearly mark the coordinates of the two maximum points and the two minimum points. Clearly mark and state the equations of the vertical asymptotes and the oblique asymptote.</span></p>
</div>
<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;">The graph of \(y = f(x){\text{ for }} - 2 \leqslant x \leqslant 8\) is shown.</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;"> </span></p>
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jZGSpBICZS4DFI+n6/+ISlBsouSuURCWFDz85aUmGUymcFg0OfxhkOhtkAhJUikBM3n8dYcLiElSLZQEp+KN/peJU6JKJfLhUMhn8c7GAyCh1FICRIpQQuHQjWHS0gJknVKmjhiyJ5XYoIijt02fzApQSIlaBsvg0RKkKxQUi6Xw6FQr9/fZH9EjhJRNShN7nlESpBICdrGyyCREiQrlMSi0V6/v+YQVU1WKBGZoDQ6hkJKkEhJG9VsZpMSJGlK4lPxlo4Y9p043wQUUoJEStooHApVf4JASpDkKBHHR5DPWeyiRFQNirkFSkqQSEkb1RwuISVIEpQ0P85ak72UiJaXl/sDAfHZc7FYJCVIpKSNag6XkBKkdilJpVK4I4bKbwZnMhkBSix2KDS63/b5V0dKkPShxLj5cAkpQWqLEnFz1eZ3fa9JHSXG9e/73HPX3bdvvq2tZ9VupARJK0qqD5eQEiScEnFK8cbzWVv8LfXXK/no0sc7djwGnoQiFylB0oqS6sMlpAQJpCSfz/f6/XUvMdXiL3bq0kfFYlEcxBkMBiW+yNM8UoKkFSXVh0tICRJCSblcHgwGB4PBupfjbl7HKBH/XfcjHuuREiStKDGqDpeQEiSEEnFKq9xvZocpEeVyueqPeKwvgpQg6UaJebiElCC1pETsNbQ8Fa1RXaFEJC56IC6eIrE9VR0pQdKNEvNwCSlBak6J+MjGyrHMLlJiXP+IR4Bi5VWQEiTdKBGHS/L5PClBakKJeCcb3UcVrLuUiMrlsti2av61wCaREiTdKDGuX+qVlCA1oqRYLMp9ZFOTEygRFYvF5l8LbBIpQdKQEnGpV1KCVJcS8ZFNfyBg8RCD4SRKROZHPG0dkSUlSBpSIi71SkqQ6lICfusXyWmUiMyT7ucSCYRLUoKkISXivMyR539ESlq2kZJ2v2XTPGdSIjKPyLb8iIeUIGlIiWEYvX7/4zt3kZKW1VBiy6HW6pxMibHhrj2NHkZKkPSkJD4V/8637yclLaumpFwu23KotTqHUyIyT7rvDwTqHpElJUh6UiJOiCAlLaumxK5DrdW5ghJRk494SEmTcrmceLv0pKRYLG68nYW9aUaJuJGNLYdaq3MRJaK63+IhJU0y9xBfe+3Y9Mwbts+/y5QYhnHXHXe+9tpr6uavEyXWz2ptlOsoEdVcqI2UNElQYk7iPtA2btt2n5IdOx574vtPqJu/NpS8f/5Cr9/f7oVIwFxKicj8zPiZZ/d+dOljFYsw04aSGlOsf52yBSXTM2/EYocWkkl10/e+1yc+jFA0f/MtU/cSYrFD+/b9o9J36bH+x//uPn/f9773s7ffVjH/kydPPr5z18mTJ5W+S4/v3KVu/uFQ6Lavb/qr224Ph0KK3qWFZFL1uyRWhKL3py4l5jQYDM4lEsvLy3KbKi0o2b17oPniOXHipN/UHwjEotGFZBL/7gIp4cSJ041JbJtkMpl2D+13/1hJZGz87NmzvX6/ovmb75Gi+RuKj5WIc4J37HhM6Ypw9bESs+rhWv09YxsPVOt3rES8RfGpuPSujcgRlHySzfoUfMApcjUl5XK5PxAYDAal784Hph8louqz2mwBRSdK+gOBuUSind+7L1eTz/o8Xp9nSzD5WcVYX12K79nk9X1jIvulMyjJFwr9gcCCmtsauJoScY3FcrlMSpAaDVcbQdGAknvuuvuZZ/fK/8td+Wzh6S2+TeGFD+Ojs38oXf+xUyiJT8UHg0EV83cvJWLdi+NepASp+XC1BRRXUxIOhTKZjOXhWllLR3o83p6nk4XKjZ86hZLl5WWfxyuxq1ZZXUqE+3web89A/Her6xsf4FJKxBf2zC01UoKEDNcaUNodcu6lxMyG4bqWHt20efvsSpUkCCXpdy5OPNXj8fo8m7eF57P1fl2tJNZNuVz2SZzHWVqa3PrI/nShYqyvpl/etvVotlSpeYgbKan6wt5a7sOjezZ5fR6v796BRHa19uXZVEco+e3ITn9P32xO0Wto518+E5Q2L0+9/vzTu2fGdvZ4vD3h9JqV59oglZRUjaWvfnfyw1yp9V9pNKf06Cav7+ZV2YqSt15/9fnwxdyaYRiV1czRgS2+er+uVjJHcPUt+7C+yM3urFqjn6fD99dIabiTkutf2Pt/hffjRz/MlQwjsv/Hs88/3ON5ZDJ71cYFmXWAkn+5cOjev6gdf/bW7ka0AAW93n1l9XdHnvrLv/jW2Mx7OVt/BapTRsl6ITncs+npUytri5evzP3zi/u33j+a/lxmTpVC+qXw6I/7fJsi6bUb70MLSuaOjP/zhd/emEludrvnPsln0CBzBIuLqrXxNysrp/r+psqOylo6svEfPddRcuMLe5X/uHTpz+LVRMbGFzPnTvVtVvqPoUJKSkvPbd91/ze/7ihKRNXXu59LJBqcQn41O/FIz0D8B/845sodnMqKOXgWL1+ZP5O4+Z9hvPVCMrw3+dmX1yj4dOX09PnCutH2sZK19OgmVZSIEyjws+s2uCaOBu08lfui+mHuokQcM9r4VenI2Pji5V+lw/e5k5Kr2Yl9z735zu7vbHYgJSITFF+dC8pWStmj2zYNLxTWXXus5PN0+D6xS7F4+cr8mRPp8CNt/iJ/kZvd6fP0jSZXSoY4trDZtzWykLs2Htun5BvDCwU7D5dUrxvxQTf29zbC4XpKxOXjY9Hoxj+6TklfMPmZit9ElZRUStljeyaWfn35V06mxMz8cuCN66FUVk71/c228BvvTTz1lx6v7+s7LR1oaJoySiql7NFtHm/PQOLt9G+mIk+MHsnYe+CtLUoqa+kD2yeW7H0Tq0fwXCJh3pa8VRpS0uSOv5Gx8cWL8dE+mw9UmSmkpLQ0OXAsW6osuoQSUSaTEd9/GwwGf/uzF7d7+kZPL61WjMjY6MeHv6/6oJWKORvmGWWer31rcNL2ldAOJdfHhL3PoHoEi8178PvOmu3gNL98fGQs/OZzuxQNX0MhJVezRw6JzVh3USLK5XKx6EGfx/tV7x3/9NZ74lzB/J8/PtWn6oWopKRSyr0/OXFo4N6v+Tzf3J8u2Pv0v5Kb3dnwuz19sxNvmevmavbIy6dW7N9PrxnB4j5b0N+srJzqqz4K/UVudufGFewKSlpd06jyP55+dCByVtFGtaGKkkopO73/+h6ZGykxDMMwvvjdsYDvq//l7nv9Po/3oYce/iS7mJvdWfP5hV0p3MFZSQSfSRYqxmLmgxd3fNNn94YVuFWyXnh/OqHAEWMDJbFoFL4AsiYfBufz+ebXNKoUfv4PO/b+2n2nqH2eDt9X71+p2rOb7EoNJde2dk/876upVOquO+70ebw/3H6n7+9edNNWSdXHnYuXr8yfOZmdeMRe1hFKfruanj56Y3NobeX0/CX7PK4ZweLfZ/SUIfefoibusNfoEIlhGJXV9NEj6ef3i7NdK6WVd09dsvMTNFFHTlFz6VaJ+OBSfIHNiIyNp1LvDj7wVz6P97FAYCGZtH79sepUUVJ1furi5SvzZ87UPXPCSi0pmT0X27Ntw46Pjc+gZgS3e9rrtRPnGp+J63BKYtFor9/fYDhWSrnk6NbNN/+TXnswyJZISdNunN8VGRv7/f+K7uk7mv6/n88lEuLD41g0Kne/9I0p28G5mp14xHf9FLX5Uz89NfCgvZ8GNqdk/fV9/7BJ8dbpxhEs7iVs1/ydTIm4w16jUVgpJIOb6hzAUrddTUoaZ5517rt/6PXqf7TMz3r67dhIUXjYtbKaPR2+tllg8cT5ejnl63zVPxG/YHbN37GUtHWHPX6dD6mLV5wvFovmRor4Aq7cZYTc8XW+ejmREnFzHLu2GJ1JSbt32CMlSE64ecXy8nIsGhVfFBSXJmtr/qREvrrrZjAYtOsuDc6kZDAYFNc0Ah9PSpCcQImoXC6nUqnBYLDdi5WREvnqrpu2v9rXOAdSInGHPVKC5BxKzPL5vDiHGzSFlMhXd93YuI/jNErk7rBHSpAcSIkZaAopka/RurFrH8dRlLQ8G61RpATJyZSY1ZhSczyFlMjXaN3YtY/jHErMy8dLHNsnJUiuoMQsn88vJJPieIo4OSWVSn2SzZISyZp8umbLPo5zKGl6NlqLSAmSuygxKxaLmUxGDA+fx3vXHXfGp+K23Me3brccJYZN+zgOoaT68vESkRIkl1JS3SfZ7OM7d5n39xV7QKlUysa7RN2KlIh9HCt3DDOcQUmja6PhkRIkDSipPlaSy+XEHSfE1oo4+W0hmVxeXrYiy61IidjHsXgPpK5TIn2otTpSgqQZJTf9PJ/PZDJziYS5wSJkiU/FBS74Bu+tSIlhx/dxukuJlUOt1ZESpK4PV+uBn+AIWcQ2i3ngVpxiGw6FxMaLmHJViYMvtygl7V1zoF7dpURsnVo/fkZKkLo+XK0n/WFwsVjM5XKpVGohmYxPxQUo5p5R9RSLHeoCJbNvnn777XOLl6+om4aGR4aGRxr96UeXPvZ5vK+9dkx6/uY72PmX8MKPX/B5vO+fv6D6XbJr/h1YhLr567EIdfP/5a9S5869c+7cOyffmJ1983SnKZk5OTs988b8mbPqJvHeNXnAnj0/2Pbgg9LzNynp8Es4MD7u83gPjI935l2yZf4dWIS6+euxiA68hOmZN2ZOznaaknkHbDG2de3ojXVlB0dcQMCubyQa3MHBcsJwtRjPdpUPWTft3B+nts5TIj6ywS8ggERKkBwyXK1ESuRD1k0798eprcOUtLxWq1ykBMkhw9VKpEQ+ZN1YOYm+w5SIC5HYfsozKUFyyHC1EimRD1w34VBI7gSTTlISn4q3eyESMFKC5JzhKh0pkQ9cN+IEE4l/7TtGifiWjcVzcxtFSpCcM1ylIyXy4eum1+9fSCbbnX9nKBEf/Vr5lk3zSAmSo4arXKREPnzdiCt6tzv/DlCSmJu396PfjZESJEcNV7lIiXz4upH7dp9qSvL5/D133b1nzw8UzV9ESpAcNVzlIiXytbVu2rmj8LWUUiJOIQns2HFKwZnI1ZESJKcNV4lIiXxtrRtxFmlbFxBSR4n5rd+PLn2sYt1UR0qQnDZcJSIl8rW7bsR1pfDHK6Kk+lQ0ReumOlKC5MDh2m6kRL521027nwqroEQ40uv3i1NISAkSKUEiJfJJrJu2Nkxsp6TGEYOUYJESJFIin8S6ESeDgRsm9lKy0RGDlGCREiRSIp/EuhH37gY3TGykpK4jBinBIiVIpEQ+uXWDb5jYRUkjRwxSgkVKkEiJfHLrBt8wsYWSJo4YpASLlCCREvmk1w34UY51Spo7YpASLFKCRErks7Ju+gOBlie/WqREnM86GAw2uXQAKUEiJUikRD4r60Zc9rX5ya9WKDEdaX5JNFKCREqQSIl8FtdNOBTqDwSa/KpLUyJ2oMKhUMtLK5ISJFKCRErks7huisVi8+uYyFEyl0jg1w0gJUikBImUyGdp3ZRyFyee+rrH6/N4vz04kV1d3/iQdikpl8vitqz41QxICRIpQSIl8smvm8pn549MX8ytGYYRGtr9rdu/6vvuq9lSpeZRbVGSy+V6/f4mH9bUjZQgkRIkUiKf9LqpfJq5VLi2GVIsFnu3/PWdnttG05/XPAykpFwui50a5OBITaQEiZQgkRL57Fo3uX+d83m8fSM/rfk5Qkkul+sPBHr9frmLs5ISJFKCRErks23drKWf3PxffR5vzX38mlNSLBZj0ajYGJG+eQ0pQSIlSKREPpvWTWUtfWD7xNLFTKbmk5dGlBSLRXHbmv5AQO5mXWakBImUIJES+exZN6WlyYFj4pirOB9kMBgUp67VUFIulzOZjPiMpj8QsOW2NaQEiZQgaUvJ6bn58xc+yBcK6qah4ZGh4ZF6f/Qfvzn8qAlB7fTA4d/82Xzkn34x/sLhj/+P+Xc/yWaH9u71ebxPfP8J868cPHjQ/N/R8GgqlVL/EjrwLtk5/w4swtXvUsdWhNKXcP7CB6fn5jtNyfETM7FDr0bGxtVN4r2zMIexkf/26APSP14AAAZKSURBVI69oxv/6Nmh5x566GGTku98+/7Hd+4KDj4TGt3vsJfQ/UWYlLj3JeixiA68hNihV4+fmOk0JY7fYlxfTU8fTReun0yytnJ6/tLaTaeWdPj244qKcAcHyPHDtXV5XXdwnL1u1nLJyLaaHZ++2dzNJ6mREiRSAkZKGuVeStYLyeGe2mMom7fPrtSc7kpKkEgJGClplHspQSMlSKQEjJQ0ipTYEClBIiVIpEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrtpScvzETOzQq5GxcXXT0PDI0PCIuvmblLj3JXRgEWL+HViEq9+ljq0IpS8hdujV4ydmOk3J6bn58xc+yBcK6ibx3qmbv0mJe19CBxZhUuLel6DHIjrwEs5f+OD03HynKdFgi5E7OEh57uBgqX6X8rru4GiwbkgJEikBIyWNIiU2REqQSAkSKZGPlCCREiQNhispkY+UIJESJA2GKymRj5QgkRIkDYYrKZGPlCCREiQNhispkY+UIJESJA2GKymRj5QgkRIkDYYrKZGPlCCREiQNhispkY+UIJESJA2GKymRj5QgkRIkDYYrKZGPlCCREiQNhispkY+UIJESJA2GKymRj5QgkRIkDYYrKZGPlCCREiQNhispkY+UIJESJA2GKymRj5QgkRIkDYYrKZGPlCCREiQNhispkY+UIJESJA2GKymRj5QgkRIkDYYrKZGPlCCREiQNhispkY+UIJESJA2GKymRz751s15IDvf0zeYqN/2UlCCREjBS0ih9KKkUksFNXh8pkYqUgJGSRulCSWlpcmDf6MAWUiIXKQEjJY3Sg5Kr2Yl9k9lcOnwfKZGLlICRkkZpQEmllD22Z2KpZHxOSqQjJWCkpFHup6S0NDlwLFuqGKTEQqQEjJQ0yu2UXM0eObRQWDcMg5RYiZSAkZJGtaBk5uTs9Mwb82fOqpuGhkeGhkfq/tHc4R9+6zoEG6bewcOnT77ydGDf63PXHn/ixUf/2rflhxNv3TQT86905SW4ZRFi/h1YhKvfpY6tCKUvYXrmjZmTs52mZPbN02+/fW7x8hV1k3jvpP7ur6aevKueMl/7duTcv1x/mPlzR74EpyzCpMS9L0GPRXTgJbz99rnZN093mpJ5N20xcgdHPu7ggHEHp1GkxIZICRIpQSIl8pESJFKC5KrhWj9SIh+/zodESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpA0GK6kRD5SgkRKkDQYrqREPlKCREqQNBiupEQ+UoJESpCcP1zFLYGbPICUyEdKkEgJkvOHKylRGClBIiVIGgxXnSn5+S9+mS8U1E3x49O/X15WN3+TEnWLSKcvvfPue6rfpXT6krr5/355+fiJGaUrIp2+FD8+rfRdeufd95S+S3n1w/UP//Zvifm3lL6En//il92hJDI2rnR65WBsNPKSuvmblKhbxIGXX/nJK1HV79KBl19RN//RyEs/nZhUuiIOvPzKKwdjSt+ln7wSVfouRdQP18jY+MzJU0rnHxkbV0HJ/wcCnZTclmYhpgAAAABJRU5ErkJggg==" alt></p>
<p style="font: normal normal normal 23px/normal Helvetica; text-align: center; margin: 0px;"> </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;">On the set of axes provided, sketch the graph of \(y = \frac{1}{{f(x)}}\), clearly showing any asymptotes and indicating the coordinates of any local maxima or minima.</span></p>
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<p style="font: normal normal normal 23px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
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