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</div><h2>SL Paper 2</h2><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 4{\tan ^2}x - 4\sin x\) , \( - \frac{\pi }{3} \le x \le \frac{\pi }{3}\) .</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;">On the grid below, sketch the graph of \(y = f(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bad_day.png" alt></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;">Solve the equation \(f(x) = 1\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider \(f(x) = 2 - {x^2}\) , for \( - 2 \le x \le 2\) and \(g(x) = \sin {{\rm{e}}^x}\) , for \( - 2 \le x \le 2\) . The graph </span><span style="font-family: times new roman,times; font-size: medium;">of <em>f</em> is given below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/crash.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the diagram above, sketch the graph of <em>g</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \(f(x) = g(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the set of values of <em>x</em> such that \(f(x) > g(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3x\) , \(g(x) = 2x - 5\) and \(h(x) = (f \circ g)(x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(h(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({h^{ - 1}}(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . The graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/troy.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> crosses the <em>x</em>-axis at \(x = a\) , \(x = 0\) and \(x = b\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>a</em> and of <em>b</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> has a maximum value when \(x = c\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>c</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region under the graph of <em>f</em> from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about </span><span style="font-family: times new roman,times; font-size: medium;">the <em>x</em>-axis. Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the curve, the <em>x</em>-axis and the line \(x = c\) , </span><span style="font-family: times new roman,times; font-size: medium;">between \(x = a\) and \(x = c\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {{\text{e}}^{x + 1}} + 2\), for \( - 4 \le x \le 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following grid, sketch the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_12.27.30.png" alt></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"><span class="s1">The graph of \(f\) </span>is translated by the vector \(\left( {\begin{array}{*{20}{c}} 3 \\ { - 1} \end{array}} \right)\) <span class="s1">to obtain the graph of a function \(g\)</span>.</p>
<p class="p1">Find an expression for \(g(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = a\cos (b(x - c))\) . The diagram below shows part of the graph of <em>f</em> , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le x \le 10\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q5.jpg" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph has a local maximum at P(3, 5) , a local minimum at Q(7, − 5) , and crosses the <em>x</em>-axis at R.</span></p>
<p align="LEFT"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(c\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>b</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><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of R.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in two colonies, \(\rm{A}\) and \(\rm{B}\), starts increasing at the same time.</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 number of bacteria in colony \(\rm{A}\) after \(t\) hours is modelled by the function \(\rm{A}(t) = 12{{\text{e}}^{0.4t}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of bacteria in colony \({\text{A}}\) after four hours.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of bacteria in colony \({\text{A}}\) after four hours.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function \(B(t) = 24{{\text{e}}^{kt}}\).</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;">After four hours, there are \(60\) bacteria in colony \({\text{B}}\). Find the value of \(k\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function \(B(t) = 24{{\text{e}}^{kt}}\).</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 number of bacteria in colony \({\text{A}}\) first exceeds the number of bacteria in colony \({\text{B}}\) after \(n\) hours, where \(n \in \mathbb{Z}\). Find the value of \(n\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2} - 1\) and \(g(x) = {x^2} - 2\), for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the following grid, sketch the graph of \((f \circ g)(x)\), for \(0 \leqslant x \leqslant 2.25\).</p>
<p style="text-align: left;"><img src="images/Schermafbeelding_2017-08-15_om_08.00.33.png" alt="M17/5/MATME/SP2/ENG/TZ2/06.b"></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>The equation \((f \circ g)(x) = k\) has exactly two solutions, for \(0 \leqslant x \leqslant 2.25\). Find the possible values of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graph of \(f(x) = {{\rm{e}}^{ - {x^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/berlin.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points A, B, C, D and E lie on the graph of <em>f</em> . Two of these are points of inflexion.</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;">Identify the <strong>two</strong> points of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of each point of inflexion.</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><span style="font-family: times new roman,times; font-size: medium;">Use the second derivative to show that one of these points is a point of inflexion.</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({u_4}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the sum of the infinite sequence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider an arithmetic sequence with <em>n</em> terms, with first term (\( - 36\)) and eighth </span><span style="font-family: times new roman,times; font-size: medium;">term (\( - 8\)) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the common difference.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \({S_n} = 2{n^2} - 38n\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The sum of the infinite geometric sequence is equal to twice the sum of the </span><span style="font-family: times new roman,times; font-size: medium;">arithmetic sequence. Find <em>n</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(q\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(p\).</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>Write down the probability of drawing three blue marbles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \,\,{\text{sin}}\,\left( {{e^x}} \right)\) for 0 ≤ \(x\) ≤ 1.5. The following diagram shows the graph of \(f\).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <em>x</em>-intercept of the graph 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>The region enclosed by the graph of \(f\), the<em> y</em>-axis and the <em>x</em>-axis is rotated 360° about the <em>x</em>-axis.</p>
<p>Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of \(f(x) = - 2{x^3} + 5.1{x^2} + 3.6x - 0.4\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_05.42.47.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of the local minimum point.</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">The graph of \(f\) is translated to the graph of \(g\) by the vector \(\left( {\begin{array}{*{20}{c}} 0 \\ k \end{array}} \right)\). Find all values of \(k\) so that \(g(x) = 0\) has exactly one solution.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">An environmental group records the numbers of coyotes and foxes in a wildlife reserve after \(t\) <span class="s1">years, starting on 1 January 1995</span>.</p>
<p class="p1">Let \(c\) be the number of coyotes in the reserve after \(t\) years. The following table shows the number of coyotes after \(t\) years.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_08.53.25.png" alt></p>
<p class="p1">The relationship between the variables can be modelled by the regression equation \(c = at + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\) and of \(b\).</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">Use the regression equation to estimate the number of coyotes in the reserve when \(t = 7\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(f\) be the number of foxes in the reserve after \(t\) years. The number of foxes can be modelled by the equation \(f = \frac{{2000}}{{1 + 99{{\text{e}}^{ - kt}}}}\), where \(k\) <span class="s1">is a constant.</span></p>
<p class="p2">Find the number of foxes in the reserve on 1 January 1995.</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">After five years, there were 64 </span>foxes in the reserve. Find \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">During which year were the number of coyotes the same as the number of foxes?</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f'(x) = - 24{x^3} + 9{x^2} + 3x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There are two points of inflexion on the graph of <em>f</em> . Write down the <em>x</em>-coordinates </span><span style="font-family: times new roman,times; font-size: medium;">of these points.</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;">Let \(g(x) = f''(x)\) . Explain why the graph of <em>g</em> has no points of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><strong>All lengths in this question are in metres.</strong></p>
<p class="p1">Let \(f(x) = - 0.8{x^2} + 0.5\), for \( - 0.5 \leqslant x \leqslant 0.5\). Mark uses \(f(x)\) as a model to create a barrel. The region enclosed by the graph of \(f\), the \(x\)-axis, the line \(x = - 0.5\) and the line \(x = 0.5\) is rotated <span class="s1">360°</span> about the \(x\)-axis. This is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_15.49.19.png" alt="N16/5/MATME/SP2/ENG/TZ0/06"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the model to find the volume of the barrel.</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"><span class="s1">The empty barrel is being filled with water. The volume \(V{\text{ }}{{\text{m}}^3}\) </span>of water in the barrel after \(t\) minutes is given by \(V = 0.8(1 - {{\text{e}}^{ - 0.1t}})\). How long will it take for the barrel to be half-full?</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider a function \(f\), for \(0 \le x \le 10\). The following diagram shows the graph of \(f'\), the derivative of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_07.44.17.png" alt></p>
<p class="p1">The graph of \(f'\) passes through \((2,{\text{ }} - 2)\) and \((5,{\text{ }}1)\), and has \(x\)-intercepts at \(0\), \(4\) and \(6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(f\) has a local maximum point when \(x = p\). State the value of \(p\), and justify your answer.</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">Write down \(f'(2)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(g(x) = \ln \left( {f(x)} \right)\) and \(f(2) = 3\).</p>
<p class="p1">Find \(g'(2)\).</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">Verify that \(\ln 3 + \int_2^a {g'(x){\text{d}}x = g(a)} \), where \(0 \le a \le 10\).</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">The following diagram shows the graph of \(g'\), the derivative of \(g\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_07.59.38.png" alt></p>
<p class="p1">The shaded region \(A\) is enclosed by the curve, the <em>\(x\)</em>-axis and the line \(x = 2\), and has area \({\text{0.66 unit}}{{\text{s}}^{\text{2}}}\).</p>
<p class="p1">The shaded region \(B\) is enclosed by the curve, the \(x\)-axis and the line \(x = 5\), and has area \({\text{0.21 unit}}{{\text{s}}^{\text{2}}}\)<span class="s1">.</span></p>
<p class="p2">Find \(g(5)\)<span class="s2">.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = - 0.5{x^4} + 3{x^2} + 2x\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.09.00.png" alt="M17/5/MATME/SP2/ENG/TZ2/08"></p>
<p> </p>
<p>There are \(x\)-intercepts at \(x = 0\) and at \(x = p\). There is a maximum at A where \(x = a\), and a point of inflexion at B where \(x = b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</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>Write down the coordinates of A.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the rate of change of \(f\) at A.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the the rate of change of \(f\) at B.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and the line \(x = a\). The region \(R\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\cos (x - \sin x)\) , \(0 \le x \le 3\) .</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;">Sketch the graph of <em>f</em> on the following set of axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/marvin.png" alt></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> intersects the <em>x</em>-axis when \(x = a\) , \(a \ne 0\) . Write down the </span><span style="font-family: times new roman,times; font-size: medium;">value of <em>a</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is revolved \(360^\circ \) about the <em>x</em>-axis from \(x = 0\) to \(x = a\) . </span><span style="font-family: times new roman,times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A population of rare birds, \({P_t}\), can be modelled by the equation \({P_t} = {P_0}{{\text{e}}^{kt}}\), where \({P_0}\) is the initial population, and \(t\) is measured in decades. After one decade, it is estimated that \(\frac{{{P_1}}}{{{P_0}}} = 0.9\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Find the value of \(k\).</p>
<p class="p1">(ii) Interpret the meaning of the value of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the least number of <strong>whole </strong>years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The price of a used car depends partly on the distance it has travelled. The following table shows the distance and the price for seven cars on <span class="s1">1 </span>January <span class="s1">2010</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_17.58.43.png" alt="M16/5/MATME/SP2/ENG/TZ2/08"></p>
<p class="p1">The relationship between \(x\) and \(y\) can be modelled by the regression equation \(y = ax + b\).</p>
</div>
<div class="specification">
<p class="p1">On <span class="s1">1 </span>January <span class="s1">2010</span>, Lina buys a car which has travelled \(11\,000{\text{ km}}\).</p>
</div>
<div class="specification">
<p class="p1">The price of a car decreases by <span class="s1">5% </span>each year.</p>
</div>
<div class="specification">
<p class="p1">Lina will sell her car when its price reaches \(10\,000\)<span class="s1"> </span>dollars.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the correlation coefficient.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(a\) and 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">Use the regression equation to estimate the price of Lina’s car, giving your answer to the nearest <span class="s1">100 </span>dollars.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the price of Lina’s car after <span class="s1">6 </span>years.</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">Find the year when Lina sells her car.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = \frac{{2x - 6}}{{1 - x}}\), for \(x \ne 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For the graph of \(f\)</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>find the \(x\)-intercept;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>write down the equation of the vertical asymptote;</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>find the equation of the horizontal asymptote.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\mathop {\lim }\limits_{x \to \infty } f(x)\).</p>
<p class="p1"> </p>
<p class="p1"> </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The number of bacteria, <em>n</em> , in a dish, after <em>t</em> minutes is given by \(n = 800{{\rm{e}}^{0.13t}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>n</em> when \(t = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the rate at which <em>n</em> is increasing when \(t = 15\) .</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;">
<address><span style="font-family: times new roman,times; font-size: medium;">After <em>k</em> minutes, the rate of increase in <em>n</em> is greater than \(10000\) bacteria </span><span style="font-family: times new roman,times; font-size: medium;">per minute. Find the least value of <em>k</em> , where \(k \in \mathbb{Z}\) .</span></address>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Let </span><span style="font-family: TimesNewRomanPS-ItalicMT;">\(f(x) = {x^3} - 2x - 4\)</span><span style="font-family: TimesNewRomanPSMT;"> . The following diagram shows part of the curve of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">f </span></em><span style="font-family: TimesNewRomanPSMT;">.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/N12P2Q3.jpg" alt></span></span></p>
<p><span style="font-family: TimesNewRomanPSMT;"><span style="font-family: TimesNewRomanPSMT;"></span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">The curve crosses the </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">x</span></em><span style="font-family: TimesNewRomanPSMT;">-axis at the point P.</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Write down the </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">x</span></em><span style="font-family: TimesNewRomanPSMT;">-coordinate of P.</span></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;">Write down the gradient of the curve at P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the equation of the normal to the curve at P, giving your equation in the </span><span style="font-family: times new roman,times; font-size: medium;">form \(y = ax + b\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the following circle with centre O and radius <em>r</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/Jamie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P, R and Q are on the circumference, \({\rm{P}}\widehat {\rm{O}}{\rm{Q}} = 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) . </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the cosine rule to show that \({\rm{PQ}} = 2r\sin \theta \) .</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><span style="font-family: times new roman,times; font-size: medium;">Let <em>l</em> be the length of the arc PRQ .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(1.3{\rm{PQ}} - l = 0\) , find the value of \(\theta \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(\theta ) = 2.6\sin \theta - 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down the root of \(f(\theta ) = 0\) .</span></p>
<p> </p>
<div class="marks">[4]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the graph of <em>f</em> to find the values of \(\theta \) for which \(l < 1.3{\rm{PQ}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\cos x\) , for \(0 \le x \le 6\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[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;">On the grid below, sketch the graph of \(y = f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The quadratic equation \(k{x^2} + (k - 3)x + 1 = 0\) has two equal real roots.</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 possible values of <em>k</em>.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Write down</strong> the values of <em>k</em> for which \({x^2} + (k - 3)x + k = 0\) has two equal </span><span style="font-family: times new roman,times; font-size: medium;">real roots.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \frac{1}{2}x\sin x\) , for \(0 \le x \le 4\) .</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;">Sketch the graph of <em>g</em> on the following set of axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/spinning.png" alt></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><span style="font-family: times new roman,times; font-size: medium;">Hence find the value of <em>x</em> for which \(g(x) = - 1\) .</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;">Let \(f(x) = (x - 1)(x - 4)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-intercepts of the graph of \(f\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360^\circ\) about the \(x\)-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 5\cos \frac{\pi }{4}x\) and \(g(x) = - 0.5{x^2} + 5x - 8\) for \(0 \le x \le 9\) .</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;">On the same diagram, sketch the graphs of <em>f</em> and <em>g</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of \(f\) . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the <em>x</em>-intercept that lies between \(x = 0\) and \(x = 3\) ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the amplitude.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of <em>g</em> . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the two <em>x</em>-intercepts;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the equation of the axis of symmetry.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> . Find the area of <em>R</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows two ships A and B. At noon, ship A was 15 km due </span><span style="font-family: times new roman,times; font-size: medium;">north of ship B. Ship A was moving south at 15 km h<sup>–1</sup> and ship B was moving east at </span><span style="font-family: times new roman,times; font-size: medium;">11 km h<sup>–1</sup>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cloudy.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the distance between the ships</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) at 13:00;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) at 14:00.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(s(t)\) be the distance between the ships <em>t</em> hours after noon, for \(0 \le t \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(s(t) = \sqrt {346{t^2} - 450t + 225} \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(s(t)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Due to poor weather, the captain of ship A can only see another ship if they are </span><span style="font-family: times new roman,times; font-size: medium;">less than 8 km apart. Explain why the captain cannot see ship B between noon </span><span style="font-family: times new roman,times; font-size: medium;">and 16:00.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = \frac{1}{{x - 1}} + 2\), for \(x > 1\).</p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = a{e^{ - x}} + b\), for \(x \geqslant 1\). The graphs of \(f\) and \(g\) have the same horizontal asymptote.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the equation of the horizontal asymptote of the graph 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">Find \(f'(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \(b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(g'(1) = - e\), find the value of \(a\).</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">There is a value of \(x\)<span class="s1">, for \(1 < x < 4\)</span>, for which the graphs of \(f\) and \(g\) have the same gradient. Find this gradient.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
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<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(t) = 2{t^2} + 7\) , where \(t > 0\) . The function <em>v</em> is obtained when the graph of <em>f</em> is </span><span style="font-family: times new roman,times; font-size: medium;">transformed by</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">a stretch by a scale factor of \(\frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">parallel to the <em>y</em>-axis,</span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">followed by a translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 4}<br>\end{array}} \right)\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(v(t)\) , giving your answer in the form \(a{(t - b)^2} + c\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle moves along a straight line so that its velocity in ms<sup>−1</sup> , at </span><span style="font-family: times new roman,times; font-size: medium;">time <em>t </em>seconds, is given by<em> v</em> . Find the distance the particle travels between </span><span style="font-family: times new roman,times; font-size: medium;">\(t = 5.0\) and \(t = 6.8\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Jose takes medication. After <em>t</em> minutes, the concentration of medication left in his </span><span style="font-family: times new roman,times; font-size: medium;">bloodstream is given by \(A(t) = 10{(0.5)^{0.014t}}\) , where <em>A</em> is in milligrams per litre.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(A(0)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the concentration of medication left in his bloodstream after 50 minutes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At 13:00, when there is no medication in Jose’s bloodstream, he takes his first </span><span style="font-family: times new roman,times; font-size: medium;">dose of medication. He can take his medication again when the concentration </span><span style="font-family: times new roman,times; font-size: medium;">of medication reaches 0.395 milligrams per litre. What time will Jose be able to </span><span style="font-family: times new roman,times; font-size: medium;">take his medication again?</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graphs of \(f(x) = \ln (3x - 2) + 1\) and \(g(x) = - 4\cos (0.5x) + 2\) , for \(1 \le x \le 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/laurie.png" alt></span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>A</em> be the area of the region <strong>enclosed</strong> by the curves of <em>f</em> and <em>g</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for <em>A</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Calculate the value of <em>A</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (ii).</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f'(x)\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find \(g'(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There are two values of <em>x</em> for which the gradient of <em>f</em> is equal to the gradient </span><span style="font-family: times new roman,times; font-size: medium;">of <em>g</em>. Find both these values of <em>x</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = (x - 1)\sin x\) , for \(0 \le x \le \frac{{5\pi }}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/exhausted.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph has \(x\)-intercepts at \(0\), \(1\), \( \pi\) and \(k\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>k</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is rotated \(360^\circ \) about the <em>x</em>-axis. Let <em>V</em> be the volume of the </span><span style="font-family: times new roman,times; font-size: medium;">solid formed.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down an expression for <em>V</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is rotated \(360^\circ \) about the <em>x</em>-axis. Let <em>V</em> be the volume of the </span><span style="font-family: times new roman,times; font-size: medium;">solid formed.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>V</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 4x - {{\rm{e}}^{x - 2}} - 3\) , for \(0 \le x \le 5\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-intercepts of the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chops.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the gradient of the graph of <em>f</em> at \(x = 3\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = a{x^3} + b{x^2} + c\) , where <em>a</em> , <em>b</em> and <em>c</em> are real numbers. The graph of <em>f</em> passes </span><span style="font-family: times new roman,times; font-size: medium;">through the point (2, 9) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(8a + 4b + c = 9\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> has a local minimum at \((1{\text{, }}4)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find two other equations in <em>a</em> , <em>b</em> and <em>c</em> , giving your answers in a similar form to </span><span style="font-family: times new roman,times; font-size: medium;">part (a).</span></p>
<p> </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><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>a</em> , of <em>b</em> and of <em>c</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the graph of \(f(x) = \frac{{{{\text{e}}^x}}}{{5x - 10}} + 3\), for \(x \ne 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(y\)-intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the vertical asymptote.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the minimum value of \(f(x)\) for \(x > 2\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = {x^2} - 4x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> , for \( - 1 \le x \le 5\) .</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><span style="font-family: times new roman,times; font-size: medium;">This function can also be written as \(f(x) = {(x - p)^2} - 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>p </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is obtained by reflecting the graph of <em>f</em> in the <em>x</em>-axis, followed by a </span><span style="font-family: times new roman,times; font-size: medium;">translation of \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>6<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(g(x) = - {x^2} + 4x + 5\) .</span> </p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g </em>is obtained by reflecting the graph of <em>f </em>in the <em>x</em>-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>6<br>\end{array}} \right)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graphs of <em>f</em> and <em>g</em> intersect at two points.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-coordinates of these two points.</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><span style="font-family: times new roman,times; font-size: medium;">The graph of \(g\) is obtained by reflecting the graph of \(f\) in the <em>x</em>-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 6 <br>\end{array}} \right)\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2{x^2} - 8x - 9\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the coordinates of the vertex.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence or otherwise, express the function in the form \(f(x) = 2{(x - h)^2} + k\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \(f(x) = 0\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \frac{{6{x^2} - 4}}{{{{\text{e}}^x}}}\), for \(0 \leqslant x \leqslant 7\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(x\)-intercept of the graph 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>The graph of \(f\) has a maximum at the point A. Write down the coordinates of A.</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>On the following grid, sketch the graph of \(f\).</p>
<p><img src="images/Schermafbeelding_2018-02-12_om_11.45.37.png" alt="N17/5/MATME/SP2/ENG/TZ0/02.c"></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{100}}{{(1 + 50{{\rm{e}}^{ - 0.2x}})}}\) . Part of the graph of \(f\) is shown below.</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(f(0)\) .</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;">Solve \(f(x) = 95\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the range of \(f\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum rate of change of \(f\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \frac{{8x - 5}}{{cx + 6}}\) for \(x \ne - \frac{6}{c},\,\,c \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <em>x</em> = 3 is a vertical asymptote to the graph of <em>f</em>. Find the value of<em> c</em>.</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>Write down the equation of the horizontal asymptote to the graph of <em>f</em>.</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>The line <em>y</em> = <em>k</em>, where \(k \in \mathbb{R}\) intersects the graph of \(\left| {f\left( x \right)} \right|\) at exactly one point. Find the possible values of <em>k</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 2\ln (x - 3)\), for \(x > 3\). The following diagram shows part of the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_17.05.00.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the equation of the vertical asymptote to the graph 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">Find the \(x\)-intercept of the graph 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">The region enclosed by the graph of \(f\), the \(x\)-axis and the line \(x = 10\) <span class="s1">is rotated \(360\)° </span>about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the region enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of \(f\) shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/paolo.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following four diagrams show <strong>images</strong> of <em>f</em> under different transformations.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/paolo2.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the <strong>same</strong> grid sketch the graph of \(y = f( - x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Complete the following table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/britney.png" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of the transformation that gives the image in </span><span style="font-family: times new roman,times; font-size: medium;">Diagram A.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Consider </span><span style="font-family: TimesNewRomanPS-ItalicMT;">\(f(x) = x\ln (4 - {x^2})\)</span><span style="font-family: TimesNewRomanPSMT;"> , for </span><span style="font-family: Times New Roman;" lang="JA">\( - 2 < x < 2\)</span><span style="font-family: TimesNewRomanPSMT;"> . The graph of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">f </span></em><span style="font-family: TimesNewRomanPSMT;">is given below.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/witch.png" alt></span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let P and Q be points on the curve of <em>f</em> where the tangent to the graph of <em>f</em> is </span><span style="font-family: times new roman,times; font-size: medium;">parallel to the <em>x</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the <em>x</em>-coordinate of P and of Q.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Consider \(f(x) = k\) . Write down all values of <em>k</em> for which there are </span><span style="font-family: times new roman,times; font-size: medium;">exactly two solutions.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(g'(x) = \frac{{ - 2{x^4}}}{{4 - {x^2}}} + 3{x^2}\ln (4 - {x^2})\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</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; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(g'\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider \(g'(x) = w\) . Write down all values of <em>w</em> for which there are exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></p>
<p align="LEFT"> </p>
<p align="LEFT"> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 0.225{x^3} - 2.7x\), for \( - 3 \leqslant x \leqslant 3\). There is a local minimum point at <span class="s1">A</span>.</p>
</div>
<div class="specification">
<p class="p1">On the following grid,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of <span class="s1">A</span><span class="s2">.</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 class="p1">(i) <span class="Apple-converted-space"> </span>sketch the graph of \(f\), clearly indicating the point <span class="s1">A</span><span class="s2">;</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>sketch the tangent to the graph of \(f\) at <span class="s1">A</span>.</p>
<p class="p1" style="text-align: left;"><img src="images/Schermafbeelding_2017-03-03_om_17.19.46.png" alt="N16/5/MATME/SP2/ENG/TZ0/02.b"></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = x{{\text{e}}^{ - x}}\) and \(g(x) = - 3f(x) + 1\).</p>
<p class="p1">The graphs of \(f\) and \(g\) intersect at \(x = p\) and \(x = q\), where \(p < q\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(p\) <span class="s1">and of \(q\).</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 class="p1">Hence, find the area of the region enclosed by the graphs of \(f\) <span class="s1">and \(g\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>f</em>(<em>x</em>) = ln <em>x</em> − 5<em>x</em> , for <em>x</em> > 0 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f '</em>(<em>x</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f "</em>(<em>x</em>).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve<em> f '</em>(<em>x</em>)<em> = f "</em>(<em>x</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 6 - \ln ({x^2} + 2)\), for \(x \in \mathbb{R}\). The graph of \(f\) passes through the point \((p,{\text{ }}4)\), where \(p > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</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 following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_13.30.18.png" alt="N17/5/MATME/SP2/ENG/TZ0/05.b"></p>
<p>The region enclosed by the graph of \(f\), the \(x\)-axis and the lines \(x = - p\) and \(x = p\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f\) and \(g\) be functions such that \(g(x) = 2f(x + 1) + 5\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(a) The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">vertical stretch by a factor of \(k\) , followed by a translation \(\left( \begin{array}{l}<br>p\\<br>q<br>\end{array} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \(k\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \(p\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) \(q\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = - g(3x)\) . The point A(\(6\), \(5\)) on the graph of \(g\) is mapped to the </span><span style="font-family: times new roman,times; font-size: medium;">point \({\rm{A}}'\) on the graph of \(h\) . Find \({\rm{A}}'\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">vertical stretch by a factor of \(k\) , followed by a translation \(\left( \begin{array}{l}<br>p\\<br>q<br>\end{array} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \(k\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \(p\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) \(q\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = - g(3x)\) . The point A(\(6\), \(5\)) on the graph of \(g\) is mapped to the </span><span style="font-family: times new roman,times; font-size: medium;">point \({\rm{A}}'\) on the graph of \(h\) . Find \({\rm{A}}'\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{3x}}{{x - q}}\), where \(x \ne q\).</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;">Write down the equations of the vertical and horizontal asymptotes of the graph of \(f\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(q\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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 vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The point \({\text{P}}(x,{\text{ }}y)\) lies on the graph of \(f\). Show that \({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the coordinates of the points on the graph of \(f\) that are closest to \((1,3)\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A particle P </span>moves along a straight line so that its velocity, \(v\,{\text{m}}{{\text{s}}^{ - 1}}\), after \(t\) seconds, is given by \(v = \cos 3t - 2\sin t - 0.5\)<span class="s1">, for \(0 \leqslant t \leqslant 5\). The initial displacement of P from a fixed point O is 4 </span>metres.</p>
</div>
<div class="specification">
<p class="p1">The following sketch shows the graph of \(v\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.51.25.png" alt="M16/5/MATME/SP2/ENG/TZ1/09.b+c+d+e"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the displacement of <span class="s1">P </span>from <span class="s1">O </span>after <span class="s1">5 </span>seconds.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find when <span class="s1">P </span>is first at rest.</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">Write down the number of times <span class="s1">P </span>changes direction.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the acceleration of <span class="s1">P </span>after 3 seconds.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the maximum speed of <span class="s1">P</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}(1 - {x^2})\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Part of the graph of \(y = f(x)\), for \( - 6 \le x \le 2\) , is shown below. The <em>x</em>-coordinates of the local minimum and maximum points are <em>r</em> and <em>s</em> respectively. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/aching.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) . </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;">Write down the <strong>equation</strong> of the horizontal asymptote.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> and of <em>s</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Let <em>L</em> be the normal to the curve of <em>f</em> at \({\text{P}}(0{\text{, }}1)\) . Show that <em>L</em> has equation \(x + y = 1\) .</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><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the curve \(y = f(x)\) and the line <em>L</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for the area of <em>R</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Calculate the area of <em>R</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {\log _3}\frac{x}{2} + {\log _3}16 - {\log _3}4\) , for \(x > 0\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f(x) = {\log _3}2x\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(f(0.5)\) and of \(f(4.5)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function <em>f</em> can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>a</em> and of <em>b</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence on graph paper, <strong>sketch</strong> the graph of <em>f</em> , for \( - 5 \le x \le 5\) , \( - 5 \le y \le 5\) , </span><span style="font-family: times new roman,times; font-size: medium;">using a scale of 1 cm to 1 unit on each axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the equation of the asymptote.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \({f^{ - 1}}(0)\) .</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point A lies on the graph of <em>f</em> . At A, \(x = 4.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">On your diagram, sketch the graph of \({f^{ - 1}}\) , noting clearly the image of point A.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2{x^2} + 4x - 6\) .</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;">Express \(f(x)\) in the form \(f(x) = 2{(x - h)^2} + k\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the equation of the axis of symmetry of the graph of <em>f</em> .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express \(f(x)\) in the form \(f(x) = 2(x - p)(x - q)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a quadrilateral ABCD with obtuse angles \({\rm{A}}\widehat {\rm{B}}{\rm{C}}\) and \({\rm{A}}\widehat {\rm{D}}{\rm{C}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/footie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">AB = 5 cm, BC = 4 cm, CD = 4 cm, AD = 4 cm , \({\rm{B}}\widehat {\rm{A}}{\rm{C}} = {30^ \circ }\) , \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {x^ \circ }\) , \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = {y^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the cosine rule to show that \({\rm{AC}} = \sqrt {41 - 40\cos x} \) .</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;">Use the sine rule in triangle ABC to find another expression for AC.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Hence, find <em>x</em>, giving your answer to two decimal places.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find AC .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find <em>y</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, or otherwise, find the area of triangle ACD.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{{2x - 1}}{{x + 1}}\) , \(x \ne - 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({h^{ - 1}}(x)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of <em>h</em> for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any </span><span style="font-family: times new roman,times; font-size: medium;">asymptotes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the equations of the asymptotes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the <em>x</em>-intercept of the graph of <em>h</em> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region in the first quadrant enclosed by the graph of <em>h</em> , the <em>x</em>-axis </span><span style="font-family: times new roman,times; font-size: medium;">and the line \(x = 3\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the area of <em>R</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down an expression for the volume obtained when <em>R</em> is revolved </span><span style="font-family: times new roman,times; font-size: medium;">through \({360^ \circ }\) about the <em>x</em>-axis.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \ln x\) and \(g(x) = 3 + \ln \left( {\frac{x}{2}} \right)\), for \(x > 0\).</p>
<p>The graph of \(g\) can be obtained from the graph of \(f\) by two transformations:</p>
<p>\[\begin{array}{*{20}{l}} {{\text{a horizontal stretch of scale factor }}q{\text{ followed by}}} \\ {{\text{a translation of }}\left( {\begin{array}{*{20}{c}} h \\ k \end{array}} \right).} \end{array}\]</p>
</div>
<div class="specification">
<p>Let \(h(x) = g(x) \times \cos (0.1x)\), for \(0 < x < 4\). The following diagram shows the graph of \(h\) and the line \(y = x\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.34.27.png" alt="M17/5/MATME/SP2/ENG/TZ1/10.b.c"></p>
<p>The graph of \(h\) intersects the graph of \({h^{ - 1}}\) at two points. These points have \(x\) coordinates 0.111 and 3.31 correct to three significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(q\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(h\);</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>Write down the value of \(k\).</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>Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\). There is a point \({\text{P}}(a,{\text{ }}b)\) on the graph of \(h\) where \(d\) is a maximum.</p>
<p>Find the coordinates of P, where \(0.111 < a < 3.31\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3\sin x + 4\cos x\) , for \( - 2\pi \le x \le 2\pi \) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the amplitude;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) the <em>x</em>-intercept that lies between \( - \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">and 0.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence write \(f(x)\) in the form \(p\sin (qx + r)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down one value of <em>x</em> such that \(f'(x) = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the two values of <em>k</em> for which the equation \(f(x) = k\) has exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of <em>x</em>, between \(0\) and \(1\), </span><span style="font-family: times new roman,times; font-size: medium;">for which the gradient of <em>f</em> is equal to the gradient of <em>g</em>. Find this value of <em>x</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
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<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3{x^2}\) . The graph of <em>f</em> is translated 1 unit to the right and 2 units down. </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is the image of the graph of <em>f</em> after this translation.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the coordinates of the vertex of the graph of <em>g</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express <em>g</em> in the form \(g(x) = 3{(x - p)^2} + q\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>h</em> is the reflection of the graph of <em>g</em> in the <em>x</em>-axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the coordinates of the vertex of the graph of <em>h</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = A{{\rm{e}}^{kx}} + 3\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ryan.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>y</em>-intercept is at (0, 13) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(A = 10\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of <em>k</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Using your value of <em>k</em> , find \(f'(x)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, explain why <em>f</em> is a decreasing function.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the equation of the horizontal asymptote of the graph <em>f</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i), (ii) and (iii).</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = - {x^2} + 12x - 24\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{20x}}{{{{\rm{e}}^{0.3x}}}}\) , for \(0 \le x \le 20\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the <em>x</em>-coordinate of the maximum point on the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the interval where <em>f</em> is increasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the interval where the rate of change of <em>f</em> is increasing.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^{\frac{x}{4}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(g(x) = mx\) , where \(m \ge 0\) , and \( - 5 \le x \le 5\) . Let \(R\) be the region </span><span style="font-family: times new roman,times; font-size: medium;">enclosed by the \(y\)-axis, the graph of \(f\) , and the graph of \(g\) .</span></p>
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<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(m = 1\).</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graphs of \(f\) and \(g\) on the same axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the area of \(R\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the area of \(R\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the </span><span style="font-family: times new roman,times; font-size: medium;">value of \(m\) that gives the greatest value for the area of \(R\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(G(x) = 95{{\text{e}}^{( - 0.02x)}} + 40\), for \(20 \le x \le 200\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following grid, sketch the graph of \(G\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_05.12.49.png" alt></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">Robin and Pat are planning a wedding banquet. The cost per guest, \(G\) dollars, is modelled by the function \(G(n) = 95{{\text{e}}^{( - 0.02n)}} + 40\), for \(20 \le n \le 200\), where \(n\) is the number of guests.</p>
<p class="p1">Calculate the <strong>total</strong> cost for \(45\) guests.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2} + 2x + 1\) and \(g(x) = x - 5\), for \(x \in \mathbb{R}\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f(8)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \((g \circ f)(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve \((g \circ f)(x) = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The velocity <em>v</em> ms<sup>−1</sup> of an object after <em>t</em> seconds is given by \(v(t) = 15\sqrt t - 3t\) , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le t \le 25\) .</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of <em>v</em> , clearly indicating the maximum point.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br></span><img src="data:image/png;base64,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" alt></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for <em>d</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, write down the value of <em>d</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first three terms of a geometric sequence are \({u_1} = 0.64,{\text{ }}{u_2} = 1.6\), and \({u_3} = 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \({S_6}\).</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">Find the least value of \(n\) such that \({S_n} > 75\,000\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question">
<p>Jill plays the game nine times. Find the probability that she wins exactly two prizes.</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were 280 taxis in the city. After <em>n</em> years the number of taxis, <em>T</em>, in the city is given by\[T = 280 \times {1.12^n} .\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find the number of taxis in the city at the end of 2005. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the year in which the number of taxis is double the number of taxis there were at the end of 2000.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">At the end of 2000 there were \(25600\) people in the city who used taxis. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">After <em>n</em> years the number of people, <em>P</em>, in the city who used taxis is given by\[P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1n}}}} .\](i) Find the value of <em>P</em> at the end of 2005, giving your answer to the nearest whole number. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) After seven complete years, will the value of <em>P</em> be double its value at the end of 2000? Justify your answer.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the ratio of the number of people using taxis in the city to the number of taxis. The city will reduce the number of taxis if \(R < 70\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of <em>R</em> at the end of 2000. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) After how many complete years will the city first reduce the number of taxis?</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}\sin 2x + 10\) , for \(0 \le x \le 4\) . Part of the graph of <em>f</em> is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/apple.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is an <em>x</em>-intercept at the point A, a local maximum point at M, where \(x = p\) and </span><span style="font-size: medium;"><span style="font-family: times new roman,times;">a local minimum point at N, where \(x = q\)</span><span style="font-family: times new roman,times;"> .</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-coordinate of A.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_p^q {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> . Explain why this is not the area of the shaded region.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The points A and B <span class="s1">lie on a line \(L\)</span>, and have position vectors \(\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ 2 \end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 1} \end{array}} \right)\) respectively. Let O <span class="s1">be the origin. This is shown on the following diagram.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-01_om_15.56.14.png" alt="M16/5/MATME/SP2/ENG/TZ1/10"></span></p>
</div>
<div class="specification">
<p class="p1">The point C <span class="s1">also lies on \(L\)</span>, such that \(\overrightarrow {{\text{AC}}} = 2\overrightarrow {{\text{CB}}} \).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let \(\theta \) </span>be the angle between \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{OC}}} \).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let D be a point such that \(\overrightarrow {{\text{OD}}} = k\overrightarrow {{\text{OC}}} \)</span>, where \(k > 1\)<span class="s1">. Let E </span>be a point on \(L\) <span class="s1">such that \({\rm{C\hat ED}}\) </span>is a right angle. This is shown on the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_16.39.34.png" alt="M16/5/MATME/SP2/ENG/TZ1/10.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\overrightarrow {{\text{AB}}} \).</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 \(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\).</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\theta \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\left| {\overrightarrow {{\text{DE}}} } \right| = (k - 1)\left| {\overrightarrow {{\text{OC}}} } \right|\sin \theta \).</p>
<p class="p2"><span class="s1">(ii) <span class="Apple-converted-space"> </span>The distance from D </span>to line \(L\) <span class="s1">is less than 3 </span>units. Find the possible values of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2}\) and \(g(x) = 3\ln (x + 1)\), for \(x > - 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \(f(x) = g(x)\).</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 area of the region enclosed by the graphs of \(f\) and \(g\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle’s displacement, in metres, is given by \(s(t) = 2t\cos t\) , for \(0 \le t \le 6\) , </span><span style="font-family: times new roman,times; font-size: medium;">where <em>t</em> is the time in seconds.</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;">On the grid below, sketch the graph of \(s\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q7.jpg" alt></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><span style="font-family: times new roman,times; font-size: medium;">Find the maximum velocity of the particle.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = p\cos qx + r\) , for \( - 5 \le x \le 14\) , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/weather.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a minimum point at (0, −3) and a maximum point at (4, 7) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>r</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The equation \(y = k\) has exactly <strong>two</strong> solutions. Write down the value of <em>k</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 2x + 3\) and \(g(x) = {x^3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \((f \circ g)(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the equation \((f \circ g)(x) = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = - {x^4} + 2{x^3} - 1\), for \(0 \le x \le 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(f\) on the following grid.</p>
<p style="text-align: center;"><img src="image_3.html" alt></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">Solve \(f(x) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360°\) about the <em>\(x\)</em>-axis.</p>
<p class="p1">Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos ({{\rm{e}}^x})\) , for \( - 2 \le x \le 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P2TZ2Q2.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a square \(ABCD\), and a sector \(OAB\) of a circle centre \(O\), radius \(r\). Part of the square is shaded and labelled \(R\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_15.28.33.png" alt></p>
<p>\[{\rm{A\hat OB}} = \theta {\text{, where }}0.5 \ \le \ \theta < \pi .\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the area of the square \(ABCD\) is \(2{r^2}(1 - \cos \theta )\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>When \(\theta = \alpha \), the area of the square \(ABCD\) is equal to the area of the sector \(OAB\).</p>
<p>(i) Write down the area of the sector when \(\theta = \alpha \).</p>
<p>(ii) Hence find \(\alpha \).</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>When \(\theta = \beta \), the area of \(R\) is more than twice the area of the sector.</p>
<p>Find all possible values of \(\beta \).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1"><span class="s1">Let \(f(x) = k{x^2} + kx\) and \(g(x) = x - 0.8\)</span>. The graphs of \(f\) and \(g\) intersect at two distinct points.</p>
<p class="p1">Find the possible values of \(k\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The velocity of a particle in ms<sup>−1</sup> is given by \(v = {{\rm{e}}^{\sin t}} - 1\) , for \(0 \le t \le 5\) .</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;">On the grid below, sketch the graph of \(v\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ronan.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the total distance travelled by the particle in the first five seconds.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the positive \(t\)-intercept.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Consider a geometric sequence where the first term is 768 and the second term is 576.</p>
<p>Find the least value of \(n\) such that the \(n\)th term of the sequence is less than 7.</p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A rock falls off the top of a cliff. Let \(h\) be its height above ground in metres, </span><span style="font-family: times new roman,times; font-size: medium;">after \(t\) seconds.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The table below gives values of \(h\) and \(t\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/paige.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable </span><span style="font-family: times new roman,times; font-size: medium;">model for the data. Use Jane’s model to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) write down the height of the cliff;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) find the height of the rock after 4.5 seconds;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) find after how many seconds the height of the rock is \(30{\text{ m}}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Kevin thinks that the function \(g(t) = - 5.2{t^2} + 9.5t + 100\) is a better model for </span><span style="font-family: times new roman,times; font-size: medium;">the data. Use Kevin’s model to find when the rock hits the ground.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) On graph paper, using a scale of 1 cm to 1 second, and 1 cm to 10 m, </span><span style="font-family: times new roman,times; font-size: medium;">plot the data given in the table.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) By comparing the graphs of <em>f</em> and <em>g</em> with the plotted data, explain which </span><span style="font-family: times new roman,times; font-size: medium;">function is a better model for the height of the falling rock.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A farmer wishes to create a rectangular enclosure, ABCD, of area 525 m<sup>2</sup>, as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/friends.png" alt></span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The fencing used for side AB costs \(\$ 11\) per metre. The fencing for the other three sides </span><span style="font-family: times new roman,times; font-size: medium;">costs \(\$ 3\) per metre. The farmer creates an enclosure so that the cost is a minimum. Find this minimum cost.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first two terms of a geometric sequence \({u_n}\) are \({u_1} = 4\) and \({u_2} = 4.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the common ratio.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find \({u_5}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Another sequence \({v_n}\) is defined by \({v_n} = a{n^k}\), where \(a,{\text{ }}k \in \mathbb{R}\), and \(n \in {\mathbb{Z}^ + }\), such that \({v_1} = 0.05\) and \({v_2} = 0.25\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(a\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(k\).</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">Find the smallest value of \(n\) for which \({v_n} > {u_n}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = {{\text{e}}^{2\,{\text{sin}}\left( {\frac{{\pi x}}{2}} \right)}}\), for <em>x</em> > 0.</p>
<p>The <em>k </em>th maximum point on the graph of <em>f</em> has <em>x</em>-coordinate <em>x<sub>k</sub></em> where \(k \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>x<sub>k</sub></em><sub> + 1</sub> = <em>x<sub>k</sub></em> + <em>a</em>, find <em>a</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <em>n</em> such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{3x}}{2} + 1\) , \(g(x) = 4\cos \left( {\frac{x}{3}} \right) - 1\) . Let \(h(x) = (g \circ f)(x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for \(h(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the period of \(h\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \(h\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right),{\text{ for }} - 4 \leqslant x \leqslant 4.\)</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 \(f\).</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 values of \(x\) where the function is decreasing.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(a\);</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(c\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after \(t\) seconds is given by</p>
<p class="p1">\[v = 6t - 6,{\text{ for }}0 \leqslant t \leqslant 2.\]</p>
<p class="p1">After \(p\) <span class="s1">seconds, the particle is 2 m </span>from its initial position. Find the possible values of \(p\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the expansion of \({(x + 2)^{11}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the number of terms in this expansion.</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;">Find the term containing \({x^2}\) .</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;">A particle moves along a straight line such that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\), is given by \(v(t) = 10t{{\text{e}}^{ - 1.7t}}\), for \(t \geqslant 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On the grid below, sketch the graph of \(v\), for \(0 \leqslant t \leqslant 4\).</span></p>
<p style="font: normal normal normal 17px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_5a.png" alt></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 distance travelled by the particle in the first three seconds.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the velocity of the particle when its acceleration is zero.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2x + 4\) and \(g(x) = 7{x^2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({f^{ - 1}}(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(3.5)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a circle with centre O and radius \(r\) cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/scooby.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Points A and B are on the circumference of the circle and \({\rm{A}}\hat {\rm{O}}{\rm{B}} = 1.4\) radians .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point C is on [OA] such that \({\rm{B}}\hat {\rm{C}}{\rm{O}} = \frac{\pi }{2}\) radians</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \({\rm{OC}} = r\cos 1.4\) .</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 area of the shaded region is \(25\) cm<sup>2</sup> . Find the value of \(r\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of a function \(y = f(x)\), for \( - 6 \leqslant x \leqslant - 2\).</p>
<p>The points \(( - 6,{\text{ }}6)\) and \(( - 2,{\text{ }}6)\) lie on the graph of \(f\). There is a minimum point at \(( - 4,{\text{ }}0)\).</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p style="text-align: left;">Let \(g(x) = f(x - 5)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the grid above, sketch the graph of \(g\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the domain of \(g\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \({{\rm{e}}^x} = 4\sin x\) , for \(0 \le x \le 2\pi \) .</span></p>
</div>
<br><hr><br><div class="specification">
<p>Let <em>g</em>(<em>x</em>) = −(<em>x</em> − 1)<sup>2</sup> + 5.</p>
</div>
<div class="specification">
<p>Let <em>f</em>(<em>x</em>) = x<sup>2</sup>. The following diagram shows part of the graph of <em>f</em>.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The graph of <em>g</em> intersects the graph of <em>f</em> at <em>x</em> = −1 and <em>x</em> = 2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of the vertex of the graph of <em>g</em>.</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>On the grid above, sketch the graph of g for −2 ≤ <em>x</em> ≤ 4.</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 area of the region enclosed by the graphs of <em>f</em> and <em>g</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a geometric series, \({u_1} = \frac{1}{{81}}\) and \({u_4} = \frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</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 \(r\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the smallest value of <em>n</em> for which \({S_n} > 40\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {x^3} - 4x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Expand \({(x + h)^3}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the formula \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}\) </span><span style="font-family: times new roman,times; font-size: medium;">to show that </span><span style="font-family: times new roman,times; font-size: medium;">the derivative of \(f(x)\) is \(3{x^2} - 4\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} - 2)\) is parallel to the tangent at </span><span style="font-family: times new roman,times; font-size: medium;">a point Q. Find the coordinates of Q.</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><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is decreasing for \(p < x < q\) . Find the value of <em>p</em> and of <em>q</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><span style="font-family: times new roman,times; font-size: medium;">Write down the range of values for the gradient of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = 5 - {x^2}\). Part of the graph of \(f\)is shown in the following diagram.</span></p>
<p style="margin: 0px; font-style: normal; font-variant: normal; font-weight: normal; font-size: 21px; line-height: normal; font-family: 'Times New Roman'; text-align: center;"><img src="images/maths_2.png" alt></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 graph crosses the \(x\)-axis at the points \(\rm{A}\) and \(\rm{B}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the \(x\)-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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