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</div><h2>SL Paper 1</h2><div class="specification">
<p>The following diagram shows the graph of \(f’\), the derivative of \(f\).</p>
<p><img src="images/Schermafbeelding_2017-08-11_om_08.50.59.png" alt="M17/5/MATME/SP1/ENG/TZ1/06"></p>
<p>The graph of \(f’\) has a local minimum at A, a local maximum at B and passes through \((4,{\text{ }} - 2)\).</p>
</div>
<div class="specification">
<p>The point \({\text{P}}(4,{\text{ }}3)\) lies on the graph of the function, \(f\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the gradient of the curve of \(f\) at P.</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>Find the equation of the normal to the curve of \(f\) at P.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the concavity of the graph of \(f\) when \(4 < x < 5\) <strong>and </strong>justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The values of the functions \(f\) and \(g\) and their derivatives for \(x = 1\) and \(x = 8\) are shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_16.42.43.png" alt="M17/5/MATME/SP1/ENG/TZ2/06"></p>
<p>Let \(h(x) = f(x)g(x)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(h(1)\).</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 \(h'(8)\).</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 following diagram shows part of the graph of a quadratic function <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tent.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>x</em>-intercepts are at \(( - 4{\text{, }}0)\) and \((6{\text{, }}0)\) , and the <em>y</em>-intercept is at \((0{\text{, }}240)\) .</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 \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .</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;">Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .</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;">A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\) , at time <em>t</em> seconds is </span><span style="font-family: times new roman,times; font-size: medium;">given by \(v = 240 + 20t - 10{t^2}\) , for \(0 \le t \le 6\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of<em> t</em> when the speed of the particle is greatest.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the acceleration of the particle when its speed is zero.</span></p>
<div class="marks">[7]</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 \(g(x) = 2x\sin x\) .</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 \(g'(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the gradient of the graph of <em>g</em> at \(x = \pi \) .</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 \(f(x) = \frac{{ax}}{{{x^2} + 1}}\) , \( - 8 \le x \le 8\) , \(a \in \mathbb{R}\) .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/bernie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The region between \(x = 3\) and \(x = 7\) is shaded.</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 \(f( - x) = - f(x)\) .</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;">Given that \(f''(x) = \frac{{2ax({x^2} - 3)}}{{{{({x^2} + 1)}^3}}}\) , find the coordinates of all points of inflexion.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">It is given that \(\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C\) </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 the area of the shaded region, giving your answer in the form \(p\ln q\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of \(\int_4^8 {2f(x - 1){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2} - x\), for \(x \in \mathbb{R}\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.25.10.png" alt="N17/5/MATME/SP1/ENG/TZ0/08"></p>
<p>The graph of \(f\) crosses the \(x\)-axis at the origin and at the point \({\text{P}}(1,{\text{ }}0)\).</p>
</div>
<div class="specification">
<p>The line <em>L</em> is the normal to the graph of <em>f</em> at P.</p>
</div>
<div class="specification">
<p>The line \(L\) intersects the graph of \(f\) at another point Q, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.27.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/08.c.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f’(1) = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of \(L\) in the form \(y = ax + b\).</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>Find the \(x\)-coordinate of Q.</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>Find the area of the region enclosed by the graph of \(f\) and the line \(L\).</p>
<div class="marks">[6]</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;">Let \(f(x) = \frac{{\cos x}}{{\sin x}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , for \(\sin x \ne 0\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In the following table, \(f'\left( {\frac{\pi }{2}} \right) = p\) and \(f''\left( {\frac{\pi }{2}} \right) = q\) . The table also gives approximate values of \(f'(x)\) and \(f''(x)\) near \(x = \frac{\pi }{2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/batman.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;">Use the quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><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">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>p</em> and of <em>q</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use information from the table to explain why there is a point of inflexion on the </span><span style="font-family: times new roman,times; font-size: medium;">graph of <em>f</em> where \(x = \frac{\pi }{2}\) .</span></p>
<div class="marks">[2]</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;">Consider \(f(x) = {x^2} + \frac{p}{x}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(x \ne 0\) , where <em>p</em> is a constant.</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;">There is a minimum value of \(f(x)\) when \(x = - 2\) . Find the value of \(p\) .</span></p>
<div class="marks">[4]</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;">The following diagram shows the graphs of the <strong>displacement</strong>, <strong>velocity</strong> and </span><span style="font-family: times new roman,times; font-size: medium;"><strong>acceleration</strong> of a moving object as functions of time, <em>t</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/333.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;">Complete the following table by noting which graph A, B or C corresponds to </span><span style="font-family: times new roman,times; font-size: medium;">each function.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/444.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;">Write down the value of <em>t</em> when the velocity is greatest.</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 \(f(x) = \cos x + \sqrt 3 \sin x\) , \(0 \le x \le 2\pi \) . The following diagram shows the graph of \(f\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/movie.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The \(y\)-intercept is at (\(0\), \(1\)) , there is a minimum point at A (\(p\), \(q\)) and a maximum </span><span style="font-family: times new roman,times; font-size: medium;">point at 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 \(f'(x)\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Hence</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) show that \(q = - 2\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) verify that A is a minimum point.</span></p>
<div class="marks">[10]</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 maximum value of \(f(x)\) .</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;">The function \(f(x)\) can be written in the form \(r\cos (x - a)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> and of <em>a</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider a function \(f\). The line <em>L</em><sub>1</sub> with equation \(y = 3x + 1\) is a tangent to the graph of \(f\) when \(x = 2\)</p>
</div>
<div class="specification">
<p>Let \(g\left( x \right) = f\left( {{x^2} + 1} \right)\) and P be the point on the graph of \(g\) where \(x = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \(f'\left( 2 \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f\left( 2 \right)\).</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>Show that the graph of <em>g</em> has a gradient of 6 at P.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <em>L</em><sub>2</sub> be the tangent to the graph of <em>g</em> at P. <em>L</em><sub>1</sub> intersects <em>L</em><sub>2</sub> at the point Q.</p>
<p>Find the y-coordinate of Q.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graph of \(f(x) = a\sin (b(x - c)) + d\) , for \(2 \le x \le 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/deanna.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum point at P(4, 12) and a minimum point at Q(8, −4) .</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;">Use the graph to write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>a</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>c</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>d</em> .</span></p>
<div class="marks">[3]</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;">Show that \(b = \frac{\pi }{4}\) .</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 \(f'(x)\) .</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;">At a point R, the gradient is \( - 2\pi \) . Find the <em>x</em>-coordinate of R.</span></p>
<div class="marks">[6]</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;">Given that \(f(x) = \frac{1}{x}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , answer the following.</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 first four derivatives of \(f(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Write an expression for \({f^{(n)}}(x)\) in terms of <em>x</em> and <em>n</em> .</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 \(f(x) = 6 + 6\sin x\) . 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/abba.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the curve of <em>f</em> , the <em>x</em>-axis, and the <em>y</em>-axis.</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;">Solve for \(0 \le x < 2\pi \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(6 + 6\sin x = 6\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(6 + 6\sin x = 0\) .</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;">Write down the exact value of the <em>x</em>-intercept of <em>f</em> , for \(0 \le x < 2\pi \) .</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 area of the shaded region is <em>k</em> . Find the value of <em>k</em> , giving your answer in </span><span style="font-family: times new roman,times; font-size: medium;">terms of \(\pi \) .</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;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of this transformation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_p^{p + \frac{{3\pi }}{2}} {g(x){\rm{d}}x} = k\) and \(0 \le p < 2\pi \) , write down the two values of <em>p</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2}\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_17.08.23.png" alt="M17/5/MATME/SP1/ENG/TZ2/10"></p>
<p>The line \(L\) is the tangent to the graph of \(f\) at the point \({\text{A}}( - k,{\text{ }}{k^2})\), and intersects the \(x\)-axis at point B. The point C is \(( - k,{\text{ }}0)\).</p>
</div>
<div class="specification">
<p>The region \(R\) is enclosed by \(L\), the graph of \(f\), and the \(x\)-axis. This is shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_17.07.29.png" alt="M17/5/MATME/SP1/ENG/TZ2/10.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \(f'(x)\).</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>Find the gradient of \(L\).</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>Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).</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>Find the area of triangle ABC, giving your answer in terms of \(k\).</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>Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = \sqrt {4x + 5} \), for \(x \geqslant - 1.25\).</p>
</div>
<div class="specification">
<p class="p1">Consider another function \(g\)<span class="s1">. Let R </span>be a point on the graph of \(g\). The \(x\)<span class="s1">-coordinate of R is 1. The equation of the tangent to the graph at R </span>is \(y = 3x + 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f'(1)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down \(g'(1)\).</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 \(g(1)\).</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">Let \(h(x) = f(x) \times g(x)\). Find the equation of the tangent to the graph of \(h\) at the point where \(x = 1\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \cos x\).</p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = {x^k}\), where \(k \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="specification">
<p>Let \(k = 21\) and \(h(x) = \left( {{f^{(19)}}(x) \times {g^{(19)}}(x)} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the first four derivatives of \(f(x)\)<span class="s1">.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find \({f^{(19)}}(x)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the first three derivatives of \(g(x)\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that \({g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})\), find \(p\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(h'(x)\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence, show that \(h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}\).</p>
<div class="marks">[7]</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;">The diagram shows part of the graph of \(y = f'(x)\) . The <em>x</em>-intercepts are at points A and C. </span><span style="font-family: times new roman,times; font-size: medium;">There is a minimum at B, and a maximum at D.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/friday.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;">(i) Write down the value of \(f'(x)\) at C.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) </span><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong>, show that C corresponds to a minimum on the graph of <em>f</em> , i.e. it </span><span style="font-family: times new roman,times; font-size: medium;">has the same <em>x</em>-coordinate.</span></p>
<p> </p>
<div class="marks">[3]</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;">Which of the points A, B, D corresponds to a maximum on 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;">Show that B corresponds to a point of inflexion on the graph of <em>f</em> .</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 \(f(x) = {{\rm{e}}^{6x}}\) .</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 \(f'(x)\) .</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;">The tangent to the graph of <em>f</em> at the point \({\text{P}}(0{\text{, }}b)\) has gradient <em>m</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(m = 6\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find <em>b</em> .</span></p>
<p> </p>
<div class="marks">[4]</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;">Hence, write down the equation of this tangent.</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;">Consider the functions \(f(x)\) , \(g(x)\) and \(h(x)\) . The following table gives some values </span><span style="font-family: times new roman,times; font-size: medium;">associated with these functions.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/omt.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows parts of the graphs of \(h\) and \(h''\) .</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/jls.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a point of inflexion on the graph of \(h\) at P, when \(x = 3\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(h(x) = f(x) \times g(x)\) ,<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(g(3)\) , of \(f'(3)\) , and of \(h''(2)\) .</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;">Explain why P is a point of inflexion.</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 \(y\)-coordinate of P.</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><span style="font-family: times new roman,times; font-size: medium;">find the equation of the normal to the graph of \(h\) at P.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\int {\frac{1}{{2x + 3}}} {\rm{d}}x\) .</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;">Given that \(\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \ln \sqrt P \) , find the value of <em>P</em>.</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 following diagram shows the graph of a quadratic function <em>f</em> , for \(0 \le x \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q8.jpg" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph passes through the point P(0, 13) , and its vertex is the point V(2, 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;">The function can be written in the form \(f(x) = a{(x - h)^2} + k\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>h</em> and of <em>k</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(a = 3\) .</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;">Find \(f(x)\) , giving your answer in the form \(A{x^2} + Bx + C\) .</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;">Calculate the area enclosed by the graph of <em>f</em> , the <em>x</em>-axis, and the lines \(x = 2\) and \(x = 4\) .</span></p>
<div class="marks">[8]</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 this question <em>s</em> represents displacement in metres and <em>t</em> represents time in seconds.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The velocity <em>v</em> m s<sup>–1</sup> of a moving body is given by \(v = 40 - at\) where <em>a</em> is a non-zero </span><span style="font-family: times new roman,times; font-size: medium;">constant.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Trains approaching a station start to slow down when they pass a point P. As a train </span><span style="font-family: times new roman,times; font-size: medium;">slows down, its velocity is given by \(v = 40 - at\) , where \(t = 0\) at P. The station is 500 m </span><span style="font-family: times new roman,times; font-size: medium;">from P.</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) If \(s = 100\) when \(t = 0\) , find an expression for <em>s</em> in terms of <em>a</em> and <em>t</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) If \(s = 0\) when \(t = 0\) , write down an expression for <em>s</em> in terms of <em>a</em> and<em> t</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A train M slows down so that it comes to a stop at the station.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the time it takes train M to come to a stop, giving your answer in terms </span><span style="font-family: times new roman,times; font-size: medium;">of <em>a</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence show that \(a = \frac{8}{5}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For a different train N, the value of <em>a</em> is 4.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that this train will stop <strong>before</strong> it reaches the station.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int {x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f(x)\), given that \(f’(x) = x{{\text{e}}^{{x^2} - 1}}\) and \(f( - 1) = 3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = k{x^4}\) . The point \({\text{P}}(1{\text{, }}k)\) lies on the curve of <em>f</em> . At P, the normal to the curve </span><span style="font-family: times new roman,times; font-size: medium;">is parallel to \(y = - \frac{1}{8}x\) </span><span style="font-family: times new roman,times; font-size: medium;">. Find the value of <em>k</em>.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of the function \(y = f(x)\) passes through the point \(\left( {\frac{3}{2},4} \right)\) . The gradient function of <em>f</em> is given as \(f'(x) = \sin (2x - 3)\) . Find \(f(x)\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;"> The following diagram shows part of the graph of<span style="line-height: normal; background-color: #f7f7f7;"> \(y = f(x)\).</span></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_6.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 has a local maximum at \(A\), where \(x = - 2\), and a local minimum at \(B\), where \(x = 6\).</span></p>
<p> </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;">On the following axes, sketch the graph of \(y = f'(x)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the following in order from least to greatest: \(f(0),{\text{ }}f'(6),{\text{ }}f''( - 2)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(y = f(x)\)<span class="s1">, for \( - 0.5 \le \) x \( \le \) \(6.5\)</span>. The following diagram shows the graph of \(f'\), the derivative of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_13.43.06.png" alt></p>
<p class="p1">The graph of \(f'\) has a local maximum when \(x = 2\), a local minimum when \(x = 4\), and it crosses the <em>\(x\)-</em><span class="s1">axis at the point \((5,{\text{ }}0)\)</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the graph of \(f\) has a local minimum when \(x = 5\).</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 set of values of \(x\) for which the graph of \(f\) is concave down.</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 following diagram shows the shaded regions \(A\), \(B\) and \(C\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_13.46.59.png" alt></p>
<p class="p1">The regions are enclosed by the graph of \(f'\), the \(x\)-axis, the \(y\)-axis, and the line \(x = 6\).</p>
<p class="p1">The area of region \(A\) <span class="s1">is 12</span>, the area of region \(B\) <span class="s1">is 6.75 </span>and the area of region \(C\) <span class="s1">is 6.75</span>.</p>
<p class="p1">Given that \(f(0) = 14\), find \(f(6)\).</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">The following diagram shows the shaded regions \(A\), \(B\) and \(C\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_13.46.59.png" alt></p>
<p class="p1">The regions are enclosed by the graph of \(f'\), the <em>x</em>-axis, the <em>y</em>-axis, and the line \(x = 6\).</p>
<p class="p1">The area of region \(A\) <span class="s1">is 12</span>, the area of region \(B\) <span class="s1">is 6.75 </span>and the area of region \(C\) <span class="s1">is 6.75</span>.</p>
<p class="p1">Let \(g(x) = {\left( {f(x)} \right)^2}\). Given that \(f'(6) = 16\), find the equation of the tangent to the graph of \(g\) at the point where \(x = 6\).</p>
<div class="marks">[6]</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 \(\int_1^5 {3f(x){\rm{d}}x = 12} \) .</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 \(\int_5^1 {f(x){\rm{d}}x = - 4} \) .</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 \(\int_1^2 {(x + f(x)){\rm{d}}x + } \int_2^5 {(x + f(x)){\rm{d}}x} \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A function \(f\) has its derivative given by \(f'(x) = 3{x^2} - 2kx - 9\), where \(k\) is a constant.</p>
</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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(f\) has a point of inflexion when \(x = 1\).</p>
<p class="p1">Show that \(k = 3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f'( - 2)\).</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 equation of the tangent to the curve of \(f\) at \(( - 2,{\text{ }}1)\), giving your answer in the form \(y = ax + b\).</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>Given that \(f'( - 1) = 0\), explain why the graph of \(f\) has a local maximum when \(x = - 1\).</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;">Consider \(f(x) = {x^2}\sin x\) .</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 \(f'(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the gradient of the curve of \(f\) at \(x = \frac{\pi }{2}\) .</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;">Consider \(f(x) = \frac{1}{3}{x^3} + 2{x^2} - 5x\) . Part of the graph of <em>f</em> is shown below. There is a maximum point at M, and a point of inflexion at N. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/abc.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;">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;">Find the <em>x</em>-coordinate of M.</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;">Find the <em>x</em>-coordinate of N.</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;">The line <em>L</em> is the tangent to the curve of <em>f</em> at \((3{\text{, }}12)\). Find the equation of <em>L</em> in the form \(y = ax + b\) .</span></p>
<div class="marks">[4]</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;">Let \(f(x) = \frac{1}{4}{x^2} + 2\) </span><span style="font-family: times new roman,times; font-size: medium;"> . The line <em>L</em> is the tangent to the curve of <em>f</em> at (4, 6) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \frac{{90}}{{3x + 4}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, for \(2 \le x \le 12\) . The following diagram shows the graph of <em>g</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/mickey.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;">Find the equation of <em>L</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the area of the region enclosed by the curve of <em>g</em> , the <em>x</em>-axis, and </span><span style="font-family: times new roman,times; font-size: medium;">the lines \(x = 2\) and \(x = 12\) . Give your answer in the form \(a\ln b\) , where \(a,b \in \mathbb{Z}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is reflected in the<em> x</em>-axis to give the graph of <em>h</em> . The area of the </span><span style="font-family: times new roman,times; font-size: medium;">region enclosed by the lines <em>L</em> , \(x = 2\) , \(x = 12\) and the <em>x</em>-axis is 120 \(120{\text{ c}}{{\text{m}}^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area enclosed by the lines<em> L</em> , \(x = 2\) , \(x = 12\) and the graph of <em>h</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Fred makes an open metal container in the shape of a cuboid, as shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_09.43.42.png" alt="M16/5/MATME/SP1/ENG/TZ2/09"></p>
<p class="p1">The container has height \(x{\text{ m}}\), width \(x{\text{ m}}\) and length \(y{\text{ m}}\)<span class="s1">. The volume is \(36{\text{ }}{{\text{m}}^3}\)</span>.</p>
<p class="p1">Let \(A(x)\) be the outside surface area of the container.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(A(x) = \frac{{108}}{x} + 2{x^2}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(A'(x)\).</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">Given that the outside surface area is a minimum, find the height of the container.</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">Fred paints the outside of the container. A tin of paint covers a surface area of \({\text{10 }}{{\text{m}}^{\text{2}}}\) and costs $20<span class="s1">. Find the total cost of the tins needed to paint the container.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of \(f(x) = 2x\sqrt {{a^2} - {x^2}} \)<span class="s1">, for \( - 1 \leqslant x \leqslant a\)</span>, where \(a > 1\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_10.04.36.png" alt="M16/5/MATME/SP1/ENG/TZ2/10"></p>
<p class="p1">The line \(L\) is the tangent to the graph of \(f\) <span class="s1">at the origin, O. The point \({\text{P}}(a,{\text{ }}b)\) </span>lies on \(L\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The point \({\text{Q}}(a,{\text{ }}0)\) </span>lies on the graph of \(f\). Let \(R\) be the region enclosed by the graph of \(f\) and the \(x\)-axis. This information is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_10.14.09.png" alt="M16/5/MATME/SP1/ENG/TZ2/10.b+c"></p>
<p class="p1">Let \({A_R}\) be the area of the region \(R\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Given that \(f'(x) = \frac{{2{a^2} - 4{x^2}}}{{\sqrt {{a^2} - {x^2}} }}\), for \( - 1 \leqslant x < a\), <span class="s1">find the equation of \(L\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find an expression for \(b\) in terms of \(a\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \({A_R} = \frac{2}{3}{a^3}\).</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 class="p1">Let \({A_T}\) <span class="s1">be the area of the triangle OPQ</span>. Given that \({A_T} = k{A_R}\), find the value of \(k\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{2x}}{{{x^2} + 5}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use the quotient rule to show that \(f'(x) = \frac{{10 - 2{x^2}}}{{{{({x^2} + 5)}^2}}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {\frac{{2x}}{{{x^2} + 5}}{\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows part of the graph of \(f\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_10c.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 shaded region is enclosed by the graph of \(f\), the \(x\)-axis, and the lines \(x = \sqrt 5 \) and \(x = q\). This region has an area of \(\ln 7\). Find the value of \(q\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\int_\pi ^a {\cos 2x{\text{d}}x} = \frac{1}{2}{\text{, where }}\pi < a < 2\pi \). Find the value of \(a\).</span></p>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A rocket moving in a straight line has velocity \(v\) km s<sup>–1</sup> and displacement \(s\) km at </span><span style="font-family: times new roman,times; font-size: medium;">time \(t\) seconds. The velocity \(v\) is given by \(v(t) = 6{{\rm{e}}^{2t}} + t\) . When \(t = 0\) , \(s = 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for the displacement of the rocket in terms of \(t\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of a function \(f\). There is a local minimum point at \(A\), where \(x > 0\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_05.51.23.png" alt></p>
<p class="p1">The derivative of \(f\) is given by \(f'(x) = 3{x^2} - 8x - 3\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-coordinate of \(A\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The \(y\)-intercept of the graph is at (\(0,6\)). Find an expression for \(f(x)\).</p>
<p class="p1">The graph of a function \(g\) is obtained by reflecting the graph of \(f\) in the \(y\)-axis, followed by a translation of \(\left({\begin{array}{*{20}{c}}m\\n\end{array}}\right)\).</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 class="p1">Find the \(x\)-coordinate of the local minimum point on the graph of \(g\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 1 + {{\text{e}}^{ - x}}\) and \(g(x) = 2x + b\), for \(x \in \mathbb{R}\), where \(b\) is a constant.</p>
</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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3\), find the value of \(b\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}\). Given that \(f(0) = 1\), find \(f(x)\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A function <em>f</em> has its first derivative given by \(f'(x) = {(x - 3)^3}\) .</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;">Find the second derivative.</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 \(f'(3)\) and \(f''(3)\) .</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 point P on the graph of <em>f</em> has <em>x</em>-coordinate \(3\). Explain why P is not a point </span><span style="font-family: times new roman,times; font-size: medium;">of inflexion.</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 \(f(x) = \frac{{6x}}{{x + 1}}\) , 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;">Find \(f'(x)\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(g'(x) = \frac{1}{{x(x + 1)}}\) .</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;">Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of <em>h</em> , the <em>x</em>-axis and </span><span style="font-family: times new roman,times; font-size: medium;">the lines \(x = \frac{1}{5}\) </span><span style="font-family: times new roman,times; font-size: medium;"> and \(x = k\) is \(\ln 4\) . Given that \(k > \frac{1}{5}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , find the value of <em>k</em> .</span></p>
<div class="marks">[7]</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;">A toy car travels with velocity <em>v</em> ms<sup>−1</sup> for six seconds. This is shown in the </span><span style="font-family: times new roman,times; font-size: medium;">graph below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/kbells.png" alt></span></p>
</div>
<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 following diagram shows the graph of \(y = f(x)\), for </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\( - 4 \le x \le 5\).</span></span></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_3.png" 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 the car’s velocity at \(t = 3\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(f( - 3)\);</span></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><span style="font-family: times new roman,times; font-size: medium;">Find the car’s acceleration at \(t = 1.5\) .</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 total distance travelled.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \({L_x}\) be a family of lines with equation given by \(r\) \( = \left( {\begin{array}{*{20}{c}} x \\ {\frac{2}{x}} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} {{x^2}} \\ { - 2} \end{array}} \right)\), where \(x > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the equation of \({L_1}\).</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">A line \({L_a}\) crosses the \(y\)-axis at a point \(P\).</p>
<p class="p1">Show that \(P\) has coordinates \(\left( {0,{\text{ }}\frac{4}{a}} \right)\).</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 class="p1">The line \({L_a}\) crosses the \(x\)-axis at \({\text{Q}}(2a,{\text{ }}0)\). Let \(d = {\text{P}}{{\text{Q}}^2}\).</p>
<p class="p1">Show that \(d = 4{a^2} + \frac{{16}}{{{a^2}}}\).</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">There is a minimum value for \(d\). Find the value of \(a\) that gives this minimum value.</p>
<div class="marks">[7]</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 a semicircle centre O, diameter [AB], with radius 2.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let P be a point on the circumference, with \({\rm{P}}\widehat {\rm{O}}{\rm{B}} = \theta \) radians.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/three_mins.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>S</em> be the total area of the two segments shaded in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flo.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;">Find the area of the triangle OPB, in terms of \(\theta \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Explain why the area of triangle OPA is the same as the area triangle OPB.</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;">Show that \(S = 2(\pi - 2\sin \theta )\) .</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;">Find the value of \(\theta \) when <em>S</em> is a local minimum, justifying that it is a minimum.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find a value of \(\theta \) for which <em>S</em> has its greatest value.</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;">Consider a function \(f(x)\) such that \(\int_1^6 {f(x){\text{d}}x = 8} \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_1^6 {2f(x){\text{d}}x} \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_1^6 {\left( {f(x) + 2} \right){\text{d}}x} \).</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;">A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bike.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point P(<em>x</em> , <em>y</em>) is a vertex of the rectangle and also lies on the circle. The angle </span><span style="font-family: times new roman,times; font-size: medium;">between (OP) and the <em>x</em>-axis is \(\theta \) radians, where \(0 \le \theta \le \frac{\pi }{2}\) .</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;">Write down an expression in terms of \(\theta \) for</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(x\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(y\) .</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;">Let the area of the rectangle be <em>A</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(A = 18\sin 2\theta \) .</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) Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) </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;">(ii) Hence, find the exact value of \(\theta \) which maximizes the area of </span><span style="font-family: times new roman,times; font-size: medium;">the rectangle.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Use the second derivative to justify that this value of \(\theta \) does give </span><span style="font-family: times new roman,times; font-size: medium;">a maximum.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{{{(\ln x)}^2}}}{2}\), for \(x > 0\).</span></p>
</div>
<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 \(g(x) = \frac{1}{x}\). The following diagram shows parts of the graphs of \(f'\) and <em>g</em>.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; min-height: 25px; text-align: center; margin: 0px;"><img src="images/maths_10b.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 of \(f'\) has an <em>x</em>-intercept at \(x = p\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f'(x) = \frac{{\ln x}}{x}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">There is a minimum on the graph of \(f\). Find the \(x\)-coordinate of this minimum.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(p\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(g\) intersects the graph of \(f'\) when \(x = q\).</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">[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 graph of \(g\) intersects the graph of \(f'\) when \(x = q\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(R\) be the region enclosed by the graph of \(f'\), the graph of \(g\) and the line \(x = p\).</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;">Show that the area of \(R\) is \(\frac{1}{2}\).</span></p>
<div class="marks">[5]</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;">The following is the graph of a function \(f\) , for \(0 \le x \le 6\) .</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/charlie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The first part of the graph is a quarter circle of radius \(2\) with centre at the origin.</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;">(a) Find \(\int_0^2 {f(x){\rm{d}}x} \) .</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;">The shaded region is enclosed by the graph of \(f\) , the \(x\)-axis, the \(y\)-axis and the </span><span style="font-family: times new roman,times; font-size: medium;">line \(x = 6\) . The area of this region is \(3\pi \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_2^6 {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">.</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 \(\int_0^2 {f(x){\rm{d}}x} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the graph of \(f\) , the \(x\)-axis, the \(y\)-axis and the </span><span style="font-family: times new roman,times; font-size: medium;">line \(x = 6\) . The area of this region is \(3\pi \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_2^6 {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</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;">Consider \(f(x) = \ln ({x^4} + 1)\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The second derivative is given by \(f''(x) = \frac{{4{x^2}(3 - {x^4})}}{{{{({x^4} + 1)}^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The equation \(f''(x) = 0\) has only three solutions, when \(x = 0\) , \( \pm \sqrt[4]{3}\) \(( \pm 1.316 \ldots )\) .</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 \(f(0)\) .</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 set of values of \(x\) for which \(f\) is increasing.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f''(1)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong>, show that there is no point of inflexion on the graph of \(f\) at \(x = 0\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is a point of inflexion on the graph of \(f\) at \(x = \sqrt[4]{3}\) \((x = 1.316 \ldots )\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(f\) , for \(x \ge 0\) .</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;">The acceleration, \(a{\text{ m}}{{\text{s}}^{ - 2}}\), of a particle at time <em>t</em> seconds is given by \(a = 2t + \cos t\) .</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 acceleration of the particle at \(t = 0\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the velocity, <em>v</em>, at time<em> t</em>, given that the initial velocity of the particle </span><span style="font-family: times new roman,times; font-size: medium;">is \({\text{m}}{{\text{s}}^{ - 1}}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_0^3 {v{\rm{d}}t} \)</span><span style="font-family: times new roman,times; font-size: medium;"> , giving your answer in the form \(p - q\cos 3\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">What information does the answer to part (c) give about the motion of </span><span style="font-family: times new roman,times; font-size: medium;">the particle?</span></p>
<div class="marks">[2]</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 \(f:x \mapsto {\sin ^3}x\) .</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;">(i) Write down the range of the function <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Consider \(f(x) = 1\) , \(0 \le x \le 2\pi \) . Write down the number of solutions to </span><span style="font-family: times new roman,times; font-size: medium;">this equation. Justify your answer.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where \(a{\text{, }}p{\text{, }}q \in \mathbb{Z}\) .</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;">Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">. Find the volume generated when the </span><span style="font-family: times new roman,times; font-size: medium;">curve of <em>g</em> is revolved through \(2\pi \) about the <em>x</em>-axis.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A function <em>f</em> is defined for \( - 4 \le x \le 3\) . 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/poo.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph has a local maximum when \(x = 0\) , and local minima when \(x = - 3\) , \(x = 2\) .</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;">Write down the <em>x</em>-intercepts of the graph of the <strong>derivative</strong> function, \(f'\) .</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><span style="font-family: times new roman,times; font-size: medium;">Write down all values of <em>x</em> for which \(f'(x)\) is positive.</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;">At point D on the graph of <em>f</em> , the <em>x</em>-coordinate is \( - 0.5\). Explain why \(f''(x) < 0\) </span><span style="font-family: times new roman,times; font-size: medium;">at D.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of a function <em>h </em>passes through the point \(\left( {\frac{\pi }{{12}}, 5} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(h'(x) = 4\cos 2x\), find \(h(x)\).</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^{ - 3x}}\) and \(g(x) = \sin \left( {x - \frac{\pi }{3}} \right)\) .</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;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(g'(x)\) .</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;">Let \(h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)\) . Find the exact value of \(h'\left( {\frac{\pi }{3}} \right)\) .</span></p>
<div class="marks">[4]</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;">The velocity <em>v</em> ms<sup>−1</sup> of a particle at time <em>t</em> seconds, is given by \(v = 2t + \cos 2t\) , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le t \le 2\) .</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;">Write down the velocity of the particle when \(t = 0\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t = k\) , the acceleration is zero.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(k = \frac{\pi }{4}\)</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;">(ii) Find the exact velocity when \(t = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[8]</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;">When \(t < \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> and when \(t > \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\) </span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch a graph of <em>v</em> against <em>t</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>d</em> be the distance travelled by the particle for \(0 \le t \le 1\) .</span></p>
<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) Represent <em>d</em> on your sketch.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In this question, you are given that \(\cos \frac{\pi }{3} = \frac{1}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , and \(\sin \frac{\pi }{3} = \frac{{\sqrt 3 }}{2}\)</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;">The displacement of an object from a fixed point, O is given by \(s(t) = t - \sin 2t\) for</span><span style="font-family: times new roman,times; font-size: medium;"> \(0 \le t \le \pi \) .</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 \(s'(t)\) .</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;">In this interval, there are only two values of <em>t</em> for which the object is not moving. </span><span style="font-family: times new roman,times; font-size: medium;">One value is \(t = \frac{\pi }{6}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the other value.</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;">Show that \(s'(t) > 0\) between these two values of <em>t</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the distance travelled between these two values of <em>t</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The following diagram shows the graph of \(f(x) = \frac{x}{{{x^2} + 1}}\), for \(0 \le x \le 4\), and the line \(x = 4\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_17.54.57.png" alt></p>
<p class="p1">Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis and the line \(x = 4\).</p>
<p class="p1">Find the area of \(R\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \sqrt x \) . Line <em>L</em> is the normal to the graph of <em>f</em> at the point (4, 2) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the diagram below, the shaded region <em>R</em> is bounded by the <em>x</em>-axis, the graph of <em>f</em> and </span><span style="font-family: times new roman,times; font-size: medium;">the line <em>L</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ring.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 the equation of <em>L</em> is \(y = - 4x + 18\) .</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;">Point A is the <em>x</em>-intercept of <em>L</em> . Find the <em>x</em>-coordinate of A.</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;"><span style="font-size: medium;">Find an expression for the area of <em>R</em></span> .</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;">The region <em>R</em> is rotated \(360^\circ \) about the <em>x</em>-axis. Find the volume of the solid formed, </span><span style="font-family: times new roman,times; font-size: medium;">giving your answer in terms of \(\pi \) .</span></p>
<div class="marks">[8]</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;"><span>Let </span><span>\(f(x) = {x^3}\)</span><span>. The following diagram shows part of the graph of </span><span><em>f</em> </span><span>.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/gone.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point \({\rm{P}}(a,f(a))\) , where \(a > 0\) , lies on the graph of <em>f</em> . The tangent at P crosses the <em>x</em>-axis at the point \({\rm{Q}}\left( {\frac{2}{3},0} \right)\) . This tangent intersects the graph of <em>f</em> at the point R(−2, −8) .</span></p>
<p> </p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The equation of the tangent at P is \(y = 3x - 2\) . Let <em>T</em> be the region enclosed by </span><span style="font-family: times new roman,times; font-size: medium;">the graph of <em>f</em> , the tangent [PR] and the line \(x = k\) , between \(x = - 2\) and \(x = k\) </span><span style="font-family: times new roman,times; font-size: medium;">where \( - 2 < k < 1\) . This is shown in the diagram below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chad.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;">(i) Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(f'(a)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (iii) Hence show that \(a = 1\) .</span></p>
<div class="marks">[7]</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;">Given that the area of <em>T</em> is \(2k + 4\) , show that <em>k</em> satisfies the equation </span><span style="font-family: times new roman,times; font-size: medium;">\({k^4} - 6{k^2} + 8 = 0\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A quadratic function \(f\) can be written in the form \(f(x) = a(x - p)(x - 3)\). The graph of \(f\) has axis of symmetry \(x = 2.5\) and \(y\)-intercept at \((0,{\text{ }} - 6)\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\).</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 line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).</p>
<div class="marks">[8]</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;">Part of the graph of \(f(x) = a{x^3} - 6{x^2}\) is shown below.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/1.jpg" alt></span></p>
<p> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">The point P lies on the graph of </span><span style="font-family: 'times new roman', times; font-size: medium;">\(f\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> . At P, <em>x</em> = 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 \(f'(x)\) .</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;">The graph of \(f\) has a gradient of \(3\) at the point P. Find the value of \(a\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f(x) = \cos x\), for \(0\) \(\le \) \(x\) \( \le \) \(2\pi \). The following diagram shows the graph of \(f\).</p>
<p class="p1">There are <em>\(x\)</em>-intercepts at \(x = \frac{\pi }{2},{\text{ }}\frac{{3\pi }}{2}\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_07.44.53.png" alt></p>
<p class="p1">The shaded region \(R\) is enclosed by the graph of \(f\), the line \(x = b\), where \(b > \frac{{3\pi }}{2}\), and the \(x\)-axis. The area of \(R\) <span class="s1">is \(\left( {1 - \frac{{\sqrt 3 }}{2}} \right)\)</span>. Find the value of \(b\).</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = p{x^3} + p{x^2} + qx\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f'(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f'(x) \geqslant 0\), show that \({p^2} \leqslant 3pq\).</span></p>
<div class="marks">[5]</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 \(f(x) = 3 + \frac{{20}}{{{x^2} - 4}}\) , for \(x \ne \pm 2\) . The graph 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/fajita.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>y</em>-intercept is at the point A.</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) Find the coordinates of A.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(f'(x) = 0\) at A.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The second derivative \(f''(x) = \frac{{40(3{x^2} + 4)}}{{{{({x^2} - 4)}^3}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> . Use this to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) justify that the graph of <em>f</em> has a local maximum at A;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) explain why the graph of <em>f</em> does <strong>not</strong> have a point of inflexion.</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;">Describe the behaviour of the graph of \(f\) for large \(|x|\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \(f\) .</span></p>
<div class="marks">[2]</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 \(f(x) = \frac{x}{{ - 2{x^2} + 5x - 2}}\) for \( - 2 \le x \le 4\) , \(x \ne \frac{1}{2}\) , \(x \ne 2\) . The graph of \(f\) is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q10.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f\) has a local minimum at A(\(1\), \(1\)) and a local maximum at 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;">Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence find the coordinates of B.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that the line \(y = k\) does not meet the graph of <em>f</em> , find the possible values </span><span style="font-family: times new roman,times; font-size: medium;">of <em>k</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f'(x) = 6{x^2} - 5\). Given that \(f(2) = - 3\), find \(f(x)\).</p>
</div>
<br><hr><br><div class="question">
<p>Consider \(f(x) = \log k(6x - 3{x^2})\), for \(0 < x < 2\), where \(k > 0\).</p>
<p>The equation \(f(x) = 2\) has exactly one solution. Find the value of \(k\).</p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f'(x) = 3{x^2} + 2\) . Given that \(f(2) = 5\) , find \(f(x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \int {\frac{{12}}{{2x - 5}}} {\rm{d}}x\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(x > \frac{5}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> . The graph of \(f\) passes through (\(4\), \(0\)) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f(x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f'(x) = {\sin ^3}(2x)\cos (2x)\). Find \(f(x)\), given that \(f\left( {\frac{\pi }{4}} \right) = 1\).</p>
</div>
<br><hr><br><div class="specification">
<p>A closed cylindrical can with radius r centimetres and height h centimetres has a volume of 20\(\pi \) cm<sup>3</sup>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The material for the base and top of the can costs 10 cents per cm<sup>2</sup> and the material for the curved side costs 8 cents per cm<sup>2</sup>. The total cost of the material, in cents, is <em>C</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>h</em> in terms of <em>r.</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>Show that \(C = 20\pi {r^2} + \frac{{320\pi }}{r}\).</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>Given that there is a minimum value for <em>C</em>, find this minimum value in terms of \(\pi \).</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\int {\frac{{{{\rm{e}}^x}}}{{1 + {{\rm{e}}^x}}}} {\rm{d}}x\) .</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 \(\int {\sin 3x\cos 3x{\rm{d}}x} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_4^{10} {(x - 4){\rm{d}}x} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Part of the graph of \(f(x) = \sqrt {{x^{}} - 4} \) , for \(x \ge 4\) , is shown below. The shaded region </span><em style="font-family: 'times new roman', times; font-size: medium;">R</em><span style="font-family: 'times new roman', times; font-size: medium;"> is enclosed by the graph of \(f\) , the line \(x = 10\) , and the </span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;">-axis.</span></p>
<p><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">The region <em>R</em> is rotated \({360^ \circ }\) about the <em>x</em>-axis. Find the volume of the solid </span><span style="font-family: 'times new roman', times; font-size: medium;">formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\)at time <em>t</em> seconds is given by \(v = 6{{\rm{e}}^{3t}} + 4\) . When \(t = 0\) , the displacement, <em>s</em>, of the particle is 7 metres. Find an expression for <em>s</em> in terms of <em>t</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the function <em>f</em> with second derivative \(f''(x) = 3x - 1\) . The graph of <em>f</em> has a </span><span style="font-family: times new roman,times; font-size: medium;">minimum point at A(2, 4) and a maximum point at \({\rm{B}}\left( { - \frac{4}{3},\frac{{358}}{{27}}} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the second derivative to justify that B is a maximum.</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;">Given that \(f'(x) = \frac{3}{2}{x^2} - x + p\)</span><span style="font-family: times new roman,times; font-size: medium;"> , show that \(p = - 4\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \(f(x)\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f'(x) = \frac{{6 - 2x}}{{6x - {x^2}}}\), for \(0 < x < 6\).</p>
<p class="p1"><span class="s1">The graph of \(f\) </span>has a maximum point at P<span class="s1">.</span></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The \(y\)</span>-coordinate of P is \(\ln 27\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-coordinate of <span class="s1">P</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">Find \(f(x)\), expressing your answer as a single logarithm.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">The graph of \(f\) </span>is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\). The image of P under this transformation has coordinates \((a,{\text{ }}b)\).</p>
<p class="p1">Find the value of \(a\) and of \(b\), where \(a,{\text{ }}b \in \mathbb{N}\).</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = {x^2}\).</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 \(\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows part of the graph of \(f\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/maths_3b.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The shaded region \(R\)<em> </em>is enclosed by the graph of \(f\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">Find the volume of the solid formed when </span><span style="font-family: 'times new roman', times; font-size: medium;">\(R\)</span><em> </em><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">is revolved \({360^ \circ }\) about the </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="background-color: #ffffff;">\(x\)</span>-axis.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_0^5 {\frac{2}{{2x + 5}}} {\rm{d}}x = \ln k\)</span><span style="font-family: times new roman,times; font-size: medium;"> , find the value of <em>k</em> .</span></p>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \frac{1}{{\sqrt {2x - 1} }}\), for \(x > \frac{1}{2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} \).</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>Part of the graph of <em>f</em> is shown in the following diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The shaded region <em>R</em> is enclosed by the graph of <em>f</em>, the <em>x</em>-axis, and the lines <em>x</em> = 1 and <em>x</em> = 9 . Find the volume of the solid formed when <em>R</em> is revolved 360° about the <em>x</em>-axis.</p>
<div class="marks">[4]</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 \(f(x) = \frac{1}{2}{x^3} - {x^2} - 3x\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sheldon.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum point at A and a minimum point at B(3, − 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;">Find the coordinates of A.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the coordinates of</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) the image of B after reflection in the <em>y</em>-axis;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the image of B after translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>{ - 2}\\<br>5<br>\end{array}} \right)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the image of B after reflection in the <em>x</em>-axis followed by </span><span style="font-family: times new roman,times; font-size: medium;">a horizontal stretch with scale factor \(\frac{1}{2}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = \sqrt x \) between \(x = 0\) and \(x = a\) is rotated \(360^\circ \) about the <em>x</em>-axis. </span><span style="font-family: times new roman,times; font-size: medium;">The volume of the solid formed is \(32\pi \) . Find the value of <em>a</em>.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = 6{x^2} - 3x\). The graph of \(f\) is shown in the following diagram.</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 \(\int {\left( {6{x^2} - 3x} \right){\text{d}}x} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of \(f\), the <em>x</em>-axis and the lines <em>x</em> = 1 and <em>x</em> = 2 .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows the probability distribution of a discrete random variable \(A\), in terms of an angle \(\theta \).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\cos \theta = \frac{3}{4}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\tan \theta > 0\), find \(\tan \theta \).</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>Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between \(x = \theta \) and \(x = \frac{\pi }{4}\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</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 part of the graph of the function \(f(x) = 2{x^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/curve.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The line <em>T</em> is the tangent to the graph of <em>f</em> at \(x = 1\) .</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;">Show that the equation of <em>T</em> is \(y = 4x - 2\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-intercept of <em>T</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region <em>R</em> is enclosed by the graph of <em>f</em> , the line <em>T</em> , and the <em>x</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for the area of <em>R</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the area of <em>R</em> .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">c(i) and (ii).</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \frac{{\ln x}}{{{x^2}}}\) , for \(x > 0\) .</span></p>
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<p><span style="font-family: times new roman,times; font-size: medium;">Use the quotient rule to show that \(g'(x) = \frac{{1 - 2\ln x}}{{{x^3}}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> has a maximum point at A. Find the <em>x</em>-coordinate of A.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p>Consider <em>f</em>(<em>x</em>), <em>g</em>(<em>x</em>) and <em>h</em>(<em>x</em>), for x∈\(\mathbb{R}\) where <em>h</em>(<em>x</em>) = \(\left( {f \circ g} \right)\)(<em>x</em>).</p>
<p>Given that <em>g</em>(3) = 7 , <em>g′</em> (3) = 4 and <em>f ′ </em>(7) = −5 , find the gradient of the normal to the curve of <em>h</em> at <em>x</em> = 3.</p>
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<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f(x) = \sqrt {16 - 4{x^2}} \) , for \( - 2 \le x \le 2\) , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/crying.png" alt></span></p>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region enclosed by the curve of <em>f</em> and the <em>x</em>-axis is rotated \(360^\circ \) about the <em>x</em>-axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the volume of the solid formed.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = {{\text{e}}^{2x}}\). The line \(L\) is the tangent to the curve of \(f\) at \((1,{\text{ }}{{\text{e}}^2})\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of \(L\) in the form \(y = ax + b\).</span></p>
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<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}\cos x\) . Find the gradient of the normal to the curve of <em>f</em> at \(x = \pi \) .</span></p>
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<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{{6x}}{{\cos x}}\) . Find \(h'(0)\) .</span></p>
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<p>A function <em>f </em>(<em>x</em>) has derivative <em>f ′</em>(<em>x</em>) = 3<em>x</em><sup>2</sup> + 18<em>x</em>. The graph of <em>f</em> has an <em>x</em>-intercept at <em>x</em> = −1.</p>
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<p>Find <em>f </em>(<em>x</em>).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
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<p>The graph of <em>f</em> has a point of inflexion at <em>x</em> = <em>p</em>. Find <em>p</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p>Find the values of <em>x</em> for which the graph of <em>f</em> is concave-down.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \sin x + \frac{1}{2}{x^2} - 2x\) , for \(0 \le x \le \pi \) .</span></p>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g\) be a quadratic function such that \(g(0) = 5\) . The line \(x = 2\) is the axis of </span><span style="font-family: times new roman,times; font-size: medium;">symmetry of the graph of \(g\) .</span></p>
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<p><span style="font-family: times new roman,times; font-size: medium;">The function \(g\) can be expressed in the form \(g(x) = a{(x - h)^2} + 3\) .</span></p>
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<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>
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<p><span style="font-family: times new roman,times; font-size: medium;">Find \(g(4)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of \(h\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of \(a\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(x\) for which the tangent to the graph of \(f\) is parallel to the </span><span style="font-family: times new roman,times; font-size: medium;">tangent to the graph of \(g\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
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