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</div><h2>SL Paper 2</h2><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 4{\tan ^2}x - 4\sin x\) , \( - \frac{\pi }{3} \le x \le \frac{\pi }{3}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(y = f(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bad_day.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \(f(x) = 1\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/badday2.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for passing through \((0{\text{, }}0)\), <em><strong>A1</strong></em> for correct shape, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for a range of approximately \( - 1\) to 15.</span></p>
<p align="LEFT"><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to solve \(f(x) = 1\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. line on sketch, using \(\tan x = \frac{{\sin x}}{{\cos x}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = - 0.207\) , \(x = 0.772\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (a), some did not realize that they should copy the curve from their GDC, paying </span><span style="font-family: times new roman,times; font-size: medium;">attention to domain and range.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Not using their GDC, and trying to solve the equation </span><span style="font-family: times new roman,times; font-size: medium;">analytically in part (b) proved to be very difficult for many. A common error was to substitute </span><span style="font-family: times new roman,times; font-size: medium;">\(x = 1\) .</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider \(f(x) = 2 - {x^2}\) , for \( - 2 \le x \le 2\) and \(g(x) = \sin {{\rm{e}}^x}\) , for \( - 2 \le x \le 2\) . The graph </span><span style="font-family: times new roman,times; font-size: medium;">of <em>f</em> is given below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/crash.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the diagram above, sketch the graph of <em>g</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \(f(x) = g(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the set of values of <em>x</em> such that \(f(x) > g(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ib.png" alt></span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1A1 N3</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = - 1.32\) , \(x = 1.68\) (accept \(x = - 1.41\) , \(x = 1.39\) if working in degrees) <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( - 1.32 < x < 1.68\) (accept \( - 1.41 < x < 1.39\) if working in degrees) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question was answered well by a pleasing number of candidates.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For part (a), many good graphs were seen, though a significant number of candidates either </span><span style="font-family: times new roman,times; font-size: medium;">used degrees or a function such as \(\sin {{\rm{e}}^x}\) </span><span style="font-family: times new roman,times; font-size: medium;">. There were students who lost marks for poor </span><span style="font-family: times new roman,times; font-size: medium;">diagrams. For example, the shape was correct but the maximum and minimum were not </span><span style="font-family: times new roman,times; font-size: medium;">accurate enough.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There were candidates who struggled in vain to solve the equation in part (b) algebraically </span><span style="font-family: times new roman,times; font-size: medium;">instead of using a GDC. Those that did use their GDCs to solve the equation frequently gave </span><span style="font-family: times new roman,times; font-size: medium;">their answers inaccurately, suggesting that they did not know how to use the "zero" function </span><span style="font-family: times new roman,times; font-size: medium;">on their calculator but found a rough solution using the "trace" function.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (c) they often gave the correct solution, or obtained follow-through marks on their </span><span style="font-family: times new roman,times; font-size: medium;">incorrect results to part (b).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3x\) , \(g(x) = 2x - 5\) and \(h(x) = (f \circ g)(x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(h(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({h^{ - 1}}(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to form composite <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(2x - 5)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h(x) = 6x - 15\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">interchanging <em>x</em> and <em>y</em> <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct manipulation <strong><em> (A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y + 15 = 6x\) , \(\frac{x}{6} = y - \frac{5}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({h^{ - 1}}(x) = \frac{{x + 15}}{6}\) <strong><em>A1 N3</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates handled this question with ease. Some were not familiar with the notation of composite functions assuming that \((f \circ g)(x)\) implied finding the composition and then multiplying this by <em>x</em> . Others misunderstood part (b) and found the reciprocal function or the derivative, indicating they were not familiar with the notation for an inverse function. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates handled this question with ease. Some were not familiar with the notation of composite functions assuming that \((f \circ g)(x)\) implied finding the composition and then multiplying this by <em>x</em> . Others misunderstood part (b) and found the reciprocal function or the derivative, indicating they were not familiar with the notation for an inverse function. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . The graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/troy.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> crosses the <em>x</em>-axis at \(x = a\) , \(x = 0\) and \(x = b\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>a</em> and of <em>b</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> has a maximum value when \(x = c\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>c</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region under the graph of <em>f</em> from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about </span><span style="font-family: times new roman,times; font-size: medium;">the <em>x</em>-axis. Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the curve, the <em>x</em>-axis and the line \(x = c\) , </span><span style="font-family: times new roman,times; font-size: medium;">between \(x = a\) and \(x = c\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = 0\) , graph</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = - 1.73\) , \(b = 1.73\) \((a = - \sqrt 3 {\text{, }}b = \sqrt 3 )\) <em><strong>A1A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find max <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. setting \(f'(x) = 0\) , graph</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(c = 1.15\) (accept (1.15, 1.13)) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute either limits or the function into formula <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \pi {\int_0^c {\left[ {f(x)} \right]} ^2}{\rm{d}}x\) , \(\pi {\int {\left[ {x\ln (4 - {x^2})} \right]} ^2}\) , \(\pi \int_0^{1.149 \ldots } {{y^2}{\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(V = 2.16\) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach recognizing 2 regions <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. finding 2 areas</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^{ - 1.73 \ldots } {f(x){\rm{d}}x + } \int_0^{1.149 \ldots } {f(x){\rm{d}}x} \) , \( - \int_{ - 1.73 \ldots }^0 {f(x){\rm{d}}x + } \int_0^{1.149 \ldots } {f(x){\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area \( = 2.07\) (accept 2.06) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by many candidates. If there were problems, it was often with incorrect or inappropriate GDC use. For example, some candidates used the trace feature to answer parts (a) and (b), which at best, only provides an approximation. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by many candidates. If there were problems, it was often with incorrect or inappropriate GDC use. For example, some candidates used the trace feature to answer parts (a) and (b), which at best, only provides an approximation. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to set up correct expressions for parts (c) and (d) and if they had used their calculators, could find the correct answers. Some candidates omitted the important parts of the volume formula. Analytical approaches to (c) and (d) were always futile and no marks were gained. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to set up correct expressions for parts (c) and (d) and if they had used their calculators, could find the correct answers. Some candidates omitted the important parts of the volume formula. Analytical approaches to (c) and (d) were always futile and no marks were gained. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {{\text{e}}^{x + 1}} + 2\), for \( - 4 \le x \le 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following grid, sketch the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_12.27.30.png" alt></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">The graph of \(f\) </span>is translated by the vector \(\left( {\begin{array}{*{20}{c}} 3 \\ { - 1} \end{array}} \right)\) <span class="s1">to obtain the graph of a function \(g\)</span>.</p>
<p class="p1">Find an expression for \(g(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space"><img src="images/Schermafbeelding_2016-01-14_om_12.38.03.png" alt> </span><strong><em>1A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"> </p>
<p class="p3"><strong>Note: <span class="Apple-converted-space"> </span></strong>Curve must be approximately correct exponential shape (increasing and concave up). Only if the shape is approximately correct, award the following:</p>
<p class="p3"><strong><em>A1 </em></strong>for right end point in circle,</p>
<p class="p3"><strong><em>A1 </em></strong>for <em>\(y\)</em>-intercept in circle,</p>
<p class="p3"><strong><em>A1 </em></strong>for asymptotic to \(y = 2\), (must be above \(y = 2\)).</p>
<p class="p3"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid attempt to find \(g\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f(x - 3) - 1,{\text{ }}g(x) = {{\text{e}}^{x + 1 - 3}} + 2 - 1,{\text{ }}{{\text{e}}^{x + 1 - 3}},{\text{ }}2 - 1\), sketch</p>
<p class="p1">\(g(x) = {{\text{e}}^{x - 2}} + 1\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Although this question involved a straightforward use of the GDC, the graphing of this exponential function on a given grid seemed challenging for a number of candidates. Although most candidates were able to graph the correct shape, they did not take into account the domain and range of this function.</p>
<p class="p1">Many were inattentive to the asymptotic nature of the function. Very few actually drew the asymptote, which in this case was a relevant feature.</p>
<p class="p1">When finding an expression for \(g\), many reversed the direction of one or both of the transformations. The vertical translation was usually correct, but the horizontal shift was poorly done. The most common error was to obtain \(g(x) = {{\text{e}}^{x + 4}} + 1\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Although this question involved a straightforward use of the GDC, the graphing of this exponential function on a given grid seemed challenging for a number of candidates. Although most candidates were able to graph the correct shape, they did not take into account the domain and range of this function.</p>
<p class="p1">Many were inattentive to the asymptotic nature of the function. Very few actually drew the asymptote, which in this case was a relevant feature.</p>
<p class="p1">When finding an expression for \(g\), many reversed the direction of one or both of the transformations. The vertical translation was usually correct, but the horizontal shift was poorly done. The most common error was to obtain \(g(x) = {{\text{e}}^{x + 4}} + 1\).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = a\cos (b(x - c))\) . The diagram below shows part of the graph of <em>f</em> , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le x \le 10\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q5.jpg" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph has a local maximum at P(3, 5) , a local minimum at Q(7, − 5) , and crosses the <em>x</em>-axis at R.</span></p>
<p align="LEFT"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(c\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>b</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of R.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(a = 5\) (accept \( - 5\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(c = 3\) (accept \(c = 7\) , if \(a = - 5\) ) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept other correct values of <em>c</em>, such as 11, \( - 5\), etc.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find period <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 8 , \(b = \frac{{2\pi }}{{{\rm{period}}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.785398 \ldots \)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{{2\pi }}{8}\) (exact), \(\frac{\pi }{4}\) , 0.785 [\(0.785{\text{, }}0.786\)] (do not accept 45) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = 0\) , symmetry of curve </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 5\) (accept \((5{\text{ ,}}0))\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) (i) was well answered in general. There were more difficulties in finding the correct value of the parameter <em>c</em>. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Finding the correct value of <em>b</em> in part (b) also proved difficult as many did not realize the period was equal to 8. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates could handle part (c) without difficulties using their GDC or working with the symmetry of the curve although follow through from errors in part (b) was often not awarded because candidates failed to show any working by writing down the equations they entered into their GDC. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in two colonies, \(\rm{A}\) and \(\rm{B}\), starts increasing at the same time.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in colony \(\rm{A}\) after \(t\) hours is modelled by the function \(\rm{A}(t) = 12{{\text{e}}^{0.4t}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of bacteria in colony \({\text{A}}\) after four hours.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of bacteria in colony \({\text{A}}\) after four hours.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function \(B(t) = 24{{\text{e}}^{kt}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">After four hours, there are \(60\) bacteria in colony \({\text{B}}\). Find the value of \(k\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function \(B(t) = 24{{\text{e}}^{kt}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of bacteria in colony \({\text{A}}\) first exceeds the number of bacteria in colony \({\text{B}}\) after \(n\) hours, where \(n \in \mathbb{Z}\). Find the value of \(n\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 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;">correct substitution into formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(12{{\text{e}}^{0.4(0)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(12\) bacteria in the dish <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 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;">correct substitution into formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(12{{\text{e}}^{0.4(4)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(59.4363\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(59\) bacteria in the dish (integer answer only) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 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;">correct equation <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(A(t) = 400,{\text{ }}12{{\text{e}}^{0.4t}} = 400\)</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;">valid attempt to solve <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> graph, use of logs</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;">\(8.76639\)</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;">\(8.77\) (hours) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 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;">valid attempt to solve <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(n(4) = 60,{\text{ }}60 = 24{{\text{e}}^{4k}}\), use of logs</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;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> sketch of intersection, \(4k = \ln 2.5\)</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;">\(k = 0.229072\)</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;">\(k = \frac{{\ln 2.5}}{4}\) (exact), \(k = 0.229\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;"><strong>METHOD 1</strong></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;">setting up an equation or inequality (accept any variable for \(n\)) <strong><em>(M1)</em></strong></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;"><em>eg</em> \(A(t) > B(t),{\text{ }}12{{\text{e}}^{0.4n}} = 24{{\text{e}}^{0.229n}},{\text{ }}{{\text{e}}^{0.4n}} = 2{{\text{e}}^{0.229n}}\)</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;">correct working <strong><em>(A1)</em></strong></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;"><em>eg</em> sketch of intersection, \({{\text{e}}^{0.171n}} = 2\)</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;">\(4.05521\) (accept \(4.05349\)) <strong><em>(A1)</em></strong></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;">\(n = 5\) (integer answer only) <strong><em>A1 N3</em></strong></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;"><strong>METHOD 2</strong></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;">\(A(4) = 59,{\text{ }}B(4) = 60\) (from earlier work)</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;">\(A(5) = 88.668,{\text{ }}B(5) = 75.446\) <strong><em>A1A1</em></strong></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;">valid reasoning <strong><em>(R1)</em></strong></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;"><em>eg</em> \(A(4) < B(4)\) <strong>and</strong> \(A(5) > B(5)\)</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;">\(n = 5\) (integer answer only) <strong><em>A1 N3</em></strong></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;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2} - 1\) and \(g(x) = {x^2} - 2\), for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the following grid, sketch the graph of \((f \circ g)(x)\), for \(0 \leqslant x \leqslant 2.25\).</p>
<p style="text-align: left;"><img src="images/Schermafbeelding_2017-08-15_om_08.00.33.png" alt="M17/5/MATME/SP2/ENG/TZ2/06.b"></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The equation \((f \circ g)(x) = k\) has exactly two solutions, for \(0 \leqslant x \leqslant 2.25\). Find the possible values of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to form composite in either order <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f({x^2} - 2),{\text{ }}{({x^2} - 1)^2} - 2\)</p>
<p>\(({x^4} - 4{x^2} + 4) - 1\) <strong><em>A1</em></strong></p>
<p>\((f \circ g)(x) = {x^4} - 4{x^2} + 3\) <strong><em>AG</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-15_om_08.05.50.png" alt="M17/5/MATME/SP2/ENG/TZ2/06.b/M"> <strong><em>A1</em></strong></p>
<p><strong><em>A1A1</em></strong> <strong><em>N3</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for approximately correct shape which changes from concave down to concave up. Only if this <strong><em>A1 </em></strong>is awarded, award the following:</p>
<p><strong><em>A1 </em></strong>for left hand endpoint in circle <strong>and </strong>right hand endpoint in oval,</p>
<p><strong><em>A1 </em></strong>for minimum in oval.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of identifying max/min as relevant points <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(x = 0,{\text{ }}1.41421,{\text{ }}y = - 1,{\text{ }}3\)</p>
<p>correct interval (inclusion/exclusion of endpoints must be correct) <strong><em>A2</em></strong> <strong><em>N3</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 1 < k \leqslant 3,{\text{ }}\left] { - 1,{\text{ 3}}} \right],{\text{ }}( - 1,{\text{ }}3]\)</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graph of \(f(x) = {{\rm{e}}^{ - {x^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/berlin.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points A, B, C, D and E lie on the graph of <em>f</em> . Two of these are points of inflexion.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Identify the <strong>two</strong> points of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of each point of inflexion.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the second derivative to show that one of these points is a point of inflexion.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">B, D <strong><em>A1A1 N2</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x) = - 2x{{\rm{e}}^{ - {x^2}}}\) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \({{\rm{e}}^{ - {x^2}}}\) and <strong><em>A1</em></strong> for \( - 2x\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) finding the derivative of \( - 2x\) , i.e. \( - 2\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the product rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 2{{\rm{e}}^{ - {x^2}}}\) \( - 2x \times - 2x{{\rm{e}}^{ - {x^2}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( - 2{{\rm{e}}^{ - {x^2}}} + 4{x^2}{{\rm{e}}^{ - {x^2}}}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to solve the equation <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \((4{x^2} - 2) = 0\) , sketch of \(f''(x)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.707\) \(\left( { = \frac{1}{{\sqrt 2 }}} \right)\) , \(q = - 0.707\) \(\left( { = - \frac{1}{{\sqrt 2 }}} \right)\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1A1 N3</span></em></strong></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using second derivative to test values on either side of POI <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. finding values, reference to graph of \(f''\) , sign table</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. finding any two correct values either side of POI,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">checking sign of \(f''\) on either side of POI</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">reference to sign change of \(f''(x)\) <em><strong>R1 N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to recognize the points of inflexion in part (a).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to recognize the points of inflexion in part (a) and had little difficulty with the first and second derivatives in part (b). A few did not recognize the application of the product rule in part (b). </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Obtaining the <em>x</em>-coordinates of the inflexion points in (c) usually did not cause many problems. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Only the better-prepared candidates understood how to set up a second derivative test in part (d). Many of those did not show, or clearly indicate, the values of <em>x</em> used to test for a point of inflexion, but merely gave an indication of the sign. Some candidates simply resorted to showing that \(f''\left( { \pm \frac{1}{{\sqrt 2 }}} \right) = 0\) , completely missing the point of the question. The necessary condition for a point of inflexion, i.e. \(f''(x) = 0\) <strong>and</strong> the change of sign for \(f''(x)\) , seemed not to be known by the vast majority of candidates. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({u_4}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the sum of the infinite sequence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider an arithmetic sequence with <em>n</em> terms, with first term (\( - 36\)) and eighth </span><span style="font-family: times new roman,times; font-size: medium;">term (\( - 8\)) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the common difference.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \({S_n} = 2{n^2} - 38n\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The sum of the infinite geometric sequence is equal to twice the sum of the </span><span style="font-family: times new roman,times; font-size: medium;">arithmetic sequence. Find <em>n</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) correct approach <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_4} = (40){\frac{1}{2}^{(4 - 1)}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, listing terms</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_4} = 5\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into formula for infinite sum <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_\infty } = \frac{{40}}{{1 - 0.5}}\) , \({S_\infty } = \frac{{40}}{{0.5}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_\infty } = 80\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) attempt to set up expression for \({u_8}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 36 + (8 - 1)d\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 8 = - 36 + (8 - 1)d\) , \(\frac{{ - 8 - ( - 36)}}{7}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(d = 4\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into formula for sum <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_n} = \frac{n}{2}(2( - 36) + (n - 1)4)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_n} = \frac{n}{2}(4n - 76)\) , \( - 36n + 2{n^2} - 2n\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_n} = 2{n^2} - 38n\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">multiplying \({S_n}\) (AP) by 2 or dividing <em>S</em> (infinite GP) by 2 <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2{S_n}\) , \(\frac{{{S_\infty }}}{2}\) , 40</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into \(2{S_n} = {S_\infty }\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2{n^2} - 38n = 40\) , \(4{n^2} - 76n - 80\) (\( = 0\))</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to solve <strong>their</strong> quadratic (equation) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. intersection of graphs, formula</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 20\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates found part (a) straightforward, although a common error in (a)(ii) was to calculate 40 divided by \(\frac{1}{2}\) as 20. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), some candidates had difficulty with the "show that" and worked backwards from the answer given. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates obtained the correct equation in part (c), although some did not reject the negative value of <em>n</em> as impossible in this context. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(q\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the probability of drawing three blue marbles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the probability of drawing three white marbles is \(\frac{1}{6}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The bag contains a total of ten marbles of which \(w\) are white. Find \(w\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Grant plays the game until he wins two prizes. Find the probability that he wins his second prize on his eighth attempt.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into \({\text{E}}(X)\) formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0(p) + 1(0.5) + 2(0.3) + 3(q) = 1.2\)</p>
<p>\(q = \frac{1}{{30}}\), 0.0333 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of summing probabilities to 1 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(p + 0.5 + 0.3 + q = 1\)</p>
<p>\(p = \frac{1}{6},{\text{ }}0.167\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P (3 blue)}} = \frac{1}{{30}},{\text{ }}0.0333\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid reasoning <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P (3 white)}} = {\text{P(0 blue)}}\)</p>
<p>\({\text{P(3 white)}} = \frac{1}{6}\) <strong><em>AG</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid method <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{P(3 white)}} = \frac{w}{{10}} \times \frac{{w - 1}}{9} \times \frac{{w - 2}}{8},{\text{ }}\frac{{_w{C_3}}}{{_{10}{C_3}}}\)</p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{w}{{10}} \times \frac{{w - 1}}{9} \times \frac{{w - 2}}{8} = \frac{1}{6},{\text{ }}\frac{{_w{C_3}}}{{_{10}{C_3}}} = 0.167\)</p>
<p>\(w = 6\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing one prize in first seven attempts <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right),{\text{ }}{\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6}\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right){\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6},{\text{ }}0.390714\)</p>
<p>correct approach <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 1 \end{array}} \right){\left( {\frac{1}{6}} \right)^1}{\left( {\frac{5}{6}} \right)^6} \times \frac{1}{6}\)</p>
<p>0.065119</p>
<p>0.0651 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \,\,{\text{sin}}\,\left( {{e^x}} \right)\) for 0 ≤ \(x\) ≤ 1.5. The following diagram shows the graph of \(f\).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <em>x</em>-intercept of the graph of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region enclosed by the graph of \(f\), the<em> y</em>-axis and the <em>x</em>-axis is rotated 360° about the <em>x</em>-axis.</p>
<p>Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg </em> \(f\left( x \right) = 0,\,\,\,\,{e^x} = 180\) or 0…</p>
<p>1.14472</p>
<p>\(x = {\text{ln}}\,\pi \) (exact), 1.14 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either their <strong>limits</strong> or the function into formula involving \({f^2}\). <em><strong>(M1)</strong></em></p>
<p><em>eg</em> \({\int_0^{1.14} {{f^2},\,\,\pi \int {\left( {{\text{sin}}\,\left( {{e^x}} \right)} \right)} } ^2}dx,\,\,0.795135\)</p>
<p>2.49799</p>
<p>volume = 2.50 <em><strong> A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of \(f(x) = - 2{x^3} + 5.1{x^2} + 3.6x - 0.4\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_05.42.47.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of the local minimum point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(f\) is translated to the graph of \(g\) by the vector \(\left( {\begin{array}{*{20}{c}} 0 \\ k \end{array}} \right)\). Find all values of \(k\) so that \(g(x) = 0\) has exactly one solution.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(( - 0.3,{\text{ }} - 0.967)\)</p>
<p>\(x = - 0.3\) (exact), \(y = - 0.967\) (exact) <strong><em>A1A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(y\)-coordinate of local maximum is \(y = 11.2\) <strong><em>(A1)</em></strong></p>
<p>negating the \(y\)-coordinate of one of the max/min <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;y = 0.967,{\text{ }}y = - 11.2\)</p>
<p>recognizing that the solution set has two intervals <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\;\;\;\)two answers,</p>
<p>\(k < - 11.2,{\text{ }}k > 0.967\) <strong><em>A1A1 N3N2</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>If working shown, do not award the final mark if strict inequalities are not used.</p>
<p>If no working shown, award <strong><em>N2 </em></strong>for \(k \le - 11.2\) or <strong><em>N1 </em></strong>for \(k \ge 0.967\)</p>
<p><em><strong>Total [7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The coordinates of the minimum point was correctly given by most candidates, although some opted for an analytical approach which was often futile and time consuming.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), few students appreciated that the solution set consisted of two <strong>intervals </strong>often giving only one correct interval or equalities. The most common, incorrect approach was an attempt to use the discriminant.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">An environmental group records the numbers of coyotes and foxes in a wildlife reserve after \(t\) <span class="s1">years, starting on 1 January 1995</span>.</p>
<p class="p1">Let \(c\) be the number of coyotes in the reserve after \(t\) years. The following table shows the number of coyotes after \(t\) years.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_08.53.25.png" alt></p>
<p class="p1">The relationship between the variables can be modelled by the regression equation \(c = at + b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\) and of \(b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the regression equation to estimate the number of coyotes in the reserve when \(t = 7\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(f\) be the number of foxes in the reserve after \(t\) years. The number of foxes can be modelled by the equation \(f = \frac{{2000}}{{1 + 99{{\text{e}}^{ - kt}}}}\), where \(k\) <span class="s1">is a constant.</span></p>
<p class="p2">Find the number of foxes in the reserve on 1 January 1995.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">After five years, there were 64 </span>foxes in the reserve. Find \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">During which year were the number of coyotes the same as the number of foxes?</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of setup <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)correct value for \(a\) or \(b\)</p>
<p class="p1">\(13.3823\), \(137.482\)</p>
<p class="p1">\(a{\rm{ }} = {\rm{ }}13.4\), \(b{\rm{ }} = {\rm{ }}137\) <strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong> </strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into <strong>their </strong>regression equation</p>
<p class="p1"><em>eg</em>\(\;\;\;13.3823 \times 7 + 137.482\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">correct calculation</p>
<p class="p1">\(231.158\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(231\) (coyotes) (must be an integer) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing \(t = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f(0)\)</p>
<p class="p1">correct substitution into the model</p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{2000}}{{1 + 99{{\text{e}}^{ - k(0)}}}},{\text{ }}\frac{{2000}}{{100}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(20\) (foxes) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing \((5,{\text{ }}64)\) satisfies the equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f(5) = 64\)</p>
<p>correct substitution into the model</p>
<p><em>eg</em>\(\;\;\;64 = \frac{{2000}}{{1 + 99{{\text{e}}^{ - k(5)}}}},{\text{ }}64(1 + 99\(e\)^{ - 5k}}) = 2000\) <strong><em>(A1)</em></strong></p>
<p>\(0.237124\)</p>
<p>\(k = - \frac{1}{5}\ln \left( {\frac{{11}}{{36}}} \right){\text{ (exact), }}0.237{\text{ }}[0.237,{\text{ }}0.238]\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;c = f\), sketch of graphs</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{2000}}{{1 + 99{{\text{e}}^{ - 0.237124t}}}} = 13.382t + 137.482\), sketch of graphs, table of values</p>
<p class="p1">\(t = 12.0403\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(2007\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Exception to the <strong><em>FT </em></strong>rule. Award <strong><em>A1FT </em></strong>on their value of \(t\).</p>
<p class="p1"><em><strong>[4 marks]</strong></em></p>
<p class="p1"><em><strong>Total [16 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f'(x) = - 24{x^3} + 9{x^2} + 3x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There are two points of inflexion on the graph of <em>f</em> . Write down the <em>x</em>-coordinates </span><span style="font-family: times new roman,times; font-size: medium;">of these points.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = f''(x)\) . Explain why the graph of <em>g</em> has no points of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) = 0\) , the max and min of \(f'\) gives the points of inflexion on <em>f</em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 0.114{\text{, }}0.364\) (accept (\( - 0.114{\text{, }}0.811\)) and (\(0.364{\text{, }}2.13)\)) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graph of <em>g</em> is a quadratic function <em><strong>R1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">a quadratic function does not have any points of inflexion <em><strong>R1 N1</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graph of <em>g</em> is concave down over entire domain <strong><em>R1 N1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">therefore no change in concavity <em><strong>R1 N1</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 3</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(g''(x) = - 144\) <em><strong>R1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">therefore no points of inflexion as \(g''(x) \ne 0\) <em><strong> R1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were mixed results in part (a). Students were required to understand the relationships between a function and its derivative and often obtained the correct solutions with incorrect or missing reasoning. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), the question was worth two marks and candidates were required to make two valid points in their explanation. There were many approaches to take here and candidates often confused their reasoning or just kept writing hoping that somewhere along the way they would say something correct to pick up the points. Many confused \(f'\) and \(g'\) . </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><strong>All lengths in this question are in metres.</strong></p>
<p class="p1">Let \(f(x) = - 0.8{x^2} + 0.5\), for \( - 0.5 \leqslant x \leqslant 0.5\). Mark uses \(f(x)\) as a model to create a barrel. The region enclosed by the graph of \(f\), the \(x\)-axis, the line \(x = - 0.5\) and the line \(x = 0.5\) is rotated <span class="s1">360°</span> about the \(x\)-axis. This is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_15.49.19.png" alt="N16/5/MATME/SP2/ENG/TZ0/06"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the model to find the volume of the barrel.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">The empty barrel is being filled with water. The volume \(V{\text{ }}{{\text{m}}^3}\) </span>of water in the barrel after \(t\) minutes is given by \(V = 0.8(1 - {{\text{e}}^{ - 0.1t}})\). How long will it take for the barrel to be half-full?</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute correct limits or the function into the formula involving</p>
<p class="p1">\({y^2}\)</p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\pi \int_{ - 0.5}^{0.5} {{y^2}{\text{d}}x,{\text{ }}\pi \int {{{( - 0.8{x^2} + 0.5)}^2}{\text{d}}x} } \)</p>
<p class="p3">0.601091</p>
<p class="p1">volume \( = 0.601{\text{ }}({{\text{m}}^3})\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to equate half <strong>their </strong>volume to \(V\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(0.30055 = 0.8(1 - {{\text{e}}^{ - 0.1t}})\)</span>, graph</p>
<p class="p2">4.71104</p>
<p class="p1"><span class="s2">4.71 </span>(minutes) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider a function \(f\), for \(0 \le x \le 10\). The following diagram shows the graph of \(f'\), the derivative of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_07.44.17.png" alt></p>
<p class="p1">The graph of \(f'\) passes through \((2,{\text{ }} - 2)\) and \((5,{\text{ }}1)\), and has \(x\)-intercepts at \(0\), \(4\) and \(6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(f\) has a local maximum point when \(x = p\). State the value of \(p\), and justify your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down \(f'(2)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(g(x) = \ln \left( {f(x)} \right)\) and \(f(2) = 3\).</p>
<p class="p1">Find \(g'(2)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Verify that \(\ln 3 + \int_2^a {g'(x){\text{d}}x = g(a)} \), where \(0 \le a \le 10\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The following diagram shows the graph of \(g'\), the derivative of \(g\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_07.59.38.png" alt></p>
<p class="p1">The shaded region \(A\) is enclosed by the curve, the <em>\(x\)</em>-axis and the line \(x = 2\), and has area \({\text{0.66 unit}}{{\text{s}}^{\text{2}}}\).</p>
<p class="p1">The shaded region \(B\) is enclosed by the curve, the \(x\)-axis and the line \(x = 5\), and has area \({\text{0.21 unit}}{{\text{s}}^{\text{2}}}\)<span class="s1">.</span></p>
<p class="p2">Find \(g(5)\)<span class="s2">.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(p = 6\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">recognizing that turning points occur when \(f'(x) = 0\) <span class="Apple-converted-space"> </span><strong><em>R1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">eg\(\;\;\;\)correct sign diagram</p>
<p class="p1">\(f'\) changes from positive to negative at \(x = 6\) <span class="Apple-converted-space"> </span><strong><em>R1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(f'(2) = - 2\) <strong><em>A1 N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to apply chain rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln (x)' \times f'(x)\)</p>
<p>correct expression for \(g'(x)\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g'(x) = \frac{1}{{f(x)}} \times f'(x)\)</p>
<p>substituting \(x = 2\) into <strong>their</strong> \(g'\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{f'(2)}}{{f(2)}}\)</p>
<p>\( - 0.666667\)</p>
<p>\(g'(2) = - \frac{2}{3}{\text{ (exact), }} - 0.667\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of integrating \(g'(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">eg\(\;\;\;g(x)|_2^a,{\text{ }}g(x)|_a^2\)</p>
<p class="p1">applying the fundamental theorem of calculus (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">eg\(\;\;\;\int_2^a {g'(x) = g(a) - g(2)} \)</p>
<p class="p1">correct substitution into integral <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">eg\(\;\;\;\ln 3 + g(a) - g(2),{\text{ }}\ln 3 + g(a) - \ln \left( {f(2)} \right)\)</p>
<p class="p1">\(\ln 3 + g(a) - \ln 3\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(\ln 3 + \int_2^a {g'(x) = g(a)} \) <span class="Apple-converted-space"> </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substituting \(a = 5\) into the formula for \(g(a)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_2^5 {g'(x){\text{d}}x,{\text{ }}g(5) = \ln 3 + \int_2^5 {g'(x){\text{d}}x\;\;\;} } \left( {{\text{do not accept only }}g(5)} \right)\)</p>
<p>attempt to substitute areas <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln 3 + 0.66 - 0.21,{\text{ }}\ln 3 + 0.66 + 0.21\)</p>
<p>correct working</p>
<p><em>eg</em>\(\;\;\;g(5) = \ln 3 + ( - 0.66 + 0.21)\) <strong><em>(A1)</em></strong></p>
<p>\(0.648612\)</p>
<p>\(g(5) = \ln 3 - 0.45{\text{ (exact), }}0.649\) <strong><em>A1 N3</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to set up an equation for one shaded region <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_4^5 {g'(x){\text{d}}x = 0.21,{\text{ }}\int_2^4 {g'(x){\text{d}}x = - 0.66,{\text{ }}\int_2^5 {g'(x){\text{d}}x = - 0.45} } } \)</p>
<p>two correct equations <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - g(4) = 0.21,{\text{ }}g(2) - g(4) = 0.66\)</p>
<p>combining equations to eliminate \(g(4)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - [\ln 3 - 0.66] = 0.21\)</p>
<p>\(0.648612\)</p>
<p>\(g(5) = \ln 3 - 0.45{\text{ (exact), }}0.649\) <strong><em>A1 N3</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>attempt to set up a definite integral <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_2^5 {g'(x){\text{d}}x = - 0.66 + 0.21,{\text{ }}\int_2^5 {g'(x){\text{d}}x = - 0.45} } \)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - g(2) = - 0.45\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - \ln 3 = - 0.45\)</p>
<p>\(0.648612\)</p>
<p>\(g(5) = \ln 3 - 0.45{\text{ (exact), }}0.649\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas \(A\) and \(B\). Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = - 0.5{x^4} + 3{x^2} + 2x\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.09.00.png" alt="M17/5/MATME/SP2/ENG/TZ2/08"></p>
<p> </p>
<p>There are \(x\)-intercepts at \(x = 0\) and at \(x = p\). There is a maximum at A where \(x = a\), and a point of inflexion at B where \(x = b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of A.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the rate of change of \(f\) at A.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the the rate of change of \(f\) at B.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and the line \(x = a\). The region \(R\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(x) = 0,{\text{ }}y = 0\)</p>
<p>2.73205</p>
<p>\(p = 2.73\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>1.87938, 8.11721</p>
<p>\((1.88,{\text{ }}8.12)\) <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>rate of change is 0 (do not accept decimals) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using GDC)</strong></p>
<p>valid approach <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f’’ = 0\), max/min on \(f’,{\text{ }}x = - 1\)</p>
<p>sketch of either \(f’\) or \(f’’\), with max/min or root (respectively) <strong><em>(A1)</em></strong></p>
<p>\(x = 1\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p>Substituting <strong>their</strong> \(x\) value into \(f\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(1)\)</p>
<p>\(y = 4.5\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong>METHOD 2 (analytical)</strong></p>
<p>\(f’’ = - 6{x^2} + 6\) <strong><em>A1</em></strong></p>
<p>setting \(f’’ = 0\) <strong><em>(M1)</em></strong></p>
<p>\(x = 1\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p>substituting <strong>their</strong> \(x\) value into \(f\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(1)\)</p>
<p>\(y = 4.5\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing rate of change is \(f’\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(y’,{\text{ }}f’(1)\)</p>
<p>rate of change is 6 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula <strong><em>(M1)</em></strong></p>
<p>involving \({f^2}\) (accept absence of \(\pi \) and/or \({\text{d}}x\))</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\pi \int {{{( - 0.5{x^4} + 3{x^2} + 2x)}^2}{\text{d}}x,{\text{ }}\int_1^{1.88} {{f^2}} } \)</p>
<p>128.890</p>
<p>\({\text{volume}} = 129\) <strong><em>A2</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\cos (x - \sin x)\) , \(0 \le x \le 3\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> on the following set of axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/marvin.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> intersects the <em>x</em>-axis when \(x = a\) , \(a \ne 0\) . Write down the </span><span style="font-family: times new roman,times; font-size: medium;">value of <em>a</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is revolved \(360^\circ \) about the <em>x</em>-axis from \(x = 0\) to \(x = a\) . </span><span style="font-family: times new roman,times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/marvin2.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A2 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Award <em><strong>A1</strong></em> for correct domain, \(0 \le x \le 3\) . Award <em><strong>A2</strong></em> for approximately correct shape, with local maximum in circle 1 and right endpoint in circle 2.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 2.31\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[1 mark]</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(V = \pi {\int {\left[ {f(x)} \right]} ^2}{\rm{d}}x\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">fully correct integral expression <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \pi {\int_0^{2.31} {\left[ {x\cos (x - \sin x)} \right]} ^2}{\rm{d}}x\) , \(V = \pi {\int_0^{2.31} {\left[ {f(x)} \right]} ^2}{\rm{d}}x\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = 5.90\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates sketched a clear and smooth freehand curve with the local maximum, <em>x</em>-intercept and endpoints in approximately correct positions. Commonly, candidates sketched a graph across \([ - 3{\text{, }}3]\) , which neglects the given domain of the function. There were some candidates who sketched a straight line through the origin, presumably from being in the degree mode of their GDC. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates sketched a clear and smooth freehand curve with the local maximum, <em>x</em>-intercept and endpoints in approximately correct positions. Commonly, candidates sketched a graph across \([ - 3{\text{, }}3]\) , which neglects the given domain of the function. There were some candidates who sketched a straight line through the origin, presumably from being in the degree mode of their GDC. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates could set up the correct integral expression for volume, but surprisingly few were able to use their GDC to find the correct value. Some attempted to analytically integrate the square of this unusual function, expending valuable time in this effort. A small but significant number of candidates wrote a final answer as \(1.88\pi \) , which accrued the accuracy penalty.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A population of rare birds, \({P_t}\), can be modelled by the equation \({P_t} = {P_0}{{\text{e}}^{kt}}\), where \({P_0}\) is the initial population, and \(t\) is measured in decades. After one decade, it is estimated that \(\frac{{{P_1}}}{{{P_0}}} = 0.9\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Find the value of \(k\).</p>
<p class="p1">(ii) Interpret the meaning of the value of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the least number of <strong>whole </strong>years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(0.9 = {{\text{e}}^{k(1)}}\)</p>
<p class="p1">\(k = - 0.105360\)</p>
<p class="p1"><span class="Apple-converted-space">\(k = \ln 0.9{\text{ (exact), }} - 0.105\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct interpretation <span class="Apple-converted-space"> </span><strong><em>R1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)population is decreasing, growth rate is negative</p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2">valid approach (accept an equality, but do not accept 0.74<span class="s1">) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(P < 0.75{P_0},{\text{ }}{P_0}{{\text{e}}^{kt}} < 0.75{P_0},{\text{ }}0.75 = {{\text{e}}^{t\ln 0.9}}\)</p>
<p class="p1">valid approach to solve <strong>their </strong>inequality <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)logs, graph</p>
<p class="p1"><span class="Apple-converted-space">\(t > 2.73045{\text{ }}({\text{accept }}t = 2.73045){\text{ }}(2.73982{\text{ from }} - 0.105)\) </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="s2">28 </span>years <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p2">valid approach which gives both crossover values accurate to at least 2 <span class="s1">sf <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{P_{2.7}}}}{{{P_0}}} = 0.75241 \ldots ,{\text{ }}\frac{{{P_{2.8}}}}{{{P_0}}} = 0.74452 \ldots \)</p>
<p class="p1"><span class="Apple-converted-space">\(t = 2.8\) </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s2">28 </span>years <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was generally done well, with many candidates able to find the value of \(k\) correctly and to interpret its meaning. Lack of accuracy was occasionally a concern, with some candidates writing their value of \(k\) to 2 significant figures or evaluating \(\ln (0.9)\) incorrectly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Few candidates were successful in part (b) with many unable to set up an inequality or equation which would allow them to find the condition on \(t\). Some were able to find the value of \(t\) in decades but most were unable to correctly interpret their inequality in terms of the least number of whole years. While a solution through analytic methods was readily available, very few students attempted to use their GDC to solve their initial equation or inequality.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The price of a used car depends partly on the distance it has travelled. The following table shows the distance and the price for seven cars on <span class="s1">1 </span>January <span class="s1">2010</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_17.58.43.png" alt="M16/5/MATME/SP2/ENG/TZ2/08"></p>
<p class="p1">The relationship between \(x\) and \(y\) can be modelled by the regression equation \(y = ax + b\).</p>
</div>
<div class="specification">
<p class="p1">On <span class="s1">1 </span>January <span class="s1">2010</span>, Lina buys a car which has travelled \(11\,000{\text{ km}}\).</p>
</div>
<div class="specification">
<p class="p1">The price of a car decreases by <span class="s1">5% </span>each year.</p>
</div>
<div class="specification">
<p class="p1">Lina will sell her car when its price reaches \(10\,000\)<span class="s1"> </span>dollars.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the correlation coefficient.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Write down the value of \(a\) and of \(b\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the regression equation to estimate the price of Lina’s car, giving your answer to the nearest <span class="s1">100 </span>dollars.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the price of Lina’s car after <span class="s1">6 </span>years.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the year when Lina sells her car.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)correct value for \(r\) (or for \(a\) or \(b\) <span class="s1">seen in (ii))</span></p>
<p class="p2">\( - 0.994347\)</p>
<p class="p1"><span class="s1">\(r = - 0.994\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2">(ii) <span class="Apple-converted-space"> \( - 1.58095,{\text{ }}33480.3\)</span></p>
<p class="p1"><span class="s1">\(a = - 1.58,{\text{ }}b = 33500\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1">correct substitution into <strong>their </strong>regression equation</p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - 1.58095(11000){\text{ }} + 33480.3\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\(16\,089.85{\text{ }}(16\,120{\text{ from 3sf}})\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s1">\({\text{price}} = 16\,100{\text{ }}({\text{dollars}})\) </span>(must be rounded to the nearest 100 <span class="s1">dollars) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(P \times {({\text{rate}})^t}\)</p>
<p class="p1">\({\text{rate}} = 0.95\) (may be seen in their expression) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">correct expression <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(16100 \times {0.95^6}\)</p>
<p class="p1">\(11\,834.97\)</p>
<p class="p1"><span class="Apple-converted-space">\(11\,800{\text{ }}({\text{dollars}})\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">attempt to find all six terms <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\left( {(16\,100 \times 0.95) \times 0.95} \right) \ldots } \right) \times 0.95\)</span>, table of values</p>
<p class="p2">5 correct values (accept values that round correctly to the nearest dollar)</p>
<p class="p2"><span class="Apple-converted-space">\(15\,295,{\text{ }}14\,530,{\text{ }}13\,804,{\text{ }}13\,114,{\text{ }}12\,458\) </span><span class="s1"><strong><em>A2</em></strong></span></p>
<p class="p2">\(11\,835\)</p>
<p class="p1"><span class="s2">\(11\,800{\text{ }}({\text{dollars}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There may be slight differences in answers, depending on which values candidates carry through in subsequent parts. Accept answers that are consistent with their working.</p>
<p class="p1"> </p>
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">correct equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(16\,100 \times {0.95^x}{\text{ = }}10\,000\)</p>
<p class="p1">valid attempt to solve <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)<img src="images/Schermafbeelding_2017-02-03_om_08.51.35.png" alt="M16/5/MATME/SP2/ENG/TZ2/08.d/M"></span>, using logs</p>
<p class="p2">9.28453 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2">year 2019 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">valid approach using table of values <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>both </strong>crossover values (accept values that round correctly to the nearest dollar) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{P}} = 10\,147{\text{ }}({\text{1 Jan 2019}}),{\text{ P}} = 9\,639.7{\text{ }}({\text{1 Jan 2020}})\)</p>
<p class="p2">year 2019 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Although the question talked about the regression equation, a few students tried to find the values of <em>a </em>and <em>b </em>by forming two equations with the coordinates of two points from the table. A considerable number of candidates did not write the value of the correlation coefficient or gave an incorrect one. It can be that a GDC feature (Diagnostics) from some calculators was turned off.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was generally well done, with many candidates earning follow through marks. There were some difficulties in rounding the answer to the nearest 100 dollars.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) was attempted in two different ways: recognizing the correct rate 0.95 and then finding the price of the car after 6 years. Some of these candidates used a formula similar to the one for terms of a geometric sequence, \(P \times {({\text{rate)}}^{t - 1}}\), but substituted \(t\) by 6 and hence, got an incorrect result.</p>
<p class="p1">Others listed all six values to obtain the answer. When using this method, the problem was using less accurate intermediate results and hence, not getting the first 5 correct values of the car.</p>
<p class="p1">Many candidates either missed out questions 8 (c) and (d) or multiplied either \(0.05 \times 6 \times 16\,100\) or \(0.95 \times 6 \times 16\,100\) and failed to notice that the answer did not make sense. Other students tried to use the sum formula for a geometric series.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Many candidates either missed out questions 8 (c) and (d) or multiplied either \(0.05 \times 6 \times 16\,100\) or \(0.95 \times 6 \times 16\,100\) and failed to notice that the answer did not make sense. Other students tried to use the sum formula for a geometric series.</p>
<p class="p1">Part (d) was attempted using a graphical approach as well as analytically using logarithms to find the year in which Lina would sell the car, though many failed in giving the correct year. Common answers were “in the ninth year” or “in 2020”. The same happened to those candidates who used a table of values and found the price of the car after 9 years and 10 years. These candidates should be reminded to show both “crossover” values for a table method to be valid.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = \frac{{2x - 6}}{{1 - x}}\), for \(x \ne 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For the graph of \(f\)</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>find the \(x\)-intercept;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>write down the equation of the vertical asymptote;</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>find the equation of the horizontal asymptote.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\mathop {\lim }\limits_{x \to \infty } f(x)\).</p>
<p class="p1"> </p>
<p class="p1"> </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)sketch, \(f(x) = 0,{\text{ }}0 = 2x - 6\)</p>
<p>\(x = 3\) or \((3,{\text{ }}0)\) <strong><em>A1 N2</em></strong></p>
<p>(ii) \(x = 1\;\;\;\)(must be equation) <strong><em>A1 N1</em></strong></p>
<p>(iii) valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)sketch, \(\frac{{2x}}{{ - 1x}}\), inputting large values of \(x\), L’Hopital’s rule</p>
<p>\(y = - 2\;\;\;\)(must be equation) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)recognizing that \(\mathop {\lim }\limits_{x \to \infty } \) is related to the horizontal asymptote, table with large values of \(x\), their \(y\) value from (a)(iii), L’Hopital’s rule</p>
<p>\(\mathop {\lim }\limits_{x \to \infty } f(x) = - 2\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was generally well done with candidates using both algebraic and graphical approaches to obtain solutions. There are still some who do not identify their asymptotes using equations.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates rarely appreciated the relevance of the horizontal asymptote in (b), and often attempted a long, and often unsuccessful, algebraic approach to find the limit.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The number of bacteria, <em>n</em> , in a dish, after <em>t</em> minutes is given by \(n = 800{{\rm{e}}^{0.13t}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>n</em> when \(t = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the rate at which <em>n</em> is increasing when \(t = 15\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<address><span style="font-family: times new roman,times; font-size: medium;">After <em>k</em> minutes, the rate of increase in <em>n</em> is greater than \(10000\) bacteria </span><span style="font-family: times new roman,times; font-size: medium;">per minute. Find the least value of <em>k</em> , where \(k \in \mathbb{Z}\) .</span></address>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 800{{\rm{e}}^0}\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 800\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using the derivative <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n'(15) = 731\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting up inequality (accept equation or reverse inequality) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(n'(t) > 10000\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, finding derivative</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = 35.1226 \ldots \) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">least value of <em>k</em> is 36 <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n'(35) = 9842\) , <strong>and</strong> \(n'(36) = 11208\) <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">least value of <em>k</em> is 36 <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question seemed to be challenging for the great majority of the candidates.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally well answered.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question seemed to be challenging for the great majority of the candidates.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally well answered but in parts (b) and (c) they did not consider that rates of </span><span style="font-family: times new roman,times; font-size: medium;">change meant they needed to use differentiation. Most students completely missed or did not </span><span style="font-family: times new roman,times; font-size: medium;">understand that the question was asking about the instantaneous rate of change, which </span><span style="font-family: times new roman,times; font-size: medium;">resulted in the fact that most of them used the original equation. Some did attempt to find an </span><span style="font-family: times new roman,times; font-size: medium;">average rate of change over the time interval, but even fewer attempted to use the derivative.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Of those who did realize to use the derivative in (b), a vast majority calculated it by hand </span><span style="font-family: times new roman,times; font-size: medium;">instead of using their GDC feature to evaluate it.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question seemed to be challenging for the great majority of the candidates.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally well answered but in parts (b) and (c) they did not consider that rates of </span><span style="font-family: times new roman,times; font-size: medium;">change meant they needed to use differentiation. Most students completely missed or did not </span><span style="font-family: times new roman,times; font-size: medium;">understand that the question was asking about the instantaneous rate of change, which </span><span style="font-family: times new roman,times; font-size: medium;">resulted in the fact that most of them used the original equation. Some did attempt to find an </span><span style="font-family: times new roman,times; font-size: medium;">average rate of change over the time interval, but even fewer attempted to use the derivative.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Of those who did realize to use the derivative in (b), a vast majority calculated it by hand </span><span style="font-family: times new roman,times; font-size: medium;">instead of using their GDC feature to evaluate it.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The inequality for part (c) was sometimes well solved using the original function but many </span><span style="font-family: times new roman,times; font-size: medium;">failed to round their answers to the nearest integer.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Let </span><span style="font-family: TimesNewRomanPS-ItalicMT;">\(f(x) = {x^3} - 2x - 4\)</span><span style="font-family: TimesNewRomanPSMT;"> . The following diagram shows part of the curve of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">f </span></em><span style="font-family: TimesNewRomanPSMT;">.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/N12P2Q3.jpg" alt></span></span></p>
<p><span style="font-family: TimesNewRomanPSMT;"><span style="font-family: TimesNewRomanPSMT;"></span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">The curve crosses the </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">x</span></em><span style="font-family: TimesNewRomanPSMT;">-axis at the point P.</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Write down the </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">x</span></em><span style="font-family: TimesNewRomanPSMT;">-coordinate of P.</span></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the gradient of the curve at P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the equation of the normal to the curve at P, giving your equation in the </span><span style="font-family: times new roman,times; font-size: medium;">form \(y = ax + b\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 2\) (accept \((2{\text{, }}0)\)) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of finding gradient of <em>f</em> at \(x = 2\) <em><strong>(M1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(2)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">the gradient is 10 <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of negative reciprocal of gradient <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 1}}{{f'(x)}}\) , \( - \frac{1}{{10}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct substitution into equation of a line <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y - 0 = \frac{{ - 1}}{{10}}(x - 2)\) , \(0 = - 0.1(2) + b\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = - \frac{1}{{10}}x + \frac{2}{{10}}\) (accept \(a = - 0.1\) , \(b = 0.2\) ) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was generally done well. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates did not use their GDC in part (b), resulting in a variety of careless errors occasionally arising either in differentiating or substituting. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were some candidates who did not know the relationship between gradients of perpendicular lines while others found the equation of the tangent rather than the normal in part (c). </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the following circle with centre O and radius <em>r</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/Jamie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points P, R and Q are on the circumference, \({\rm{P}}\widehat {\rm{O}}{\rm{Q}} = 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) . </span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the cosine rule to show that \({\rm{PQ}} = 2r\sin \theta \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>l</em> be the length of the arc PRQ .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(1.3{\rm{PQ}} - l = 0\) , find the value of \(\theta \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(\theta ) = 2.6\sin \theta - 2\theta \) , for \(0 < \theta < \frac{\pi }{2}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Write down the root of \(f(\theta ) = 0\) .</span></p>
<p> </p>
<div class="marks">[4]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the graph of <em>f</em> to find the values of \(\theta \) for which \(l < 1.3{\rm{PQ}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into cosine rule <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = {r^2} + {r^2} - 2(r)(r)\cos (2\theta )\) , \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 2{r^2} - 2{r^2}(\cos (2\theta ))\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(1 - 2{\sin ^2}\theta \) for \(\cos 2\theta \) (seen anywhere) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 2{r^2} - 2{r^2}(1 - 2{\sin ^2}\theta )\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working towards answer <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 2{r^2} - 2{r^2} + 4{r^2}{\sin ^2}\theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing \(2{r^2} - 2{r^2} = 0\) (including crossing out) (seen anywhere) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{P}}{{\rm{Q}}^{\rm{2}}} = 4{r^2}{\sin ^2}\theta \) , \({\rm{PQ}} = \sqrt {4{r^2}{{\sin }^2}\theta } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{PQ = 2}}r{\rm{sin}}\theta \) <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{PRQ}} = r \times 2\theta \) (seen anywhere) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct set up <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1.3 \times 2r\sin \theta - r \times (2\theta ) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to eliminate <em>r</em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation in terms of the one variable \(\theta \) <em><strong>(A1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1.3 \times 2\sin \theta - 2\theta = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">1.221496215 </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 1.22\) (accept \(70.0^\circ \) (69.9)) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/fedup.png" alt> <em><strong>A1A1A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <strong><em>A1</em></strong> for approximately correct shape, <strong><em>A1</em></strong> for <em>x</em>-intercept in approximately </span><span style="font-family: times new roman,times; font-size: medium;">correct position, <strong><em>A1</em></strong> for domain. Do not penalise if sketch starts at origin.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(1.221496215\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 1.22\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[4 marks]</strong> </span></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach (may be seen earlier) <strong><em>M2 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\theta < 2.6\sin \theta \) , \(0 < f(\theta )\) , showing positive part of sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0 < \theta < 1.221496215\)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0 < \theta = 1.22\) (accept \(\theta < 1.22\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This exercise seemed to be challenging for the great majority of the candidates, in particular parts (b), (c) and (d).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally attempted using the cosine rule, but many failed to substitute correctly into the right hand side or skipped important steps. A high percentage could not arrive at the given expression due to a lack of knowledge of trigonometric identities or making algebraic errors, and tried to force their way to the given answer.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The most common errors included taking the square root too soon, and sign errors when distributing the negative after substituting \(\cos 2\theta \) by \(1 - 2{\sin ^2}\theta \) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This exercise seemed to be challenging for the great majority of the candidates, in particular parts (b), (c) and (d). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), most candidates understood what was required but could not find the correct length of the arc PRQ mainly due to substituting the angle by \(\theta \) instead of \(2\theta \) . </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Regarding part (c), many valid approaches were seen for the graph of <em>f</em>, making a good use of their GDC. A common error was finding a second or third solution outside the domain. A considerable amount of sketches were missing a scale. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There were candidates who achieved the correct equation but failed to realize they could use their GDC to solve it. </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was attempted by very few, and of those who achieved the correct answer not many were able to show the method they used. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\cos x\) , for \(0 \le x \le 6\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(y = f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the product rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x \times ( - \sin x) + 1 \times \cos x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \cos x - x\sin x\) <em><strong>A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc2.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1A1 N4</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for correct domain, \(0 \le x \le 6\) with endpoints in circles, <em><strong>A1</strong></em> for approximately correct shape, <em><strong>A1</strong></em> for local minimum in circle, <em><strong>A1</strong></em> for local maximum in circle. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This problem was well done by most candidates. There were some candidates that struggled </span><span style="font-family: times new roman,times; font-size: medium;">to apply the product rule in part (a) and often wrote nonsense like \( - x\sin x = - \sin {x^2}\) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part </span><span style="font-family: times new roman,times; font-size: medium;">(b), few candidates were able to sketch the function within the required domain and a large </span><span style="font-family: times new roman,times; font-size: medium;">number of candidates did not have their calculator in the correct mode.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The quadratic equation \(k{x^2} + (k - 3)x + 1 = 0\) has two equal real roots.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the possible values of <em>k</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Write down</strong> the values of <em>k</em> for which \({x^2} + (k - 3)x + k = 0\) has two equal </span><span style="font-family: times new roman,times; font-size: medium;">real roots.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to use discriminant <em><strong>(M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution, \({(k - 3)^2} - 4 \times k \times 1\) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting <strong>their</strong> discriminant equal to zero <em><strong>M1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({(k - 3)^2} - 4 \times k \times 1 = 0\) , \({k^2} - 10k + 9 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 1\) , \(k = 9\) <em><strong>A1A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 1\) , \(k = 9\) <em><strong> A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although some candidates correctly considered the discriminant to find the possible values of <em>k </em>, many of them did not set it equal to \(0\), writing an inequality instead.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), some students realized that the discriminants in parts (a) and (b) were the same, earning follow through marks just by writing the same (often incorrect) answers they got in part (a). Many, however, did not see the connection between the two parts. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \frac{1}{2}x\sin x\) , for \(0 \le x \le 4\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>g</em> on the following set of axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/spinning.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence find the value of <em>x</em> for which \(g(x) = - 1\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/nfl.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1A1 N4</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape, <strong><em>A1</em></strong> for left end point in circle, </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for local maximum in circle, <strong><em>A1</em></strong> for right end point in circle.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to solve \(g(x) = - 1\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. marking coordinate on graph, \(\frac{1}{2}x\sin x + 1 = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 3.71\) <strong><em>A1 N2</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by the majority of candidates. Most sketched an approximately correct shape in the given domain, though some candidates did not realize they had to set their GDC to radians, producing a meaningless sketch. Candidates need to be aware that unless otherwise specified, questions will expect radians to be used. The most confident candidates used a table to aid their graphing. Although most recognized the need of the GDC to answer part (b), some used the trace function, hence obtaining an inaccurate result, while others attempted a fruitless analytical approach. Merely stating "using GDC" is insufficient evidence of method; a sketch or an equation set equal to zero are both examples of appropriate evidence. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by the majority of candidates. Most sketched an approximately correct shape in the given domain, though some candidates did not realize they had to set their GDC to radians, producing a meaningless sketch. Candidates need to be aware that unless otherwise specified, questions will expect radians to be used. The most confident candidates used a table to aid their graphing. Although most recognized the need of the GDC to answer part (b), some used the trace function, hence obtaining an inaccurate result, while others attempted a fruitless analytical approach. Merely stating "using GDC" is insufficient evidence of method; a sketch or an equation set equal to zero are both examples of appropriate evidence. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = (x - 1)(x - 4)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-intercepts of the graph of \(f\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360^\circ\) about the \(x\)-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(f(x) = 0\), sketch of parabola showing two \(x\)-intercepts</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1,{\text{ }}x = 4{\text{ }}\left( {{\text{accept (1, 0), (4, 0)}}} \right)\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute either limits or the function into formula involving \({f^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_1^4 {{{\left( {f(x)} \right)}^2}{\text{d}}x,{\text{ }}\pi \int {{{\left( {(x - 1)(x - 4)} \right)}^2}} } \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{volume}} = 8.1\pi {\text{ (exact), 25.4}}\) <strong><em>A2 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks] </em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 5\cos \frac{\pi }{4}x\) and \(g(x) = - 0.5{x^2} + 5x - 8\) for \(0 \le x \le 9\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the same diagram, sketch the graphs of <em>f</em> and <em>g</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of \(f\) . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the <em>x</em>-intercept that lies between \(x = 0\) and \(x = 3\) ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the amplitude.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of <em>g</em> . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the two <em>x</em>-intercepts;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the equation of the axis of symmetry.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> . Find the area of <em>R</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;"><img src="images/luke.png" alt></span><em><span style="font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for <em>f</em> being of sinusoidal shape, with 2 maxima and one minimum, </span><span style="font-size: medium;"><em><strong>A1</strong></em> for <em>g</em> being a parabola opening down, </span><span style="font-size: medium;"><em><strong>A1</strong></em> for <strong>two</strong> intersection points in approximately correct position. </span></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks] </span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \((2{\text{, }}0)\) (accept \(x = 2\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{period}} = 8\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \({\text{amplitude}} = 5\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \((2{\text{, }}0)\) , \((8{\text{, }}0)\) (accept \(x = 2\) , \(x = 8\) ) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(x = 5\) (must be an equation) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">intersect when \(x = 2\) and \(x = 6.79\) (may be seen as limits of integration) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {g - f} \) , \(\int {f(x){\rm{d}}x - \int {g(x){\rm{d}}x}}\) , \(\int_2^{6.79} {\left( {( - 0.5{x^2} + 5x - 8) - \left( {5\cos \frac{\pi }{4}x} \right)} \right)}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 27.6\) <em><strong>A2 N3</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">intersect when \(x = 2\) and \(x = 6.79\) (seen anywhere) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach using a sketch of <em>g</em> and <em>f</em> , or \(g - f\) . <em><strong>(M1)</strong></em></span></p>
<p><br><img src="images/going_out.png" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. area = \(A + B - C\) , \(12.7324 + 16.0938 - 1.18129 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 27.6\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Graph sketches were much improved over previous sessions. Most candidates graphed the two functions correctly, but many ignored the domain restrictions. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found parts (b) and (c) accessible, although quite a few did not know how to find the period of the cosine function.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found parts (b) and (c) accessible, although quite a few did not know how to find the period of the cosine function.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) proved elusive to many candidates. Some used creative approaches that split the area into parts above and below the <em>x</em>-axis; while this leads to a correct result, few were able to achieve it. Many candidates were unable to use their GDCs effectively to find points of intersection and the subsequent area. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows two ships A and B. At noon, ship A was 15 km due </span><span style="font-family: times new roman,times; font-size: medium;">north of ship B. Ship A was moving south at 15 km h<sup>–1</sup> and ship B was moving east at </span><span style="font-family: times new roman,times; font-size: medium;">11 km h<sup>–1</sup>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/cloudy.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the distance between the ships</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) at 13:00;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) at 14:00.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(s(t)\) be the distance between the ships <em>t</em> hours after noon, for \(0 \le t \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(s(t) = \sqrt {346{t^2} - 450t + 225} \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(s(t)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Due to poor weather, the captain of ship A can only see another ship if they are </span><span style="font-family: times new roman,times; font-size: medium;">less than 8 km apart. Explain why the captain cannot see ship B between noon </span><span style="font-family: times new roman,times; font-size: medium;">and 16:00.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. ship A where B was, B \(11{\text{ km}}\) away </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{distance}} = 11\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. new diagram, Pythagoras, vectors </span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(s = \sqrt {{{15}^2} + {{22}^2}} \) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>(A1)</strong> </span></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\sqrt {709} = 26.62705\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(s = 26.6\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>M0A0A0</strong></em> for using the formula given in part (b). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. a table, diagram, formula \(d = r \times t\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">distance ship A travels <em>t</em> hours after noon is \(15(t - 1)\) <em><strong>(A2) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">distance ship B travels in <em>t</em> hours after noon is \(11t\) <em><strong>(A1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s(t) = \sqrt {{{\left[ {15(t - 1)} \right]}^2} + {{(11t)}^2}} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct simplification <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sqrt {225({t^2} - 2t + 1) + 121{t^2}} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s(t) = \sqrt {346{t^2} - 450t + 225} \) <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bale.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for shape, <em><strong>A1</strong></em> for minimum at approximately \((0.7{\text{, }}9)\), <em><strong>A1</strong></em> for domain.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s'(t) = 0\) , find minimum of \(s(t)\) , graph, reference to "more than 8 km" </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\min = 8.870455 \ldots \) (accept 2 or more sf) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since \({s_{\min}} > 8\) , captain cannot see ship B <em><strong>R1 N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally well done although some candidates incorrectly used the function given in part (b) to find the required values. There was evidence that some candidates are not comfortable with a 24-hour clock. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had difficulty generalizing the problem and therefore, were unable to show how the function \(s(t)\) was obtained in part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Surprisingly, the graph in part (c) was not well done. Candidates often ignored the given domain, provided no indication of scale, and drew "V" shapes or parabolas. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), candidates simply regurgitated the question without providing any significant evidence for their statements that the two ships must have been more than 8 km apart. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = \frac{1}{{x - 1}} + 2\), for \(x > 1\).</p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = a{e^{ - x}} + b\), for \(x \geqslant 1\). The graphs of \(f\) and \(g\) have the same horizontal asymptote.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the equation of the horizontal asymptote of the graph of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f'(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \(b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(g'(1) = - e\), find the value of \(a\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">There is a value of \(x\)<span class="s1">, for \(1 < x < 4\)</span>, for which the graphs of \(f\) and \(g\) have the same gradient. Find this gradient.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(y = 2\) (correct equation only) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({(x - 1)^{ - 1}} + 2,{\text{ }}f'(x) = \frac{{0(x - 1) - 1}}{{{{(x - 1)}^2}}}\)</p>
<p class="p1"><span class="Apple-converted-space">\( - {(x - 1)^{ - 2}},{\text{ }}f'(x) = \frac{{ - 1}}{{{{(x - 1)}^2}}}\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct equation for the asymptote of \(g\)</p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(y = b\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1">\(b = 2\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct derivative of <span class="s1"><em>g </em></span>(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A2)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(g'(x) = - a{{\text{e}}^{ - x}}\)</p>
<p class="p1">correct equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - {\text{e}} = - a{{\text{e}}^{ - 1}}\)</p>
<p class="p2">7.38905</p>
<p class="p1"><span class="s1">\(a = {{\text{e}}^2}{\text{ }}({\text{exact}}),{\text{ }}7.39\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to equate <strong>their </strong>derivatives <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f'(x) = g'(x),{\text{ }}\frac{{ - 1}}{{{{(x - 1)}^2}}} = - a{{\text{e}}^{ - x}}\)</p>
<p class="p1">valid attempt to solve <strong>their </strong>equation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)correct value outside the domain of \(f\) <span class="s1">such as 0.522 or 4.51,</span></p>
<p class="p2"><img src="images/Schermafbeelding_2017-02-03_om_09.34.38.png" alt="M16/5/MATME/SP2/ENG/TZ2/09.e/M"></p>
<p class="p1">correct solution (may be seen in sketch) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(x = 2,{\text{ }}(2,{\text{ }} - 1)\)</p>
<p class="p2">gradient is \( - 1\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was in general well answered. Many candidates lost the marks for writing 2 or \(y \ne 2\) instead of \(y = 2\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b) some candidates got confused and found \({f^{ - 1}}(x)\) instead of \(f'(x)\)<span class="s1"><em>. </em></span>When calculating the derivative, two types of approaches were seen. Most of the ones who rewrote the function as \(f(x) = {(x - 1)^{ - 1}} + 2\), applied the chain rule correctly. Those who tried to apply the quotient rule made various mistakes: incorrect derivative of a constant, incorrect multiplication by zero, wrong subtraction order in the numerator, omitted the negative sign in the answer.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (c), most candidates were coherent and obtained the same value as the one written in part (a).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (d) many candidates did not manage to differentiate the function g correctly. Of those who could, the equation was generally well solved algebraically.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">For part (e), not many candidates wrote a correct equation with their derivatives. There was mixed performance for this question, as those who knew they needed to use their GDC managed to obtain an answer, while many got tangled in unsuccessful attempts to solve the equation algebraically. Many candidates tried to solve quite complex equations ‘manually’ instead of trying to graph the expressions on their calculators and finding the value of \(x\) at the point of intersection. Of those students who tried to solve graphically only a small percentage actually sketched the two curves that they were considering. This sketch is particularly useful to examiners to see how the student is thinking, or what steps s/he is taking to solve the equations.</p>
<p class="p1">Only a few realized that the question asked for the gradient, which was represented by the \(y\)-coordinate of the point of intersection, rather than the \(x\)-coordinate.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(t) = 2{t^2} + 7\) , where \(t > 0\) . The function <em>v</em> is obtained when the graph of <em>f</em> is </span><span style="font-family: times new roman,times; font-size: medium;">transformed by</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">a stretch by a scale factor of \(\frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">parallel to the <em>y</em>-axis,</span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">followed by a translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 4}<br>\end{array}} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(v(t)\) , giving your answer in the form \(a{(t - b)^2} + c\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</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 in ms<sup>−1</sup> , at </span><span style="font-family: times new roman,times; font-size: medium;">time <em>t </em>seconds, is given by<em> v</em> . Find the distance the particle travels between </span><span style="font-family: times new roman,times; font-size: medium;">\(t = 5.0\) and \(t = 6.8\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">applies vertical stretch parallel to the <em>y</em>-axis factor of \(\frac{1}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. multiply by \(\frac{1}{3}\) , \(\frac{1}{3}f(t)\) , \(\frac{1}{3} \times 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">applies horizontal shift 2 units to the right <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(t - 2)\) , \(t - 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">applies a vertical shift 4 units down <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. subtracting 4, \(f(t) - 4\) , \(\frac{7}{3} - 4\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(v(t) = \frac{2}{3}{(t - 2)^2} - \frac{5}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N4</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing that distance travelled is area under the curve <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {v,\frac{2}{9}} {(t - 2)^3} - \frac{5}{3}t\) </span><span style="font-family: times new roman,times; font-size: medium;">, sketch</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/belle.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">distance = 15.576 (accept 15.6) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While a number of candidates had an understanding of each transformation, most had difficulty applying them in the correct order, and few obtained the completely correct answer in part (a). Many earned method marks for discerning three distinct transformations. Few candidates knew to integrate to find the distance travelled. Many instead substituted time values into the velocity function or its derivative and subtracted. A number of those who did recognize the need for integration attempted an analytic approach rather than using the GDC, which often proved unsuccessful.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While a number of candidates had an understanding of each transformation, most had difficulty applying them in the correct order, and few obtained the completely correct answer in part (a). Many earned method marks for discerning three distinct transformations. Few candidates knew to integrate to find the distance travelled. Many instead substituted time values into the velocity function or its derivative and subtracted. A number of those who did recognize the need for integration attempted an analytic approach rather than using the GDC, which often proved unsuccessful.</span></p>
<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;">Jose takes medication. After <em>t</em> minutes, the concentration of medication left in his </span><span style="font-family: times new roman,times; font-size: medium;">bloodstream is given by \(A(t) = 10{(0.5)^{0.014t}}\) , where <em>A</em> is in milligrams per litre.</span></p>
</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 \(A(0)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the concentration of medication left in his bloodstream after 50 minutes.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</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 13:00, when there is no medication in Jose’s bloodstream, he takes his first </span><span style="font-family: times new roman,times; font-size: medium;">dose of medication. He can take his medication again when the concentration </span><span style="font-family: times new roman,times; font-size: medium;">of medication reaches 0.395 milligrams per litre. What time will Jose be able to </span><span style="font-family: times new roman,times; font-size: medium;">take his medication again?</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A(0) = 10\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substitution into formula <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(10{(0.5)^{0.014(50)}}\) , \(A(50)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A(50) = 6.16\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">set up equation <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A(t) = 0.395\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to solve <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. graph, use of logs</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of intersection, \(0.014t\log 0.5 = \log 0.0395\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 333.00025 \ldots \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct time 18:33 or 18:34 (accept 6:33 or 6:34 but nothing else) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For a later question in Section A, a pleasing number of candidates made good progress. Some candidates believed that raising a base to the zero power gave zero which indicated that they most likely did not begin by analysing the function with their GDC. For part (c), many candidates could set up the equation correctly and had some idea to apply logarithms but became lost in the algebra. Those who used their GDC to find when the function equalled 0.395 typically did so successfully. A common error for those who obtained a correct value for time in minutes was to treat 5.55 hours as 5 hours and 55 minutes after 13:00. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For a later question in Section A, a pleasing number of candidates made good progress. Some candidates believed that raising a base to the zero power gave zero which indicated that they most likely did not begin by analysing the function with their GDC. For part (c), many candidates could set up the equation correctly and had some idea to apply logarithms but became lost in the algebra. Those who used their GDC to find when the function equalled 0.395 typically did so successfully. A common error for those who obtained a correct value for time in minutes was to treat 5.55 hours as 5 hours and 55 minutes after 13:00. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For a later question in Section A, a pleasing number of candidates made good progress. Some candidates believed that raising a base to the zero power gave zero which indicated that they most likely did not begin by analysing the function with their GDC. For part (c), many candidates could set up the equation correctly and had some idea to apply logarithms but became lost in the algebra. Those who used their GDC to find when the function equalled 0.395 typically did so successfully. A common error for those who obtained a correct value for time in minutes was to treat 5.55 hours as 5 hours and 55 minutes after 13:00. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graphs of \(f(x) = \ln (3x - 2) + 1\) and \(g(x) = - 4\cos (0.5x) + 2\) , for \(1 \le x \le 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/laurie.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>A</em> be the area of the region <strong>enclosed</strong> by the curves of <em>f</em> and <em>g</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for <em>A</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Calculate the value of <em>A</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f'(x)\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find \(g'(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There are two values of <em>x</em> for which the gradient of <em>f</em> is equal to the gradient </span><span style="font-family: times new roman,times; font-size: medium;">of <em>g</em>. Find both these values of <em>x</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) intersection points \(x = 3.77\) , \(x = 8.30\) (may be seen as the limits) <em><strong>(A1)(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction and integrals <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">fully correct expression <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{3.77}^{8.30} {(( - 4\cos (0.5x) + 2) - (\ln (3x - 2) + 1)){\rm{d}}x} \) , \(\int_{3.77}^{8.30} {g(x){\rm{d}}x - } \int_{3.77}^{8.30} {f(x){\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(A = 6.46\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x) = \frac{3}{{3x - 2}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for numerator (3), <em><strong>A1</strong></em> for denominator (\({3x - 2}\)) , but penalize </span><span style="font-family: times new roman,times; font-size: medium;">1 mark for additional terms.</span></p>
<p><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) \(g'(x) = 2\sin (0.5x)\) <em><strong>A1A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for 2, <em><strong>A1</strong></em> for \(\sin (0.5x)\) , but penalize 1 mark for additional terms.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using derivatives for gradients <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct approach <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = g'(x)\) , points of intersection</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 1.43\) , \(x = 6.10\) <em><strong>A1A1 N2N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not make good use of the GDC in this problem. Most had the correct </span><span style="font-family: times new roman,times; font-size: medium;">expression but incorrect limits. Some tried to integrate to find the area without using their </span><span style="font-family: times new roman,times; font-size: medium;">GDC. This became extremely complicated and time consuming.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (b), the chain rule was </span><span style="font-family: times new roman,times; font-size: medium;">not used by some.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates realized the relationship between the gradient and the </span><span style="font-family: times new roman,times; font-size: medium;">first derivative and set the two derivatives equal to one another. Once again many did not </span><span style="font-family: times new roman,times; font-size: medium;">realize that the intersection could be easily found on their GDC.</span></p>
<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 graph of \(y = (x - 1)\sin x\) , for \(0 \le x \le \frac{{5\pi }}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/exhausted.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph has \(x\)-intercepts at \(0\), \(1\), \( \pi\) and \(k\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>k</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is rotated \(360^\circ \) about the <em>x</em>-axis. Let <em>V</em> be the volume of the </span><span style="font-family: times new roman,times; font-size: medium;">solid formed.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down an expression for <em>V</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is rotated \(360^\circ \) about the <em>x</em>-axis. Let <em>V</em> be the volume of the </span><span style="font-family: times new roman,times; font-size: medium;">solid formed.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>V</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = 0\) , \(\sin x = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2\pi = 6.283185 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 6.28\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute either limits or the function into formula <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(accept absence of \({\rm{d}}x\) ) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \pi \int_\pi ^k {{{(f(x))}^2}{\rm{d}}x} \) , \(\pi \int {{{((x - 1)\sin x)}^2}} \) , \(\pi \int_\pi ^{6.28 \ldots } {{y^2}{\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A2 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi \int_\pi ^{6.28} {{{(x - 1)}^2}{{\sin }^2}x{\rm{d}}x} \) , \(\pi \int_\pi ^{2\pi } {{{((x - 1)\sin x)}^2}{\rm{d}}x} \) </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(V = 69.60192562 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = 69.6\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of d<em>x</em> , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of d<em>x</em> , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of d<em>x</em>, using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 4x - {{\rm{e}}^{x - 2}} - 3\) , for \(0 \le x \le 5\) .</span></p>
</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>-intercepts of the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chops.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the gradient of the graph of <em>f</em> at \(x = 3\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">intercepts when \(f(x) = 0\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(0.827, 0) (4.78, 0) (accept \(x = 0.827\), \(x = 4.78\) ) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sock.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for maximum point in circle, <em><strong>A1</strong></em> for <em>x</em>-intercepts in circles, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for correct shape (<em>y</em> approximately greater than \( - 3.14\)).</span></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">gradient is 1.28 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = a{x^3} + b{x^2} + c\) , where <em>a</em> , <em>b</em> and <em>c</em> are real numbers. The graph of <em>f</em> passes </span><span style="font-family: times new roman,times; font-size: medium;">through the point (2, 9) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(8a + 4b + c = 9\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> has a local minimum at \((1{\text{, }}4)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find two other equations in <em>a</em> , <em>b</em> and <em>c</em> , giving your answers in a similar form to </span><span style="font-family: times new roman,times; font-size: medium;">part (a).</span></p>
<p> </p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>a</em> , of <em>b</em> and of <em>c</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute coordinates in <em>f</em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(2) = 9\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a \times {2^3} + b \times {2^2} + c = 9\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(8a + 4b + c = 9\) <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \((1{\text{, }}4)\) is on the graph of <em>f</em> <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(1) = 4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a + b + c = 4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \(f' = 0\) at minimum (seen anywhere) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(1) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 3a{x^2} + 2bx\) (seen anywhere) <strong><em>A1A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into derivative <strong><em>(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3a \times {1^2} + 2b \times 1 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct simplified equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3a + 2b = 0\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid method for solving system of equations <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. inverse of a matrix, substitution </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 2\) , \(b = - 3\) , \(c = 5\) <em><strong>A1A1A1 N4</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally well done, with a few candidates failing to show a detailed substitution. Some substituted 2 in place of <em>x</em>, but didn't make it clear that they had substituted in <em>y</em> as well. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A great majority could find the two equations in part (b). However there were a significant number of candidates who failed to identify that the gradient of the tangent is zero at a minimum point, thus getting the incorrect equation \(3a + 2b = 4\) .</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A considerable number of candidates only had 2 equations, so that they either had a hard time trying to come up with a third equation (incorrectly combining some of the information given in the question) to solve part (c) or they completely failed to solve it. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite obtaining three correct equations many used long elimination methods that caused algebraic errors. Pages of calculations leading nowhere were seen. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Those who used matrix methods were almost completely successful. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the graph of \(f(x) = \frac{{{{\text{e}}^x}}}{{5x - 10}} + 3\), for \(x \ne 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(y\)-intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the vertical asymptote.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the minimum value of \(f(x)\) for \(x > 2\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(0)\), <img src="images/Schermafbeelding_2017-08-14_om_11.45.16.png" alt="M17/5/MATME/SP2/ENG/TZ1/03.a/M"></p>
<p>\(y\)-intercept is 2.9 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach involving equation or inequality <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(5x - 10 = 0,{\text{ }}2,{\text{ }}x \ne 2\)</p>
<p>\(x = 2\) (must be an equation) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>7.01710</p>
<p>\({\text{min value}} = 7.02\) <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p> </p>
<p><strong>Note:</strong> If candidate gives the minimum point as their final answer, award <strong><em>A1 </em></strong>for \((3,{\text{ }}7.02)\).</p>
<p> </p>
<p><strong><em>[</em></strong><strong><em>2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = {x^2} - 4x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> , for \( - 1 \le x \le 5\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This function can also be written as \(f(x) = {(x - p)^2} - 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>p </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is obtained by reflecting the graph of <em>f</em> in the <em>x</em>-axis, followed by a </span><span style="font-family: times new roman,times; font-size: medium;">translation of \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>6<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(g(x) = - {x^2} + 4x + 5\) .</span> </p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g </em>is obtained by reflecting the graph of <em>f </em>in the <em>x</em>-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>6<br>\end{array}} \right)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graphs of <em>f</em> and <em>g</em> intersect at two points.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-coordinates of these two points.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(g\) is obtained by reflecting the graph of \(f\) in the <em>x</em>-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 6 <br>\end{array}} \right)\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q9.jpg" alt> <em><strong>A1A1A1A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> The shape <strong>must</strong> be an approximately correct upwards parabola. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Only</strong> if the shape is approximately correct, award the following: </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for vertex \(x \approx 2\) , <em><strong>A1</strong></em> for <em>x</em>-intercepts between 0 and 1, and 3 and 4, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for correct <em>y</em>-intercept \((0{\text{, }}1)\), <em><strong>A1</strong></em> for correct domain \([ - 1{\text{, }}5]\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Scale not required on the axes, but approximate positions need to be clear. </span></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 2\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct vertical reflection, correct vertical translation <strong><em>(A1)(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - f(x)\) , \( - ({(x - 2)^2} - 3)\) , \( - y\) , \( - f(x) + 6\) , \(y + 6\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">transformations in correct order <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - ({x^2} - 4x + 1) + 6\) , \( - ({(x - 2)^2} - 3) + 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplification which clearly leads to given answer <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - {x^2} + 4x - 1 + 6\) , \( - ({x^2} - 4x + 4 - 3) + 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g(x) = - {x^2} + 4x + 5\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: If working shown, award <em><strong>A1A1A0A0</strong></em> if transformations correct, but done in reverse order, e.g. \( - ({x^2} - 4x + 1 + 6)\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, \(f = g\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 0.449489 \ldots \) , \(4.449489 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\((2 \pm \sqrt 6 )\) (exact), \( - 0.449{\text{ }}[ - 0.450{\text{, }} - 0.449]\) ; \(4.45{\text{ }}[4.44{\text{, }}4.45]\) <em><strong>A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute limits or functions into area formula (accept absence of \({\rm{d}}x\) ) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_a^b {(( - {x^2}} + 4x + 5) - ({x^2} - 4x + 1)){\rm{d}}x\) , \(\int_{4.45}^{ - 0.449} {(f - g)} \) , \(\int {( - 2{x^2}} + 8x + 4){\rm{d}}x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction of integrals/areas (accept absence of \({\rm{d}}x\) ) <strong><em> (M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_a^b {( - {x^2}} + 4x + 5) - \int_a^b {({x^2}} - 4x + 1)\) , \(\int {(f - g){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 39.19183 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 39.2\) \([39.1{\text{, }}39.2]\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of students provided a clear sketch of the quadratic function within the given domain. Some lost marks as they did not clearly indicate the approximate positions of the most important points of the parabola either by labelling or providing a suitable scale. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were few difficulties in part (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), candidates often used an insufficient number of steps to show the required result or had difficulty setting out their work logically. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was generally done well though many candidates gave at least one answer to fewer than three significant figures, potentially resulting in more lost marks. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (e), many candidates were unable to connect the points of intersection found in part (d) with the limits of integration. Mistakes were also made here either using a GDC incorrectly or not subtracting the correct functions. Other candidates tried to divide the region into four areas and made obvious errors in the process. Very few candidates subtracted \(f(x)\) from \(g(x)\) to get a simple function before integrating and there were numerous, fruitless analytical attempts to find the required integral.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2{x^2} - 8x - 9\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the coordinates of the vertex.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence or otherwise, express the function in the form \(f(x) = 2{(x - h)^2} + k\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \(f(x) = 0\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \((2{\text{, }} - 17)\) or \(x = 2\) , \(y = - 17\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. graph, completing the square, equating coefficients </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 2{(x - 2)^2} - 17\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. graph, quadratic formula </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 0.9154759 \ldots \) , \(4.915475 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = - 0.915\) , \(4.92\) <em><strong>A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by the majority of candidates. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by the majority of candidates. There were still many however who opted for an analytical approach in part (b), which often led to errors in sign and accuracy. Some candidates used the trace feature on their GDC to find the vertex which often resulted in accuracy errors. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \frac{{6{x^2} - 4}}{{{{\text{e}}^x}}}\), for \(0 \leqslant x \leqslant 7\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(x\)-intercept of the graph of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of \(f\) has a maximum at the point A. Write down the coordinates of A.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the following grid, sketch the graph of \(f\).</p>
<p><img src="images/Schermafbeelding_2018-02-12_om_11.45.37.png" alt="N17/5/MATME/SP2/ENG/TZ0/02.c"></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(x) = 0,{\text{ }} \pm 0.816\)</p>
<p>0.816496</p>
<p>\(x = \sqrt {\frac{2}{3}} \) (exact), 0.816 <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\((2.29099,{\text{ }}2.78124)\)</p>
<p>\({\text{A}}(2.29,{\text{ }}2.78)\) <strong><em>A1A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-12_om_12.46.59.png" alt="N17/5/MATME/SP2/ENG/TZ0/02.c/M"> <strong><em>A1A1A1 N3</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>A1 </em></strong>for correct domain and endpoints at \(x = 0\) and \(x = 7\) in circles,</p>
<p><strong><em>A1 </em></strong>for maximum in square,</p>
<p><strong><em>A1 </em></strong>for approximately correct shape that passes through <strong>their</strong> \(x\)-intercept in circle and has changed from concave down to concave up between 2.29 and 7.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{100}}{{(1 + 50{{\rm{e}}^{ - 0.2x}})}}\) . Part of the graph of \(f\) is shown below.</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(f(0)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \(f(x) = 95\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the range of \(f\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum rate of change of \(f\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(0) = \frac{{100}}{{51}}\) (exact), \(1.96\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">setting up equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(95 = \frac{{100}}{{1 + 50{{\rm{e}}^{ - 0.2x}}}}\) , sketch of graph with horizontal line at \(y = 95\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 34.3\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">upper bound of \(y\) is \(100\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">lower bound of \(y\) is \(0\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">range is \(0 < y < 100\) <strong><em>A1 N3 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting function ready to apply the chain rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(100{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 1}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct differentiation (must be substituted into chain rule) <strong><em>(A1)(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(u' = - 100{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 2}}\) , \(v' = (50{{\rm{e}}^{ - 0.2x}})( - 0.2)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct chain rule derivative <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = - 100{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 2}}(50{{\rm{e}}^{ - 0.2x}})( - 0.2)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working clearly leading to the required answer <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = 1000{{\rm{e}}^{ - 0.2x}}{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 2}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) <strong><em> AG N0</em> </strong></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to apply the quotient rule (accept reversed numerator terms) <strong><em> (M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{vu' - uv'}}{{{v^2}}}\) , \(\frac{{uv' - vu'}}{{{v^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct differentiation inside the quotient rule <strong><em>(A1)(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = \frac{{(1 + 50{{\rm{e}}^{ - 0.2x}})(0) - 100(50{{\rm{e}}^{ - 0.2x}} \times - 0.2)}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) , \(\frac{{100( - 10){{\rm{e}}^{ - 0.2x}} - 0}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">any correct expression for derivative (\(0\) may not be explicitly seen) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{ - 100(50{{\rm{e}}^{ - 0.2x}} \times - 0.2)}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working clearly leading to the required answer <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = \frac{{0 - 100( - 10){{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) , \(\frac{{ - 100( - 10){{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{{\rm{1000}}{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) <strong><em>AG N0 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">sketch of \(f'(x)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> </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;">recognizing maximum on \(f'(x)\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> dot on max of sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding maximum on graph of \(f'(x)\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> (\(19.6\), \(5\)) , \(x = 19.560 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum rate of increase is \(5\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD</strong></span><span style="font-family: times new roman,times; font-size: medium;"><strong> 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing \(f''(x) = 0\) <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding any correct expression for \(f''(x) = 0\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}( - 200{{\rm{e}}^{ - 0.2x}}) - (1000{{\rm{e}}^{ - 0.2x}})(2(1 + 50{{\rm{e}}^{ - 0.2x}})( - 10{{\rm{e}}^{ - 0.2x}}))}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(x = 19.560 \ldots \) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum rate of increase is \(5\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had little difficulty with parts (a), (b) and (c).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had little difficulty with parts (a), (b) and (c). Successful analytical approaches were often used in part (b) but again, this was not the most efficient or expected method.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had little difficulty with parts (a), (b) and (c). In part (c), candidates gained marks by correctly identifying upper and lower bounds but often did not express them properly using an appropriate notation.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), the majority of candidates opted to use the quotient rule and did so with some degree of competency, but failed to recognize the command term “show that” and consequently did not show enough to gain full marks. Approaches involving the chain rule were also successful but with the same point regarding sufficiency of work.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was poorly done as most were unable to interpret what was required. There were a few responses involving the use of the “trace” feature of the GDC which often led to inaccurate answers and a number of candidates incorrectly reported \(x = 19.6\) as their final answer. Some found the maximum value of \(f\) rather than \(f'\).</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \frac{{8x - 5}}{{cx + 6}}\) for \(x \ne - \frac{6}{c},\,\,c \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <em>x</em> = 3 is a vertical asymptote to the graph of <em>f</em>. Find the value of<em> c</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the horizontal asymptote to the graph of <em>f</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <em>y</em> = <em>k</em>, where \(k \in \mathbb{R}\) intersects the graph of \(\left| {f\left( x \right)} \right|\) at exactly one point. Find the possible values of <em>k</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg</em> \(cx + 6 = 0,\,\, - \frac{6}{c} = 3\)</p>
<p><em>c</em> = −2 <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg </em>\(\mathop {{\text{lim}}\,f}\limits_{x \to \infty } \left( x \right),\,\,y = \frac{8}{c}\)</p>
<p><em>y</em> = −4 (must be an equation) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to analyze modulus function <em><strong>(M1)</strong></em><br><em>eg</em> sketch, horizontal asymptote at <em>y</em> = 4, <em>y</em> = 0</p>
<p><em>k</em> = 4, <em>k</em> = 0 <em><strong>A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 2\ln (x - 3)\), for \(x > 3\). The following diagram shows part of the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_17.05.00.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the equation of the vertical asymptote to the graph of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-intercept of the graph of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The region enclosed by the graph of \(f\), the \(x\)-axis and the line \(x = 10\) <span class="s1">is rotated \(360\)° </span>about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)horizontal translation \(3\) units to the right</p>
<p class="p1">\(x = 3\) (must be an equation) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f(x) = 0,{\text{ }}{e^0} = x - 3\)</p>
<p class="p1">\(4,{\text{ }}x = 4,{\text{ }}(4,{\text{ }}0)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute either <strong>their correct </strong>limits or the function into formula involving \({f^2}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\int_4^{10} {{f^2},{\text{ }}\pi \int {{{\left( {2\ln (x - 3)} \right)}^2}{\text{d}}x} } \)</p>
<p class="p1">\(141.537\)</p>
<p class="p1">volume = \(142\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the region enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding intersection points <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = g(x)\) , \(\cos {x^2} = {{\rm{e}}^x}\) , sketch showing intersection</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = - 1.11\) , \(x = 0\) (may be seen as limits in the integral) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of approach involving integration and subtraction (in any order) <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{ - 1.11}^0 {\cos {x^2} - {{\rm{e}}^x}} \) , \(\int {(\cos {x^2} - {{\rm{e}}^x}){\rm{d}}x} \) , \(\int {g - f} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 0.282\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was poorly done by a great many candidates. Most seemed not to understand what was meant by the phrase "region enclosed by" as several candidates assumed that the limits of the integral were those given in the domain. Few realized what area was required, or that intersection points were needed. Candidates who used their GDCs to first draw a suitable sketch could normally recognize the required region and could find the intersection points correctly. However, it was disappointing to see the number of candidates who could not then use their GDC to find the required area or who attempted unsuccessful analytical approaches. </span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of \(f\) shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/paolo.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following four diagrams show <strong>images</strong> of <em>f</em> under different transformations.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/paolo2.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the <strong>same</strong> grid sketch the graph of \(y = f( - x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Complete the following table.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/britney.png" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of the transformation that gives the image in </span><span style="font-family: times new roman,times; font-size: medium;">Diagram A.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/paolo3.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A2 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/cheerios.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">translation (accept move/shift/slide etc.) with vector \(\left( {\begin{array}{*{20}{c}}<br>{ - 6}\\<br>{ - 2}<br>\end{array}} \right)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was reasonably solved by many students, though a good number confused \(f( - x)\) with \( - f(x)\) in part (a), thus reflecting the original diagram in the <em>x-</em>axis. Candidates need more practice in correctly and fully describing transformations. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates need more practice in correctly and fully describing transformations. There was often confusion between the description of the transformation and the equation that represents it. A fairly low percentage of the candidates used the term "translation". </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates need more practice in correctly and fully describing transformations. There was often confusion between the description of the transformation and the equation that represents it. A fairly low percentage of the candidates used the term "translation". </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Consider </span><span style="font-family: TimesNewRomanPS-ItalicMT;">\(f(x) = x\ln (4 - {x^2})\)</span><span style="font-family: TimesNewRomanPSMT;"> , for </span><span style="font-family: Times New Roman;" lang="JA">\( - 2 < x < 2\)</span><span style="font-family: TimesNewRomanPSMT;"> . The graph of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">f </span></em><span style="font-family: TimesNewRomanPSMT;">is given below.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/witch.png" alt></span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let P and Q be points on the curve of <em>f</em> where the tangent to the graph of <em>f</em> is </span><span style="font-family: times new roman,times; font-size: medium;">parallel to the <em>x</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the <em>x</em>-coordinate of P and of Q.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Consider \(f(x) = k\) . Write down all values of <em>k</em> for which there are </span><span style="font-family: times new roman,times; font-size: medium;">exactly two solutions.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(g'(x) = \frac{{ - 2{x^4}}}{{4 - {x^2}}} + 3{x^2}\ln (4 - {x^2})\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(g'\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider \(g'(x) = w\) . Write down all values of <em>w</em> for which there are exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></p>
<p align="LEFT"> </p>
<p align="LEFT"> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \( - 1.15{\text{, }}1.15\) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) recognizing that it occurs at P and Q <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = - 1.15\) , \(x = 1.15\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = - 1.13\) , \(k = 1.13\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the product rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(uv' + vu'\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of \({x^3}\) is \(3{x^2}\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of \(\ln (4 - {x^2})\) is \(\frac{{ - 2x}}{{4 - {x^2}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong> A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({x^3} \times \frac{{ - 2x}}{{4 - {x^2}}} + \ln (4 - {x^2}) \times 3{x^2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{{ - 2{x^4}}}{{4 - {x^2}}} + 3{x^2}\ln (4 - {x^2})\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/hill.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(w = 2.69\) , \(w < 0\) <em><strong>A1A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates correctly found the <em>x</em>-coordinates of P and Q in (a)(i) with their GDC. In </span><span style="font-family: times new roman,times; font-size: medium;">(a)(ii) some candidates incorrectly interpreted the words “exactly two solutions” as an </span><span style="font-family: times new roman,times; font-size: medium;">indication that the discriminant of a quadratic was required. Many failed to realise that the </span><span style="font-family: times new roman,times; font-size: medium;">values of <em>k</em> they were looking for in this question were the <em>y</em>-coordinates of the points found in </span><span style="font-family: times new roman,times; font-size: medium;">(a)(i).</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates were unclear in their application of the product formula in the verifying the </span><span style="font-family: times new roman,times; font-size: medium;">given derivative of <em>g</em>. Showing that the derivative was the given expression often received full </span><span style="font-family: times new roman,times; font-size: medium;">marks though it was not easy to tell in some cases if that demonstration came through </span><span style="font-family: times new roman,times; font-size: medium;">understanding of the product and chain rules or from reasoning backwards from the given </span><span style="font-family: times new roman,times; font-size: medium;">result.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Some candidates drew their graphs of the derivative in (c) on their examination papers despite </span><span style="font-family: times new roman,times; font-size: medium;">clear instructions to do their work on separate sheets. Most who tried to plot the graph in (c) </span><span style="font-family: times new roman,times; font-size: medium;">did so successfully.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Correct solutions to 10(d) were not often seen.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 0.225{x^3} - 2.7x\), for \( - 3 \leqslant x \leqslant 3\). There is a local minimum point at <span class="s1">A</span>.</p>
</div>
<div class="specification">
<p class="p1">On the following grid,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of <span class="s1">A</span><span class="s2">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>sketch the graph of \(f\), clearly indicating the point <span class="s1">A</span><span class="s2">;</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>sketch the tangent to the graph of \(f\) at <span class="s1">A</span>.</p>
<p class="p1" style="text-align: left;"><img src="images/Schermafbeelding_2017-03-03_om_17.19.46.png" alt="N16/5/MATME/SP2/ENG/TZ0/02.b"></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">\({\text{A }}(2,{\text{ }}-3.6)\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) (ii) <span class="Apple-converted-space"> <img src="images/Schermafbeelding_2017-03-03_om_17.24.16.png" alt="N16/5/MATME/SP2/ENG/TZ0/02.b/M"></span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>A1A1A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p2"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p3"> </p>
<p class="p1"><strong>Notes:</strong> (i) Award <span class="s1"><strong><em>A1</em></strong> </span>for correct cubic shape with correct curvature.</p>
<p class="p1">Only if this <span class="s1"><strong><em>A1 </em></strong></span>is awarded, award the following:</p>
<p class="p1"><span class="s1"><strong><em>A1 </em></strong></span>for passing through <strong>their </strong><span class="s2">point A </span>and the origin,</p>
<p class="p1"><span class="s1"><strong><em>A1 </em></strong></span>for endpoints,</p>
<p class="p1"><span class="s1"><strong><em>A1 </em></strong></span>for maximum.</p>
<p class="p1">(ii) Award <span class="s1"><strong><em>A1 </em></strong></span>for horizontal line through <strong>their </strong><span class="s2">A</span>.</p>
<p class="p3"> </p>
<p class="p2"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = x{{\text{e}}^{ - x}}\) and \(g(x) = - 3f(x) + 1\).</p>
<p class="p1">The graphs of \(f\) and \(g\) intersect at \(x = p\) and \(x = q\), where \(p < q\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(p\) <span class="s1">and of \(q\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, find the area of the region enclosed by the graphs of \(f\) <span class="s1">and \(g\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid attempt to find the intersection <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(f = g\)</span>, sketch, one correct answer</p>
<p class="p1">\(p = 0.357402,{\text{ }}q = 2.15329\)</p>
<p class="p2"><span class="s3">\(p = 0.357,{\text{ }}q = 2.15\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to set up an integral involving subtraction (in any order) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\int_p^q {\left[ {f(x) - g(x)} \right]{\text{d}}x,{\text{ }}} \int_p^q {f(x){\text{d}}x - } \int_p^q {g(x){\text{d}}x} \)</p>
<p class="p3">0.537667</p>
<p class="p2"><span class="s2">\({\text{area}} = 0.538\) <span class="Apple-converted-space"> </span></span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>f</em>(<em>x</em>) = ln <em>x</em> − 5<em>x</em> , for <em>x</em> > 0 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f '</em>(<em>x</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f "</em>(<em>x</em>).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve<em> f '</em>(<em>x</em>)<em> = f "</em>(<em>x</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(f'\left( x \right) = \frac{1}{x} - 5\) <em><strong>A1A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>f "</em>(<em>x</em>) = −<em>x</em><sup>−2 </sup> <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using GDC)</strong></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em><img src="data:image/png;base64,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"></p>
<p>0.558257</p>
<p><em>x</em> = 0.558 <em><strong>A1 N2</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if additional answers given.</p>
<p> </p>
<p><strong>METHOD 2 (analytical)</strong></p>
<p>attempt to solve their equation <em>f '(x) = f "</em>(<em>x</em>) (do not accept \(\frac{1}{x} - 5 = - \frac{1}{{{x^2}}}\)) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(5{x^2} - x - 1 = 0,\,\,\frac{{1 \pm \sqrt {21} }}{{10}},\,\,\frac{1}{x} = \frac{{ - 1 \pm \sqrt {21} }}{2},\,\, - 0.358\)</p>
<p>0.558257</p>
<p><em>x</em> = 0.558 <em><strong>A1 N2</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if additional answers given.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 6 - \ln ({x^2} + 2)\), for \(x \in \mathbb{R}\). The graph of \(f\) passes through the point \((p,{\text{ }}4)\), where \(p > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_13.30.18.png" alt="N17/5/MATME/SP2/ENG/TZ0/05.b"></p>
<p>The region enclosed by the graph of \(f\), the \(x\)-axis and the lines \(x = - p\) and \(x = p\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(p) = 4\), intersection with \(y = 4,{\text{ }} \pm 2.32\)</p>
<p>2.32143</p>
<p>\(p = \sqrt {{{\text{e}}^2} - 2} \) (exact), 2.32 <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <strong>either their</strong> limits <strong>or</strong> the function into volume formula (must involve \({f^2}\), accept reversed limits and absence of \(\pi \) and/or \({\text{d}}x\), but do not accept any other errors) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_{ - 2.32}^{2.32} {{f^2},{\text{ }}\pi \int {{{\left( {6 - \ln ({x^2} + 2)} \right)}^2}{\text{d}}x,{\text{ 105.675}}} } \)</p>
<p>331.989</p>
<p>\({\text{volume}} = 332\) <strong><em>A2 N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f\) and \(g\) be functions such that \(g(x) = 2f(x + 1) + 5\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(a) The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">vertical stretch by a factor of \(k\) , followed by a translation \(\left( \begin{array}{l}<br>p\\<br>q<br>\end{array} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \(k\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \(p\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) \(q\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = - g(3x)\) . The point A(\(6\), \(5\)) on the graph of \(g\) is mapped to the </span><span style="font-family: times new roman,times; font-size: medium;">point \({\rm{A}}'\) on the graph of \(h\) . Find \({\rm{A}}'\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">vertical stretch by a factor of \(k\) , followed by a translation \(\left( \begin{array}{l}<br>p\\<br>q<br>\end{array} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the value of</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \(k\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \(p\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) \(q\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = - g(3x)\) . The point A(\(6\), \(5\)) on the graph of \(g\) is mapped to the </span><span style="font-family: times new roman,times; font-size: medium;">point \({\rm{A}}'\) on the graph of \(h\) . Find \({\rm{A}}'\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) (i) \(k = 2\) <em><strong> A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(p = - 1\) <em><strong>A1 N1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(q = 5\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"><br></span></strong></em><span style="font-family: times new roman,times; font-size: medium;">(b) recognizing one transformation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> horizontal stretch by \(\frac{1}{3}\) , reflection in \(x\)-axis</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A'}}\) is (\(2\), \( - 5\)) <strong><em>A1A1 N3 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">Total [6 marks]<br></span></em></strong></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(k = 2\) <em><strong> A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(p = - 1\) <em><strong>A1 N1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(q = 5\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing one transformation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> horizontal stretch by \(\frac{1}{3}\) , reflection in \(x\)-axis</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{A'}}\) is (\(2\), \( - 5\)) <strong><em>A1A1 N3 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">Total [6 marks]<br></span></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was frequently done well but a lack of understanding of the notation \(f (x +1)\) often led to an incorrect value for \(p\). In part (b), candidates did not recognize the simplicity of the problem. Most seemed to be unable to correctly recognize the two transformations implied in the question and were thus unable to attempt a geometric solution. Flawed algebraic approaches to part (b) were common and many could not interpret the notation \(g(3x)\) as multiplying the \(x\)-value by \(\frac{1}{3}\).</span></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was frequently done well but a lack of understanding of the notation \(f (x +1)\) often led to an incorrect value for \(p\). In part (b), candidates did not recognize the simplicity of the problem. Most seemed to be unable to correctly recognize the two transformations implied in the question and were thus unable to attempt a geometric solution. Flawed algebraic approaches to part (b) were common and many could not interpret the notation \(g(3x)\) as multiplying the \(x\)-value by \(\frac{1}{3}\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was frequently done well but a lack of understanding of the notation \(f (x +1)\) often led to an incorrect value for \(p\). In part (b), candidates did not recognize the simplicity of the problem. Most seemed to be unable to correctly recognize the two transformations implied in the question and were thus unable to attempt a geometric solution. Flawed algebraic approaches to part (b) were common and many could not interpret the notation \(g(3x)\) as multiplying the \(x\)-value by \(\frac{1}{3}\).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{3x}}{{x - q}}\), where \(x \ne q\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the equations of the vertical and horizontal asymptotes of the graph of \(f\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(q\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The point \({\text{P}}(x,{\text{ }}y)\) lies on the graph of \(f\). Show that \({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the coordinates of the points on the graph of \(f\) that are closest to \((1,3)\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 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;">\(x = q,{\text{ }}y = 3\) (must be equations) <strong><em>A1A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing connection between point of intersection and asymptote <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(x = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = 1\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 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;">correct substitution into distance formula <strong><em>A1</em></strong></span></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;"><em>eg</em> \(\sqrt {{{(x - 1)}^2} + {{(y - 3)}^2}} \)</span></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;">attempt to substitute \(y = \frac{{3x}}{{x - 1}}\) <strong><em>(M1)</em></strong></span></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;"><em>eg</em> \(\sqrt {{{(x - 1)}^2} + {{\left( {\frac{{3x}}{{x - 1}} - 3} \right)}^2}} \)</span></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;">correct simplification of \(\left( {\frac{{3x}}{{x - 1}} - 3} \right)\) <em><strong>(A1)</strong></em></span></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;"><em>eg</em> \(\frac{{3x - 3x(x - 1)}}{{x - 1}}\)</span></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;">correct expression clearly leading to the required answer <strong><em>A1</em></strong></span></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;"><em>eg</em> \(\frac{{3x - 3x + 3}}{{x - 1}},{\text{ }}\sqrt {{{(x - 1)}^2} + {{\left( {\frac{{3x - 3x + 3}}{{x - 1}}} \right)}^2}} \)</span></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;">\({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \) <strong><em>AG N0</em></strong></span></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;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 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;">recognizing that closest is when \({\text{PQ}}\) is a minimum <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> sketch of \({\text{PQ}}\), \(({\text{PQ}})'(x) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = - 0.73205{\text{ }}x = 2.73205\) (seen anywhere) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find <em>y</em>-coordinates <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(f( - 0.73205)\)</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;">\((-0.73205, 1.267949) , (2.73205, 4.73205)\)</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;">\((-0.732, 1.27) , (2.73, 4.73) \) <strong><em>A1A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A particle P </span>moves along a straight line so that its velocity, \(v\,{\text{m}}{{\text{s}}^{ - 1}}\), after \(t\) seconds, is given by \(v = \cos 3t - 2\sin t - 0.5\)<span class="s1">, for \(0 \leqslant t \leqslant 5\). The initial displacement of P from a fixed point O is 4 </span>metres.</p>
</div>
<div class="specification">
<p class="p1">The following sketch shows the graph of \(v\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.51.25.png" alt="M16/5/MATME/SP2/ENG/TZ1/09.b+c+d+e"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the displacement of <span class="s1">P </span>from <span class="s1">O </span>after <span class="s1">5 </span>seconds.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find when <span class="s1">P </span>is first at rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the number of times <span class="s1">P </span>changes direction.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the acceleration of <span class="s1">P </span>after 3 seconds.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the maximum speed of <span class="s1">P</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">recognizing \(s = \int v \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">recognizing displacement of P in first 5 <span class="s1">seconds (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p2">(accept missing \({\text{d}}t\)<span class="s1">)</span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^5 {v{\text{d}}t,{\text{ }} - 3.71591} \)</p>
<p class="p1">valid approach to find total displacement <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(4 + ( - 3.7159),{\text{ }}s = 4 + \int_0^5 v \)</p>
<p class="p2">0.284086</p>
<p class="p2">0.284 (m) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">recognizing \(s = \int v \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">correct integration <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{3}\sin 3t + 2\cos t - \frac{t}{2} + c\) (do not penalize missing “\(c\)”)</p>
<p class="p1">attempt to find \(c\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(4 = \frac{1}{3}\sin (0) + 2\cos (0)--\frac{0}{2} + c,{\text{ }}4 = \frac{1}{3}\sin 3t + 2\cos t - \frac{t}{2} + c,{\text{ }}2 + c = 4\)</p>
<p class="p1">attempt to substitute \(t = 5\) into their expression with \(c\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(s(5),{\text{ }}\frac{1}{3}\sin (15) + 2\cos (5)5--\frac{5}{2} + 2\)</p>
<p class="p2">0.284086</p>
<p class="p2">0.284 (m) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing that at rest, \(v = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">\(t = 0.179900\)</p>
<p class="p1"><span class="s1">\(t = 0.180{\text{ (secs)}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing when change of direction occurs <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(v\) <span class="s1">crosses </span>\(t\) axis</p>
<p class="p1"><span class="s2">2 </span>(times) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">acceleration is \({v'}\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(v'(3)\)</p>
<p class="p2">0.743631</p>
<p class="p1"><span class="s1">\(0.744{\text{ }}({\text{m}}{{\text{s}}^{ - 2}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach involving max or min of \(v\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(v\prime = 0,{\text{ }}a = 0\)</span>, graph</p>
<p class="p1">one correct co-ordinate for min <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(1.14102,{\text{ }}-3.27876\)</p>
<p class="p1"><span class="s1">\(3.28{\text{ }}({\text{m}}{{\text{s}}^{ - 1}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. In (a), most candidates recognized the need to integrate \(v\) to find the displacement, although a significant number differentiated \(v\). Of those that integrated, many assumed incorrectly that the initial displacement was the value of the constant of integration. Some candidates integrated \(\left| v \right|\) and obtained no marks for an invalid approach. In the case where a correct definite integral was given, it was disappointing to see many candidates try to evaluate it analytically rather than using their GDC.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. In part (b), many candidates did not read the question carefully and gave the two occasions, in the given domain, where the particle was at rest.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. In part (c), many candidates did not appreciate that velocity is a vector and that the particle would change direction when its velocity changes sign. Consequently, many candidates gave the incorrect answer of four changes in directions, rather than the correct two direction changes.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. Part (d), was done very poorly, with candidates struggling to differentiate sine and cosine correctly and to evaluate their derivative. As with question 3, many candidates worked with the incorrect angle setting on their calculator.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. Few candidates attempted part (e). Of those that did, many attempted to find the largest local maximum of the graph rather than least local minimum as they did not recognise speed as \(\left| v \right|\).</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}(1 - {x^2})\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Part of the graph of \(y = f(x)\), for \( - 6 \le x \le 2\) , is shown below. The <em>x</em>-coordinates of the local minimum and maximum points are <em>r</em> and <em>s</em> respectively. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/aching.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) . </span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <strong>equation</strong> of the horizontal asymptote.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> and of <em>s</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>L</em> be the normal to the curve of <em>f</em> at \({\text{P}}(0{\text{, }}1)\) . Show that <em>L</em> has equation \(x + y = 1\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the curve \(y = f(x)\) and the line <em>L</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for the area of <em>R</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Calculate the area of <em>R</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using the product rule <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = {{\rm{e}}^x}(1 - {x^2}) + {{\rm{e}}^x}( - 2x)\) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \({{\rm{e}}^x}(1 - {x^2})\) , <em><strong>A1</strong></em> for \({{\rm{e}}^x}( - 2x)\) .</span></p>
<p align="LEFT"><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;">\(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 0\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">at the local maximum or minimum point</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 0\) \(({{\rm{e}}^x}(1 - 2x - {x^2}) = 0)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow 1 - 2x - {x^2} = 0\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(r = - 2.41\) \(s = 0.414\) <em><strong>A1A1 N2N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(0) = 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">gradient of the normal \(= - 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into an equation for a straight line <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y - 1 = - 1(x - 0)\) , \(y - 1 = - x\) , \(y = - x + 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x + y = 1\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) intersection points at \(x = 0\) and \(x = 1\) (may be seen as the limits) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction and integrals <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">fully correct expression <em><strong>A2 N4</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^1 {\left( {{{\rm{e}}^x}(1 - {x^2}) - (1 - x)} \right)} {\rm{d}}x\) , \(\int_0^1 {f(x){\rm{d}}x - \int_0^1 {(1 - x){\rm{d}}x} } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) area \(R = 0.5\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates clearly applied the product rule to correctly show the given derivative. Some </span><span style="font-family: times new roman,times; font-size: medium;">candidates missed the multiplicative nature of the function and attempted to apply a chain rule </span><span style="font-family: times new roman,times; font-size: medium;">instead.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For part (b), the equation of the horizontal asymptote was commonly written as \(x = 0\) .</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Although part (c) was a “write down” question where no working is required, a good number of </span><span style="font-family: times new roman,times; font-size: medium;">candidates used an algebraic method of solving for <em>r</em> and <em>s</em> which sometimes returned </span><span style="font-family: times new roman,times; font-size: medium;">incorrect answers. Those who used their GDC usually found correct values, although not </span><span style="font-family: times new roman,times; font-size: medium;">always to three significant figures.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (d), many candidates showed some skill showing the equation of a normal, although </span><span style="font-family: times new roman,times; font-size: medium;">some tried to work with the gradient of the tangent.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Surprisingly few candidates set up a completely correct expression for the area between </span><span style="font-family: times new roman,times; font-size: medium;">curves that considered both integration and the correct subtraction of functions. Using limits </span><span style="font-family: times new roman,times; font-size: medium;">of \( - 6\) and 2 was a common error, as was integrating on \(f(x)\) alone. Where candidates did </span><span style="font-family: times new roman,times; font-size: medium;">write a correct expression, many attempted to perform analytic techniques to calculate the </span><span style="font-family: times new roman,times; font-size: medium;">area instead of using their GDC.</span></p>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {\log _3}\frac{x}{2} + {\log _3}16 - {\log _3}4\) , for \(x > 0\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f(x) = {\log _3}2x\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(f(0.5)\) and of \(f(4.5)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function <em>f</em> can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>a</em> and of <em>b</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence on graph paper, <strong>sketch</strong> the graph of <em>f</em> , for \( - 5 \le x \le 5\) , \( - 5 \le y \le 5\) , </span><span style="font-family: times new roman,times; font-size: medium;">using a scale of 1 cm to 1 unit on each axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the equation of the asymptote.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \({f^{ - 1}}(0)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point A lies on the graph of <em>f</em> . At A, \(x = 4.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">On your diagram, sketch the graph of \({f^{ - 1}}\) , noting clearly the image of point A.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">combining 2 terms <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _3}8x - {\log _3}4\) , \({\log _3}\frac{1}{2}x + {\log _3}4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">expression which clearly leads to answer given <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _3}\frac{{8x}}{4}\) , \({\log _3}\frac{{4x}}{2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = {\log _3}2x\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute either value into <em>f</em> <strong> <em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _3}1\) , \({\log _3}9\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(0.5) = 0\) , \(f(4.5) = 2\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a = 2\) , \(b = 3\) <em><strong>A1A1 N1N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii)<br><img src="images/doc.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for sketch approximately through \((0.5 \pm 0.1{\text{, }}0 \pm 0.1)\) , </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for approximately correct shape, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for sketch asymptotic to the <em>y</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \(x = 0\) (must be an equation) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}(0) = 0.5\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/home.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1A1 N4</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for sketch approximately through \((0 \pm 0.1{\text{, }}0.5 \pm 0.1)\) , </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for approximately correct shape of the graph reflected over \(y = x\) , </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for sketch asymptotic to <em>x</em>-axis, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for point \((2 \pm 0.1{\text{, }}4.5 \pm 0.1)\) clearly marked and on curve.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few candidates had difficulty with part (a) although it was often communicated using some very sloppy applications of the rules of logarithm, writing \(\frac{{\log 16}}{{\log 4}}\) instead of \(\log \left( {\frac{{16}}{4}} \right)\) . </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) was generally done well.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) (i) was generally done well; candidates seemed quite comfortable changing bases. There were some very good sketches in (c) (ii), but there were also some very poor ones with candidates only considering shape and not the location of the <em>x</em>-intercept or the asymptote. A surprising number of candidates did not use the scale required by the question and/or did not use graph paper to sketch the graph. In some cases, it was evident that students simply transposed their graphs from their GDC without any analytical consideration. </span></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was poorly done as candidates did not consider the command term, “write down” and often proceeded to find the inverse function before making the appropriate substitution.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) eluded a great many candidates as most preferred to attempt to find the inverse analytically rather than simply reflecting the graph of <em>f</em> in the line \(y = x\) . This graph also suffered from the same sort of problems as the graph in (c) (ii). Some students did not have their curve passing through \((2{\text{, }}4.5)\) nor did they clearly indicate its position as instructed. This point was often mislabelled on the graph of <em>f</em>. The efforts in this question demonstrated that students often work tenuously from one question to the next, without considering the "big picture", thereby failing to make important links with earlier parts of the question.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2{x^2} + 4x - 6\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express \(f(x)\) in the form \(f(x) = 2{(x - h)^2} + k\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the equation of the axis of symmetry of the graph of <em>f</em> .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express \(f(x)\) in the form \(f(x) = 2(x - p)(x - q)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of obtaining the vertex <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. a graph, \(x = - \frac{b}{{2a}}\) , completing the square</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 2{(x + 1)^2} - 8\) <em><strong>A2 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[3 marks]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = - 1\) (equation must be seen) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 2(x - 1)(x + 3)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p align="LEFT"><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates answered this question with great ease. Still, some found themselves unable to correctly find the vertex algebraically, often mixing the signs of the <em>h</em> and <em>k</em> values. Using the GDC may have been a more fruitful approach. Some candidates did not write the axis of symmetry as an equation. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates answered this question with great ease. Still, some found themselves unable to correctly find the vertex algebraically, often mixing the signs of the <em>h</em> and <em>k</em> values. Using the GDC may have been a more fruitful approach. Some candidates did not write the axis of symmetry as an equation. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates answered this question with great ease. Still, some found themselves unable to correctly find the vertex algebraically, often mixing the signs of the <em>h</em> and <em>k</em> values. Using the GDC may have been a more fruitful approach. Some candidates did not write the axis of symmetry as an equation. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a quadrilateral ABCD with obtuse angles \({\rm{A}}\widehat {\rm{B}}{\rm{C}}\) and \({\rm{A}}\widehat {\rm{D}}{\rm{C}}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/footie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">AB = 5 cm, BC = 4 cm, CD = 4 cm, AD = 4 cm , \({\rm{B}}\widehat {\rm{A}}{\rm{C}} = {30^ \circ }\) , \({\rm{A}}\widehat {\rm{B}}{\rm{C}} = {x^ \circ }\) , \({\rm{A}}\widehat {\rm{D}}{\rm{C}} = {y^ \circ }\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the cosine rule to show that \({\rm{AC}} = \sqrt {41 - 40\cos x} \) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the sine rule in triangle ABC to find another expression for AC.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Hence, find <em>x</em>, giving your answer to two decimal places.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find AC .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find <em>y</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, or otherwise, find the area of triangle ACD.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(25 + 16 - 40\cos x\) , \({5^2} + {4^2} - 2 \times 4 \times 5\cos x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AC}} = \sqrt {41 - 40\cos x} \) <em><strong>AG</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{AC}}}}{{\sin x}} = \frac{4}{{\sin 30}}\) , \(\frac{1}{2}{\rm{AC}} = 4\sin x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AC}} = 8\sin x\) (accept \(\frac{{4\sin x}}{{\sin 30}}\)) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of appropriate approach using AC <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(8\sin x = \sqrt {41 - 40\cos x} \) , sketch showing intersection </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct solution \(8.682 \ldots \), \(111.317 \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">obtuse value \(111.317 \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 111.32\) to 2 dp (do <strong>not</strong> accept the radian answer 1.94 ) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) substituting value of <em>x</em> into either expression for AC <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{AC}} = 8\sin 111.32\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{AC}} = 7.45\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of choosing cosine rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos B = \frac{{{a^2} + {c^2} - {b^2}}}{{2ac}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{4^2} + {4^2} - {{7.45}^2}}}{{2 \times 4 \times 4}}\) , \({7.45^2} = 32 - 32\cos y\) , \(\cos y = - 0.734 \ldots \)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(y = 137\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into area formula <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2} \times 4 \times 4 \times \sin 137\) , \(8\sin 137\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> area \(= 5.42\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates worked comfortably with the sine and cosine rules in part (a) and (b).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates worked comfortably with the sine and cosine rules in part (a) and (b). </span><span style="font-family: times new roman,times; font-size: medium;">Equally as many did not take the cue from the word "hence" and used an alternate method to </span><span style="font-family: times new roman,times; font-size: medium;">solve the problem and thus did not receive full marks. Those who managed to set up an </span><span style="font-family: times new roman,times; font-size: medium;">equation, again did not go directly to their GDC but rather engaged in a long, laborious </span><span style="font-family: times new roman,times; font-size: medium;">analytical approach that was usually unsuccessful.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Equally as many did not take the cue from the word "hence" and used an alternate method to solve the problem and thus did not receive full marks. Those who managed to set up an equation, again did not go directly to their GDC but rather engaged in a long, laborious analytical approach that was usually unsuccessful. No matter what values were found in (c) (i) most candidates recovered and earned follow through marks for the remainder of the question. A large number of candidates worked in the wrong mode and rounded prematurely throughout this question often resulting in accuracy penalties.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Equally as many did not take the cue from the word "hence" and used an alternate method to solve the problem and thus did not receive full marks. Those who managed to set up an equation, again did not go directly to their GDC but rather engaged in a long, laborious analytical approach that was usually unsuccessful. No matter what values were found in (c) (i) most candidates recovered and earned follow through marks for the remainder of the question. A large number of candidates worked in the wrong mode and rounded prematurely throughout this question often resulting in accuracy penalties.</span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{{2x - 1}}{{x + 1}}\) , \(x \ne - 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({h^{ - 1}}(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of <em>h</em> for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any </span><span style="font-family: times new roman,times; font-size: medium;">asymptotes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the equations of the asymptotes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the <em>x</em>-intercept of the graph of <em>h</em> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region in the first quadrant enclosed by the graph of <em>h</em> , the <em>x</em>-axis </span><span style="font-family: times new roman,times; font-size: medium;">and the line \(x = 3\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the area of <em>R</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down an expression for the volume obtained when <em>R</em> is revolved </span><span style="font-family: times new roman,times; font-size: medium;">through \({360^ \circ }\) about the <em>x</em>-axis.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = \frac{{2x - 1}}{{x + 1}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">interchanging <em>x</em> and <em>y</em> (seen anywhere) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = \frac{{2y - 1}}{{y + 1}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(xy + x = 2y - 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">collecting terms <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x + 1 = 2y - xy\) , \(x + 1 = y(2 - x)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({h^{ - 1}}(x) = \frac{{x + 1}}{{2 - x}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/piper.png" alt></span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1A1A1 N4</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct intercepts, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for correct shape, <em><strong>A1</strong></em> for asymptotes, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for approximately correct domain and range.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(x = - 1\) , \(y = 2\) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \(\frac{1}{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\text{area}} = 2.06\) <em><strong>A2 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) attempt to substitute into volume formula (do not accept \(\pi \int_a^b {{y^2}{\rm{d}}x} \) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">volume \( = \pi {\int_{\frac{1}{2}}^3 {\left( {\frac{{2x - 1}}{{x + 1}}} \right)} ^2}{\rm{d}}x\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \ln x\) and \(g(x) = 3 + \ln \left( {\frac{x}{2}} \right)\), for \(x > 0\).</p>
<p>The graph of \(g\) can be obtained from the graph of \(f\) by two transformations:</p>
<p>\[\begin{array}{*{20}{l}} {{\text{a horizontal stretch of scale factor }}q{\text{ followed by}}} \\ {{\text{a translation of }}\left( {\begin{array}{*{20}{c}} h \\ k \end{array}} \right).} \end{array}\]</p>
</div>
<div class="specification">
<p>Let \(h(x) = g(x) \times \cos (0.1x)\), for \(0 < x < 4\). The following diagram shows the graph of \(h\) and the line \(y = x\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.34.27.png" alt="M17/5/MATME/SP2/ENG/TZ1/10.b.c"></p>
<p>The graph of \(h\) intersects the graph of \({h^{ - 1}}\) at two points. These points have \(x\) coordinates 0.111 and 3.31 correct to three significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(q\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(h\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(k\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\). There is a point \({\text{P}}(a,{\text{ }}b)\) on the graph of \(h\) where \(d\) is a maximum.</p>
<p>Find the coordinates of P, where \(0.111 < a < 3.31\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(q = 2\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(q = 1\), \(h = 0\), and \(k = 3 - \ln (2)\), 2.31 as candidate may have rewritten \(g(x)\) as equal to \(3 + \ln (x) - \ln (2)\).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(h = 0\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(q = 1\), \(h = 0\), and \(k = 3 - \ln (2)\), 2.31 as candidate may have rewritten \(g(x)\) as equal to \(3 + \ln (x) - \ln (2)\).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(k = 3\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(q = 1\), \(h = 0\), and \(k = 3 - \ln (2)\), 2.31 as candidate may have rewritten \(g(x)\) as equal to \(3 + \ln (x) - \ln (2)\).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2.72409</p>
<p>2.72 <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing area between \(y = x\) and \(h\) equals 2.72 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)<img src="images/Schermafbeelding_2017-08-14_om_17.00.04.png" alt="M17/5/MATME/SP2/ENG/TZ1/10.b.ii/M"></p>
<p>recognizing graphs of \(h\) and \({h^{ - 1}}\) are reflections of each other in \(y = x\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)area between \(y = x\) and \(h\) equals between \(y = x\) and \({h^{ - 1}}\)</p>
<p>\(2 \times 2.72\int_{0.111}^{3.31} {\left( {x - {h^{ - 1}}(x)} \right){\text{d}}x = 2.72} \)</p>
<p>5.44819</p>
<p>5.45 <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find \(d\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)difference in \(y\)-coordinates, \(d = h(x) - x\)</p>
<p>correct expression for \(d\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\ln \frac{1}{2}x + 3} \right)(\cos 0.1x) - x\)</p>
<p>valid approach to find when \(d\) is a maximum <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)max on sketch of \(d\), attempt to solve \(d’ = 0\)</p>
<p>0.973679</p>
<p>\(x = 0.974\) <strong><em>A2 N4 </em></strong></p>
<p>substituting <strong>their</strong> \(x\) value into \(h(x)\) <strong><em>(M1)</em></strong></p>
<p>2.26938</p>
<p>\(y = 2.27\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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 \(f(x) = 3\sin x + 4\cos x\) , for \( - 2\pi \le x \le 2\pi \) .</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;">Sketch the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the amplitude;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) the <em>x</em>-intercept that lies between \( - \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">and 0.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</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 \(f(x)\) in the form \(p\sin (qx + r)\) .</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;">Write down one value of <em>x</em> such that \(f'(x) = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the two values of <em>k</em> for which the equation \(f(x) = k\) has exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</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 \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of <em>x</em>, between \(0\) and \(1\), </span><span style="font-family: times new roman,times; font-size: medium;">for which the gradient of <em>f</em> is equal to the gradient of <em>g</em>. Find this value of <em>x</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/mike.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></p>
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>A1</strong></em> for approximately sinusoidal shape, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for end points approximately correct \(( - 2\pi {\text{, }}4)\) \((2\pi {\text{, }}4)\), </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for approximately correct position of graph, (<em>y</em>-intercept \((0{\text{, }}4)\), maximum to right of <em>y</em>-axis). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 5 <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(2\pi \) (6.28) <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \( - 0.927\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 5\sin (x + 0.927)\) (accept \(p = 5\) , \(q = 1\) , \(r = 0.927\) ) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. max/min, sketch of \(f'(x)\) indicating roots </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/wee.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">one 3 s.f. value which rounds to one of \( - 5.6\), \( - 2.5\), \(0.64\), \(3.8\) <em><strong>A1 N2 </strong></em></span></p>
<p> </p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = - 5\) , \(k = 5\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graphical approach (but must involve derivative functions) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/visit.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">each curve <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 0.511\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{1}{{x + 1}}\) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 3\cos x - 4\sin x\) \((5\cos (x + 0.927))\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to solve \(g'(x) = f'(x)\) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 0.511\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A2 N2</strong> </span></em></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[5 marks]</strong> </span></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some graphs in part (a) were almost too detailed for just a sketch but more often, the important features were far from clear. Some graphs lacked scales on the axes. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A number of candidates had difficulty finding the period in part (b)(ii).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A number of candidates had difficulty writing the correct value of <em>q</em> in part (c). </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The most common approach in part (d) was to differentiate and set \(f'(x) = 0\) . Fewer students found the values of <em>x</em> given by the maximum or minimum values on their graphs. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) proved challenging for many candidates, although if candidates answered this part, they generally did so correctly. </span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (f), many candidates were able to get as far as equating the two derivatives but fewer used their GDC to solve the resulting equation. Again, many had trouble demonstrating their method of solution. </span></p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3{x^2}\) . The graph of <em>f</em> is translated 1 unit to the right and 2 units down. </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is the image of the graph of <em>f</em> after this translation.</span></p>
</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 the vertex of the graph of <em>g</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express <em>g</em> in the form \(g(x) = 3{(x - p)^2} + q\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</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>h</em> is the reflection of the graph of <em>g</em> in the <em>x</em>-axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the coordinates of the vertex of the graph of <em>h</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((1{\text{, }} - 2)\) <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(g(x) = 3{(x - 1)^2} - 2\) (accept \(p = 1\) , \(q = - 2\) ) <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((1{\text{, }}2)\) <em><strong>A1A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates had little difficulty with this question. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates had little difficulty with this question. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates had little difficulty with this question. In part (c), a few reflected the vertex in the <em>y</em>-axis rather than the <em>x</em>-axis. </span></p>
<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 \(f(x) = A{{\rm{e}}^{kx}} + 3\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ryan.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>y</em>-intercept is at (0, 13) .</span></p>
</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 \(A = 10\) .</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 \(f(15) = 3.49\) (correct to 3 significant figures), find the value of <em>k</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</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) Using your value of <em>k</em> , find \(f'(x)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, explain why <em>f</em> is a decreasing function.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the equation of the horizontal asymptote of the graph <em>f</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = - {x^2} + 12x - 24\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">substituting (0, 13) into function <em><strong>M1 </strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(13 = A{{\rm{e}}^0} + 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(13 = A + 3\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(A = 10\) <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">substituting into \(f(15) = 3.49\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3.49 = 10{{\rm{e}}^{15k}} + 3\) , \(0.049 = {{\rm{e}}^{15k}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of solving equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, using \(\ln \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = - 0.201\) (accept \(\frac{{\ln 0.049}}{{15}}\) ) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(f(x) = 10{{\rm{e}}^{ - 0.201x}} + 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 10{{\rm{e}}^{ - 0.201x}} \times - 0.201\) \(( = - 2.01{{\rm{e}}^{ - 0.201x}})\) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \(10{{\rm{e}}^{ - 0.201x}}\) , <em><strong>A1</strong></em> for \( \times - 0.201\) , <em><strong>A1</strong></em> for the derivative of 3 is zero. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) valid reason with reference to derivative <em><strong>R1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) < 0\) , derivative always negative </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(y = 3\) <em><strong> A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">finding limits \(3.8953 \ldots \), \(8.6940 \ldots \) (seen anywhere) <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of integrating and subtracting functions <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{3.90}^{8.69} {g(x) - f(x){\rm{d}}x} \) , \(\int_{3.90}^{8.69} {\left[ {\left( { - {x^2} + 12x - 24} \right) - \left( {10{{\rm{e}}^{ - 0.201x}} + 3} \right)} \right]} {\rm{d}}x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \(= 19.5\) <em><strong>A2 N4</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question was quite well done by a great number of candidates indicating that calculus is </span><span style="font-family: times new roman,times; font-size: medium;">a topic that is covered well by most centres. Parts (a) and (b) proved very accessible to many </span><span style="font-family: times new roman,times; font-size: medium;">candidates.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question was quite well done by a great number of candidates indicating that calculus is </span><span style="font-family: times new roman,times; font-size: medium;">a topic that is covered well by most centres. Parts (a) and (b) proved very accessible to many </span><span style="font-family: times new roman,times; font-size: medium;">candidates.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The chain rule in part (c) was also carried out well. Few however, recognized the </span><span style="font-family: times new roman,times; font-size: medium;">command term “hence” and that </span><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) < 0\) guarantees a decreasing function. A common </span><span style="font-family: times new roman,times; font-size: medium;">answer for the equation of the asymptote was to give </span><span style="font-family: times new roman,times; font-size: medium;">\(y = 0\) or </span><span style="font-family: times new roman,times; font-size: medium;">\(x = 3\) .</span></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (d), it was again </span><span style="font-family: times new roman,times; font-size: medium;">surprising and somewhat disappointing to see how few candidates were able to use their GDC </span><span style="font-family: times new roman,times; font-size: medium;">effectively to find the area between curves, often not finding correct limits, and often trying to </span><span style="font-family: times new roman,times; font-size: medium;">evaluate the definite integral without the GDC, which led nowhere.</span></p>
<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 \(f(x) = \frac{{20x}}{{{{\rm{e}}^{0.3x}}}}\) , for \(0 \le x \le 20\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> .</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;">(i) Write down the <em>x</em>-coordinate of the maximum point on the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the interval where <em>f</em> is increasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the interval where the rate of change of <em>f</em> is increasing.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/phoebe.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape with inflexion/change of curvature, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for maximum skewed to the left, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong> </em>for asymptotic behaviour to the right.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(x = 3.33\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct interval, with right end point \(3\frac{1}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1 N2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0 < x \le 3.33\) , \(0 \le x < 3\frac{1}{3}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept any inequalities in the right direction.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. quotient rule, product rule</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">2 correct derivatives (must be seen in product or quotient rule) <em><strong>(A1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(20\) , \(0.3{{\rm{e}}^{0.3x}}\) or \( - 0.3{{\rm{e}}^{ - 0.3x}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into product or quotient rule <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{20{{\rm{e}}^{0.3x}} - 20x(0.3){{\rm{e}}^{0.3x}}}}{{{{({{\rm{e}}^{0.3x}})}^2}}}\) , \(20{{\rm{e}}^{ - 0.3x}} + 20x( - 0.3){{\rm{e}}^{ - 0.3x}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{20{{\rm{e}}^{0.3x}} - 6x{{\rm{e}}^{0.3x}}}}{{{{\rm{e}}^{0.6x}}}}\) , \(\frac{{{{\rm{e}}^{0.3x}}(20 - 20x(0.3))}}{{{{{\rm{(}}{{\rm{e}}^{0.3x}})}^2}}}\) , \({{\rm{e}}^{ - 0.3x}}(20 + 20x( - 0.3))\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">consideration of \(f'\) or \(f''\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(f'\) , \(f''\) is positive, \(f'' = 0\) , reference to minimum of \(f'\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct value \(6.6666666 \ldots \) \(\left( {6\frac{2}{3}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct interval, with <strong>both</strong> endpoints <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6.67 < x \le 20\) , \(6\frac{2}{3} \le x < 20\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates earned the first four marks of the question in parts (a) and (b) for correctly using their GDC to graph and find the maximum value. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates earned the first four marks of the question in parts (a) and (b) for correctly using their GDC to graph and find the maximum value. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most had a valid approach in part (c) using either the quotient or product rule, but many had difficulty applying the chain rule with a function involving e and simplifying. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was difficult for most candidates. Although many associated rate of change with derivative, only the best-prepared students had valid reasoning and could find the correct interval with both endpoints. </span></p>
<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 \(f(x) = {{\rm{e}}^{\frac{x}{4}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(g(x) = mx\) , where \(m \ge 0\) , and \( - 5 \le x \le 5\) . Let \(R\) be the region </span><span style="font-family: times new roman,times; font-size: medium;">enclosed by the \(y\)-axis, the graph of \(f\) , and the graph of \(g\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(m = 1\).</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;">(i) Sketch the graphs of \(f\) and \(g\) on the same axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the area of \(R\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of \(R\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the </span><span style="font-family: times new roman,times; font-size: medium;">value of \(m\) that gives the greatest value for the area of \(R\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong> (i)</strong></span></em></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Award <em><strong>A1</strong></em> for the graph of \(f\) positive, increasing and concave up. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> Award <strong><em>A1</em></strong> for graph of \(g\) increasing and linear with \(y\)-intercept of \(0\). </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> Penalize one mark if domain is not [\( - 5\), \(5\)] and/or if \(f\) and \(g\) do not intersect in the first quadrant. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<p> </p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"><br>(ii)<br>attempt to find intersection of the graphs of \(f\) and \(g\) (M1)<br><br>eg \({{\rm{e}}^{\frac{x}{4}}} = x\)<br><br>\(x = 1.42961 \ldots \) A1<br><br>valid attempt to find area of \(R\) (M1)<br><br>eg \(\int {(x - {{\rm{e}}^{\frac{x}{4}}}} ){\rm{d}}x\) , \(\int_0^1 {(g - f)} \) , \(\int {(f - g)} \)<br><br>area \( = 0.697\) A2 N3<br><br><br>[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find intersection of the graphs of \(f\) and \(g\)<strong> <em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({{\rm{e}}^{\frac{x}{4}}} = x\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 1.42961 \ldots \) </span><strong><span style="font-family: times new roman,times; font-size: medium;"><em>A1</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid attempt to find area of \(R\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\int {(x - {{\rm{e}}^{\frac{x}{4}}}} ){\rm{d}}x\) , \(\int_0^1 {(g - f)} \) , \(\int {(f - g)} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \( = 0.697\) <em><strong>A2 N3 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[5 marks]<br></strong></em></span></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognize that area of \(R\) is a maximum at point of tangency <strong><em>(R1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(m = f'(x)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equating functions <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f(x) = g(x)\) , \({{\rm{e}}^{\frac{x}{4}}} = mx\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{1}{4}{{\rm{e}}^{\frac{x}{4}}}\) <strong><em> (A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equating gradients <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = g'(x)\) , \(\frac{1}{4}{{\rm{e}}^{\frac{x}{4}}} = m\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve system of two equations for \(x\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{1}{4}{{\rm{e}}^{\frac{x}{4}}} \times x = {{\rm{e}}^{\frac{x}{4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 4\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \(m\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(4)\) , \(\frac{1}{4}{{\rm{e}}^{\frac{4}{4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(m = \frac{1}{4}e\) (exact), \(0.680\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3 </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a flaw with the domain noted in this question. While not an error in itself, it meant that part (b) no longer assessed what was intended. The markscheme included a variety of solutions based on candidate work seen, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were looked at during the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">While some candidates sketched accurate graphs on the given domain, the majority did not. Besides the common domain error, some exponential curves were graphed with several concavity changes.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a flaw with the domain noted in this question. While not an error in itself, it meant that part (b) no longer assessed what was intended. The markscheme included a variety of solutions based on candidate work seen, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were looked at during the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">In part (a)(ii), most candidates found the intersection correctly. Those who used their GDC to evaluate the integral numerically were usually successful, unlike those who attempted to solve with antiderivatives. A common error was to find the area of the region enclosed by \(f\) and \(g\) (although it involved a point of intersection outside of the given domain), rather than the area of the region enclosed by \(f\) and \(g\) and the \(y\)-axis.<br></span></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a flaw with the domain noted in this question. While not an error in itself, it meant that part (b) no longer assessed what was intended. The markscheme included a variety of solutions based on candidate work seen, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were looked at during the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">While some candidates were able to show some good reasoning in part (b), fewer were able to find the value of \(m\) which maximized the area of the region. In addition to the answer obtained from the restricted domain, full marks were awarded for the answer obtained by using the point of tangency.<br></span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(G(x) = 95{{\text{e}}^{( - 0.02x)}} + 40\), for \(20 \le x \le 200\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On the following grid, sketch the graph of \(G\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_05.12.49.png" alt></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Robin and Pat are planning a wedding banquet. The cost per guest, \(G\) dollars, is modelled by the function \(G(n) = 95{{\text{e}}^{( - 0.02n)}} + 40\), for \(20 \le n \le 200\), where \(n\) is the number of guests.</p>
<p class="p1">Calculate the <strong>total</strong> cost for \(45\) guests.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space"><img src="images/Schermafbeelding_2016-01-14_om_05.21.36.png" alt> </span><strong><em>A1A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Curve must be approximately correct exponential shape (concave up and decreasing). Only if the shape is approximately correct, award the following:</p>
<p class="p1"><strong><em>A1</em></strong> for left endpoint in circle,</p>
<p class="p1"><strong><em>A1</em></strong> for right endpoint in circle,</p>
<p class="p1"><strong><em>A1</em></strong> for asymptotic to \(y = 40\) (must not go below \(y = 40\)).</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to find \(G(45)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;78.6241\), value read from <strong>their </strong>graph</p>
<p class="p1">multiplying cost times number of people <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;45 \times 78.6241,{\text{ }}G(45) \times 45\)</p>
<p class="p1">\(3538.08\)</p>
<p class="p1">\(3540\) (dollars) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<p class="p1"> </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority of candidates were able to sketch the shape of the graph accurately, but graph sketching is an area of the syllabus in which candidates continue to lose marks. In this particular question, candidates often did not consider the given domain or failed to accurately show the behaviour of the graph close to the horizontal asymptote as \(x \to \infty \).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (b), most candidates were able to identify the initial approach by finding \(G(45)\), but missed the fact that function defined the cost per guest and not the total cost.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2} + 2x + 1\) and \(g(x) = x - 5\), for \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f(8)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \((g \circ f)(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve \((g \circ f)(x) = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute \(x = 8\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({8^2} + 2 \times 8 + 1\)</p>
<p class="p2"><span class="Apple-converted-space">\(f(8) = 81\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to form composition (in any order) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(f(x - 5),{\text{ }}g\left( {f(x)} \right),{\text{ }}\left( {{x^2} + 2x + 1} \right) - 5\)</p>
<p class="p3"><span class="s2">\((g \circ f)(x) = {x^2} + 2x - 4\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p3"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em> <span class="Apple-converted-space"> \(x = \frac{{ - 2 \pm \sqrt {20} }}{2}\), <img src="images/Schermafbeelding_2017-03-03_om_17.12.57.png" alt="N16/5/MATME/SP2/ENG/TZ0/01.c/M"></span></p>
<p class="p2"><span class="s2">\(1.23606,{\text{ }} - 3.23606\)</span></p>
<p class="p2"><span class="s2">\(x = 1.24,{\text{ }}x = - 3.24\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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 velocity <em>v</em> ms<sup>−1</sup> of an object after <em>t</em> seconds is given by \(v(t) = 15\sqrt t - 3t\) , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le t \le 25\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of <em>v</em> , clearly indicating the maximum point.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br></span><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAhEAAAIECAIAAAAo5+ttAAAgAElEQVR4nO2dzWucWZan/W94FYJYSSDopjZeObM2xukCY5IqGMaQNGkb5EU3STPk4JJRdxYF0z2GECU8DBQ4Q5RJSBhcIUoJWTgtAg9DTqcrVXigZ1IOehYerNLCFCYqwEKLlzOLsORQKD5u2Oe98Tvh50GblH/SjZs65zzvR3ycMgAAgDROTfsBAABAGHAGAACkgjMAACAVnAEAAKngDAAASAVnAABAKuU5o+g8+f2tG188KQb925ON2vp3fxr0TwAAIEtJzjj409Y/nvuodv9Je0ig/eTejXMf3f7uTwflPAAAAPCnDGcUne9r779f+75z8jziefPe/zzUSNcr6z8MiAEAgCIlOKPz3a33L936/sWJfyg63//q4zs/9Cjixfe3Lr1/67uO/4MAAAB/3J2x/+TOz05/fO9Z/8nDwZ++/82n7//7O0/2e79bPLlz7tTP+r4JAACapDuj6Dxp3vtvtz6q3thqF6++833t/VM/OXZK0d769PTpc8dOJrpnHqdPveL4vxY/3DlX7c8DAIAkic4o2ls3Dqf+jz7det79ZueH9Y+OGaIbG3TeUPxw59zp059unbgn/nzr0x+dOndn4NOrAABAiomuTb34/tZPjp8oFO2t/3TjlUKse2Hq1EBntLc+PX0km16G/wgAAIgx2f2M4smdc6dO9Zwu7D/Z+H3PKcLzrU9/dOr00cWr1z839PzjlTMG6gQAALSY8B54e+vT06eOLiUVz/77xh97nx81zBkjLkDhDACAMEzojOKHO+dOv5r+xf/b2nh8/GmyQ5wx6kY3zgAACMMbOeP0ja32/rOt3/+x/+V4g29OFE/unBtqBe5nAACEYdLXZ7w6k7j/w+M/Dnjbj+59iz49FO2tG6cHXLDq+YU8bwoAIAJv5oyPasPeYLDz3a33+y5DHVqh/fjevf/d/5Lv4oc756q8FBwAIASTOmP/yZ2/v7F14lXexwI/O37e8OL7Wz859f6nd7aenBRD8ezex6e5MAUAEIOc7zd1Et5vCgAgEuW9r+0/bo15n/MRb38LAACKlPb5Gd/d/ujcjXsjPj/jfu3jj/s/P+Ply5cvX74s5yEBAMDbUuJnuxZ/+h+1oZ/T98WNW79/cuIMY6PRqFbmyntIAADwNgh9HvjLly9//N571cocpxoAAJoIOWOj0eg6Y6PRmPZjAQCAAag4o3uS0b029eP33uNUAwBAEBVndE8yXr582XUGpxoAAIJIOOPoJMPMutemONUAABBEwhm9kujeA+dUAwBAkOk7o88Q3efacqoBACDI9J3xzTff9Oqh6wxONQAABJm+M7799tvHjx8f/efRa/oeP37c+30AAJg603dGH7wOHABAFpwBAACp4AwAAEgFZzizvy/3+VF/+QsfUBIP/mqgiagzNr/6uu9rxDdD5Ef8yOf1u9G3MOkS09ryFP+vlrrlPFvw2vIUH1KGfJ7aniKizgAAAEHCO2NS65ad/7d/+78T5e3wcCydSR/Sv3z3h1J//xv8SNlbzrCFsrc86V/N9Hph0i2/wRJqeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37gjOc8zgjBcG+UhugOCNi3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYv4Z0BAADZCO8MtQMNzjNSEDwWUzvo5jwjYt4ka9sXnOGcxxkpCPaV2gDFGRHzJlnbvuAM5zzOSEGwr9QGKM6ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtS3hnAABANkSdsan0hvgu+RE/wudnRNyC1JbzbMFry1N8SBnyfH5GbjjPAACQJbwzJrVu2XnuZ6QgeM237CW4n1HGEmp5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfuCM5zzOCMFwb5SG6A4I2LeJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti/hnQEAANkI7wy1Aw3OM1IQPBZTO+jmPCNi3iRr2xcXZ7RbD9auzc9VK3PV+aurje29oj9R7DaudwPdr4v11olMF5wxFpzhns+wBM4oYwm1vEnWti9v74wX27UPX8ugMletLFxYeXBcG8+by2d7AmdvNp8P+3U4Yyw4wz2fYQmcUcYSanmTrG1f3tYZxW7j75bWtlptMzNrtxorFypz1crl9db+60yr/tPlZjvtF+KMseAM93yGJXBGGUuo5U2ytn15S2fst35zp9kujn2nfrlaWbhU33n13eLpxtKZauXMtdqXmwOuWvWDM8aCM9zzGZbAGWUsoZY3ydr2xf0eeNFuriy+vvpUdLbXLvRetlputDqjxIEzxoIz3PMZlsAZZSyhljfJ2vbF3Rn7rfrl6vm17WNiKDqthxv15a48Fpcau8OtwXNtAQBk8XZG8XRj6YPrjacDpVDsfbt25czYe+DDvj6v3+37skOrD/snnbzgQ2ILCg/JJS/4kN7Bv8LRNzM8pCni64yis712aeRpRPdJt4vDb4lzngEAIIurMzqPVq/c3h55u+LVxSten9HDpAcO3M9wz2dYYtItcz8jYt4ka9sXP2d0/nV9udbcOxiXwxn94Az3348zFPI4o4wlZsUZxW7zs3/a2B0rDDPbb9WXht3wMJyRAM5wz2dYAmeUsYRa3iRr2xcPZxS7zc9+sb7Tc4ei2PvD7bsP24O8MO76Fc4YC85wz2dYAmeUsYRa3iRr25e3dkax21y5ePI5Tq/uchdPN5Y+uFa7331NRrH36G7t16OvX+GMseAM93yGJXBGGUuo5U2ytn15S2f0vZHUyXeUOno3kbnq/NXVew9Hv6DPcEYCOMM9n2EJnFHGEmp5k6xtX8K/FzoAAGQjvDPUDjQ4z0hB8FhM7aCb84yIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37gjOc8zgjBcG+UhugOCNi3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fQnvDAAAyIaoMza/+rrva8Q3Q+RH/Ej3wCT0FiZdYlpbnuL/1VK3nGcLXlue4kPKkM9T21NE1BkAACBIeGdMat2y89zPSEHwmm/ZS3A/o4wl1PImWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX8I7AwAAshHeGWoHGpxnpCB4LKZ20M15RsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37gjOc8zgjBcG+UhugOCNi3iRr25fwzgAAgGyIOmNT6Q3xXfIjfoTPz4i4Bakt59mC15an+JAy5Pn8jNxwngEAIEt4Z0xq3bLz3M9IQfCab9lLcD+jjCXU8iZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fwjsDAACyEd4ZagcanGekIHgspnbQzXlGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfuCM5zzOCMFwb5SG6A4I2LeJGvbl/DOAACAbIg6Y1PpDfFd8iN+hM/PiLgFqS3n2YLXlqf4kDLk+fyM3HCeAQAgS3hnTGrdsvPcz0hB8Jpv2UtwP6OMJdTyJlnbvuAM5zzOSEGwr9QGKM6ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV/COwMAALIR3hlqBxqcZ6QgeCymdtDNeUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fcIZzHmekINhXagMUZ0TMm2Rt+4IznPM4IwXBvlIboDgjYt4ka9uX8M4AAIBsiDpjU+kN8V3yI36Ez8+IuAWpLefZgteWp/iQMuT5/IzccJ4BACBLeGdMat2y89zPSEHwmm/ZS3A/o4wl1PImWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX8I7AwAAshHeGWoHGpxnpCB4LKZ20M15RsS8Sda2Ly7OaLcerF2bn6tW5qrzV1cb23vFiUCzfvP8QrWycGH5i+29gxG/C2eMBWe45zMsgTPKWEItb5K17cvbO+PFdu3DamWu52vhwsqDHm0c7DY+WTz/y+begVm71Vi5cH5tu9NvlSNwxlhwhns+wxI4o4wl1PImWdu+vK0zit3G3y2tbbXaZvZKCZW5auXyemv/KHB9/uzN5vPDH3i6sXT2Qu1RZ8gvxBljwRnu+QxL4IwyllDLm2Rt+/KWzthv/eZOs10c+079crWycKm+Uxz95/xKT+bkd46BM8aCM9zzGZbAGWUsoZY3ydr2xf0eeNFurixWDk8sip31iwvVi/VWMSRwApwxFpzhns+wBM4oYwm1vEnWti/uzthv1S9Xj+5YtJs35+cWl5vtnkTRql96fSLSD84YC85wz2dYAmeUsYRa3iRr2xdvZxRPN5Y+uN542vVB0apfqgx0Rv83j+D1GQAAsvg6o+hsr11aauwenkG8mTOGfX1ev9v3ZYdWH/ZPOnnBh8QWFB6SS17wIb2Df4Wjb2Z4SFPE1RmdR6tXbh97Hm3516YAACAbfs7o/Ov6cq3Z93o97oEnMOmBA/cz3PMZlph0y9zPiJg3ydr2xckZxW7zs3/a2D35Am+eazsenOH++3GGQh5nlLHETDij2G1+9ov1nZ7rT8XeH27ffdgu7NVr+j5c3X5x+E+8pq8fnOH++3GGQh5nlLFEfGcUu82ViyfvV/fcwzj23iE79aVF3jvkODjD/ffjDIU8zihjiejOeN5cPjvoOU59tyuO3sTwzLXa/dZwYRjOSABnuOczLIEzylhCLW+Ste0L74XunMcZKQj2ldoAxRkR8yZZ276EdwYAAGQjvDPUDjQ4z0hB8FhM7aCb84yIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37gjOc8zgjBcG+UhugOCNi3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYv4Z0BAADZEHXG5ldf932N+GaI/Igf6R6YhN7CpEtMa8tT/L9a6pbzbMFry1N8SBnyeWp7iog6AwAABAnvjEmtW3ae+xkpCF7zLXsJ7meUsYRa3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste1LeGcAAEA2wjtD7UCD84wUBI/F1A66Oc+ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37Et4ZAACQDVFnbCq9Ib5LfsSP8PkZEbcgteU8W/Da8hQfUoY8n5+RG84zAABkCe+MSa1bdp77GSkIXvMtewnuZ5SxhFreJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17Ut4ZwAAQDbCO0PtQIPzjBQEj8XUDro5z4iYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfsS3hkAAJANUWdsKr0hvkt+xI/w+RkRtyC15Txb8NryFB9Shjyfn5EbzjMAAGQJ74xJrVt2nvsZKQhe8y17Ce5nlLGEWt4ka9sXnOGcxxkpCPaV2gDFGRHzJlnbvuAM5zzOSEGwr9QGKM6ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtS3hnAABANsI7Q+1Ag/OMFASPxdQOujnPiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fcIZzHmekINhXagMUZ0TMm2Rt+xLeGQAAkA1RZ2wqvSG+S37Ej/D5GRG3ILXlPFvw2vIUH1KGPJ+fkRvOMwAAZAnvjEmtW3ae+xkpCF7zLXsJ7meUsYRa3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX8I7AwAAsuHkjGJv+3eN39auLl6st4pB/77buD4/V60cfg2JGecZCXCe4Z7PsATnGWUsoZY3ydr2xcMZxc760srntauLQ2XwvLl89rUwKmdvNp8P+2U4Yyw4wz2fYQmcUcYSanmTrG1f/K5NFTvrFxcGOqNo1X+63Gyn/RqcMRac4Z7PsATOKGMJtbxJ1rYv5TujeLqxdKZaOXOt9uXm9t6QK1KvwRljwRnu+QxL4IwyllDLm2Rt+1K2M4rO9tqF11elFi4sN1qdUeLAGWPBGe75DEvgjDKWUMubZG37kuPalFnRaT3cqC935bG41Ngdbg2cMRac4Z7PsATOKGMJtbxJ1rYveZxxGNn7du3KGd974AAAkI2szrDDJ90uDr8l3vP0qv6vz+t3+77s0OrD/kknL/iQ2ILCQ3LJCz6kd/CvcPTNDA9piuR2htl+q37Z8fUZAACQjfDOULugyf2MFMreMvczFPJvcESstgXuZ5xkKs5Yut54OiyFM8aCM9zzGZbAGWUsoZY3ydr2JbszOo9Wr9zeHv50W5wxFpzhns+wBM4oYwm1vEnWti8lO6N4urH0wbXa/e5rMoq9R3drv27uHYz4NThjLDjDPZ9hCZxRxhJqeZOsbV9cnNH3dlILl+o7h+Jotxorr17TN3919d7D0S/oM5yRAM5wz2dYAmeUsYRa3iRr25fw74WuVjQ4IwXBvlIboDgjYt4ka9uX8M4AAIBshHeG2oEG5xkpCB6LqR10c54RMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfuCM5zzOCMFwb5SG6A4I2LeJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2JbwzAAAgG6LO2Pzq676vEd8MkR/xI90Dk9BbmHSJaW15iv9XS91yni14bXmKDylDPk9tTxFRZwAAgCDhnTGpdcvOcz8jBcFrvmUvwf2MMpZQy5tkbfuCM5zzOCMFwb5SG6A4I2LeJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9Ce8MAADIRnhnqB1ocJ6RguCxmNpBN+cZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX8I7AwAAsiHqjE2lN8R3yY/4ET4/I+IWpLacZwteW57iQ8qQ5/MzcsN5BgCALOGdMal1y85zPyMFwWu+ZS/B/YwyllDLm2Rt+4IznPM4IwXBvlIboDgjYt4ka9sXnOGcxxkpCPaV2gDFGRHzJlnbvuAM5zzOSEGwr9QGKM6ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn0J7wwAAMhGeGeoHWhwnpGC4LGY2kE35xkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fwjsDAACyIeqMTaU3xHfJj/gRPj8j4haktpxnC15bnuJDypDn8zNyw3kGAIAs4Z0xqXXLznM/IwXBa75lL8H9jDKWUMubZG37gjOc8zgjBcG+UhugOCNi3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste1LeGcAAEA2wjtD7UCD84wUBI/F1A66Oc+ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37Et4ZAACQDVFnbCp9iIpLfsSP8JlLEbcgteU8W/Da8hQfUoY8n7mUG84zAABkCe+MSa1bdp77GSkIXvMtewnuZ5SxhFreJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17Ut4ZwAAQDacnFHsbf+u8dva1cWL9VZx8p/brWb95vmFamXhwvIX23sHI34T5xlj4TzDPZ9hCc4zylhCLW+Ste2LhzOKnfWllc9rVxcrc9UBzjjYbXyyeP6Xzb0Ds3arsXLh/Np2Z4BYuuCMseAM93yGJXBGGUuo5U2ytn3xuzZV7KxfXDjpjGK3cX3+7M3m88P/frqxdPZC7VFnyK/BGWPBGe75DEvgjDKWUMubZG37UrYz9lv1y9X5lWa7GP6dY+CMseAM93yGJXBGGUuo5U2ytn0p2RkDvlm0myuLlZ4zj+PgjLHgDPd8hiVwRhlLqOVNsrZ9KdkZ7ebN+bnF5Wa7N9iqX6osXKrvDDzRwBljwRnu+QxL4IwyllDLm2Rt+1KuM4pW/VJloDP6v3kEz7UFAJBF0RnDvj6v3+37skOrD/snnbzgQ2ILCg/JJS/4kN7Bv8LRNzM8pCkS/toUAABkg3vgznnuZ6RQ9pa5n6GQf4MjYrUtcD/jJDzX1jmPM1IQ7Cu1AYozIuZNsrZ9yfOavg9Xt18c/jev6esHZ7j/fpyhkMcZZSwx+87oe++QnfrSIu8dchyc4f77cYZCHmeUscRsOON5c/lszxOcTt7fbrcerF2bn6tWzlyr3W8NF4bhjARwhns+wxI4o4wl1PImWdu+hH8vdLWiwRkpCPaV2gDFGRHzJlnbvoR3BgAAZCO8M9QONDjPSEHwWEztoJvzjIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfuCM5zzOCMFwb5SG6A4I2LeJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti/hnQEAANkQdcbmV1/3fY34Zoj8iB/pHpiE3sKkS0xry1P8v1rqlvNswWvLU3xIGfJ5anuKiDoDAAAECe+MSa1bdp77GSkIXvMtewnuZ5SxhFreJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17Ut4ZwAAQDbCO0PtQIPzjBQEj8XUDro5z4iYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfsS3hkAAJANUWdsKr0hvkt+xI/w+RkRtyC15Txb8NryFB9Shjyfn5EbzjMAAGQJ74xJrVt2nvsZKQhe8y17Ce5nlLGEWt4ka9sXnOGcxxkpCPaV2gDFGRHzJlnbvuAM5zzOSEGwr9QGKM6ImDfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDceP+lcAAA+BSURBVFCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV/COwMAALIR3hlqBxqcZ6QgeCymdtDNeUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fcIZzHmekINhXagMUZ0TMm2Rt+4IznPM4IwXBvlIboDgjYt4ka9uX8M4AAIBsiDpjU+lDVFzyI36Ez1yKuAWpLefZgteWp/iQMuT5zKXccJ4BACBLeGdMat2y89zPSEHwmm/ZS3A/o4wl1PImWdu+4AznPM5IQbCv1AYozoiYN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX8I7AwAAshHeGWoHGpxnpCB4LKZ20M15RsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37gjOc8zgjBcG+UhugOCNi3iRr25fwzgAAgGyIOmNT6Q3xXfIjfoTPz4i4Bakt59mC15an+JAy5Pn8jNxwngEAIEt4Z0xq3bLz3M9IQfCab9lLcD+jjCXU8iZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fwjsDAACykc8ZxW7j+vxctXL4dbHeKgbEOM8YC+cZ7vkMS3CeUcYSanmTrG1fsjnjeXP57GthVM7ebD4fmMMZY8EZ7vkMS+CMMpZQy5tkbfuSyRlFq/7T5WY7IYkzxoIz3PMZlsAZZSyhljfJ2vYlizOKpxtLZ6qVM9dqX25u7w26IvUanDEWnOGez7AEzihjCbW8Sda2LxmcUXS21y68viq1cGG50eoMFQfOGAvOcM9nWAJnlLGEWt4ka9uXbPczik7r4UZ9uSuPxaXG7hBr4Iyx4Az3fIYlcEYZS6jlTbK2fcn9XNti79u1K2cc74EDAEA2pvD6jO6TbheH3BLveW5V/9fn9bt9X3Zo9WH/pJMXfEhsQeEhueQFH9I7+Fc4+maGhzRFpvKavv1W/bLX6zMAACAb4Z2hdkGT+xkplL1l7mco5N/giFhtC9zPOMm0nLF0vfF04F1wnDEWnOGez7AEzihjCbW8Sda2L9NwRufR6pXb20OeboszxoIz3PMZlsAZZSyhljfJ2valfGcUTzeWPrhWu999TUax9+hu7dfNvYNhcZwxFpzhns+wBM4oYwm1vEnWti8ZzjParcbKq9f0zV9dvfdwxAv6DGckgDPc8xmWwBllLKGWN8na9iX8e6GrFQ3OSEGwr9QGKM6ImDfJ2vYlvDMAACAb4Z2hdqDBeUYKgsdiagfdnGdEzJtkbfuCM5zzOCMFwb5SG6A4I2LeJGvbF5zhnMcZKQj2ldoAxRkR8yZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9Ce8MAADIhqgzNr/6uu9rxDdD5Ef8SPfAJPQWJl1iWlue4v/VUrecZwteW57iQ8qQz1PbU0TUGQAAIEh4Z0xq3bLz3M9IQfCab9lLcD+jjCXU8iZZ277gDOc8zkhBsK/UBijOiJg3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fwjsDAACyEd4ZagcanGekIHgspnbQzXlGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfuCM5zzOCMFwb5SG6A4I2LeJGvbF5zhnMcZKQj2ldoAxRkR8yZZ276EdwYAAGRD1BmbSh+i4pIf8SN85lLELUhtOc8WvLY8xYeUIc9nLuWG8wwAAFnCO2NS65ad535GCoLXfMtegvsZZSyhljfJ2vYFZzjncUYKgn2lNkBxRsS8Sda2LzjDOY8zUhDsK7UBijMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37Et4ZAACQjfDOUDvQ4DwjBcFjMbWDbs4zIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fcIZzHmekINhXagMUZ0TMm2Rt+4IznPM4IwXBvlIboDgjYt4ka9sXnOGcxxkpCPaV2gDFGRHzJlnbvoR3BgAAZEPUGZtKb4jvkh/xI3x+RsQtSG05zxa8tjzFh5Qhz+dn5IbzDAAAWcI7Y1Lrlp3nfkYKgtd8y16C+xllLKGWN8na9gVnOOdxRgqCfaU2QHFGxLxJ1rYvOMM5jzNSEOwrtQGKMyLmTbK2fcEZznmckYJgX6kNUJwRMW+Ste0LznDO44wUBPtKbYDijIh5k6xtX3CGcx5npCDYV2oDFGdEzJtkbfsS3hkAAJCN8M5QO9DgPCMFwWMxtYNuzjMi5k2ytn3BGc55nJGCYF+pDVCcETFvkrXtC85wzuOMFAT7Sm2A4oyIeZOsbV9whnMeZ6Qg2FdqAxRnRMybZG37gjOc8zgjBcG+UhugOCNi3iRr2xec4ZzHGSkI9pXaAMUZEfMmWdu+hHcGAABkQ9QZm0pviO+SH/EjfH5GxC1IbTnPFry2PMWHlCHP52fkhvMMAABZwjtjUuuWned+RgqC13zLXoL7GWUsoZY3ydr2BWc453FGCoJ9pTZAcUbEvEnWti84wzmPM1IQ7Cu1AYozIuZNsrZ9wRnOeZyRgmBfqQ1QnBExb5K17QvOcM7jjBQE+0ptgOKMiHmTrG1fcIZzHmekINhXagMUZ0TMm2Rt+xLeGTMAW34XYMvvCDO/6/DOUDvQeIPzjLK3LHieofZXzrDEpFuegfOMN5iealt4g0ISrG1fcIZzHmekoPZXzrAEzihjCbW8Sda2LzjDOY8zUlD7K2dYAmeUsYRa3iRr2xec4ZzHGSmo/ZUzLIEzylhCLW/Za/vx48f/+Z//udVqTfRL3gac4ZzHGSmo/ZUzLIEzylhCLW/Za/vPf/7z8s9/Xq3MLf/853nMgTOc8zgjBbW/coYlcEYZS6jlbUq13Wq1spkjvDNmALb8LsCW3xGmuOs85lB0Bl988cUXX2//9ezZM/cRLeeM6Ozv70/7IfTzl790pv0QYGL4q8Eb8PLly41G48fvvffj997baDRevnzpvgTOAAAIT5otnj9s/Ev77RbK7Iyi02quL1+sVuaq55fvbu8VeZcHAJgxks8tis727ev1nbeculmdUew2rs9f/IfmbmFFp9W4ef7frW6/yPkAAABmiVarlXYl6mBv+4ub5/9mvfW2F88zOqN4urF0ZnG5eXhmdLDb+GTx/Np2h5MNAIA34dmzZ998882Y+xadR6vnFw5vjC9certTjXzOKFr1S5WzN5vPR3wHAAD8KXbWLy70HLK/Odmcsd+qX65WLh87M2o3b87PuWwDAACG0m7enPc5QM/mjOfN5bPV+ZVmu+esqNhZv7hQvVhvvatXp4rdxvX5nudTz+D/ioO97a837tWuzV8edCF1Np8TUextbza+XL1ydshFgIPdxieLr59E/7bXCqZPp7VVu9rd0eKV2saAv2O71azfPL9QrSxcWP5ie+9gGo/SlfFbluruot1cWawM7MGJyeWMrh4GOqPvm+8Qz5vLZ3tegDN7l+n2W/X/ePPOrWvzc/2nmGY2o8+JKFp3ry/fXr1yZqgM2s2bvaMkev0fu1be/er+TY/o3rn8ZXPvwKzdaqxciH4Xc/yWTay7nzeXz3pJC2dMjaJV/+k7cV1u0GVJm/HnRBSt+qXBzthv1f92hg4ODnYby9dr91vdv1pnZ6N71tgznordxvXeqyLF042lsxdqj8K+ZHH8lk2tu4ud9Yt/7XU6y7WpKVE83Vg6U62cuVb7cnNWrskMYbAzZvs5EcOc8ep6xfzV1Xtfz8IlmmLn7q8eto9/Z/3iQs+fe79Vv3y88U9+JxTjtyzX3b6dxT3wqVB0ttcu9FzRvrDcaM3E8fUgBjpjxuthiDNebNc+7L2gcbOxE/ZwexjHjw4HHBd2r63PyMGBmZ04IFbr7qLdXFn0kzTPtZ0iRaf1cKO+3C2vxaXG7mxaY6AzZvy8c/i1KTNrt5qNV3f+K2euN57G324Pxc76xb9+felp0HHAyP85Aenb8uF3Zbr78GZG+39tNP7P2x+jZH5NX+91zJm6fv02FHvfrl05M+27ZOUxyBmzfn8rYSwe7D36r9fm498DP06x27j+V59s7L667Fa06pcqA50xIyeUdmLL/f86/e5+sV37sHp+eb3Zcjmpnd57h+zcvTb/4Qw8T8aF7mXumemi4+CMYXSfdDtLxwovtmt/03vm9A44o3/LJ5mx7s7/HoX3V6+cqVbmFq+sbbVm4/+hC/ut+uXZuCxzAq5Njch5PqFl2hSd7dvX+i7RzPi1qUFbHsBMdTfvhS7CTFXVcbgHPiI3M84oOjtf3PzsQf+ThGb5HviQLQ9gprobZ4iw36ovzdrt0FfwXNsRuZ31D//DsEvhgSj2Hvxi5XeDbvPO3HNtDxm+5ZPMVHfjDA06j1av3J7RpwOMeE3fzD4nIs0ZiVc21Cn2Hvxi+Yudnj9csfftf6l/2z1fLHYb13vvXIZ/TZ/ZuC33M1vdjTOmQfF0Y+mDa4cvJS32Ht2t/bo5A6/wGswQZ8z0cyIGOqPYbVz/q6urD7pPXznY2/5y9VfN6b/i6+0o9h78Q/8bafS9Vcax9w7ZqS9FPzIYs+VZ726cMRXarcbKq1f9zF9dvfdwRl/Q171yPeJt2mbxORF9byfVK8uj95mozC1eqf3W6bmP06R/s8PeRKvderB2bX6uWjlzNEyjMn7LM97dOAMAAFLBGQAAkArOAACAVHAGAACkgjMAACAVnAEAAKngDAAASAVnAABAKjgDAABSwRkAAJAKzgAAgFRwBgAApIIzAAAgFZwBAACp4AwAAEgFZwAAQCo4AwAAUvn/LiOOGsDwtIkAAAAASUVORK5CYII=" alt></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for <em>d</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, write down the value of <em>d</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/HSM3.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape, <em><strong>A1</strong></em> for right endpoint at \((25{\text{, }}0)\) and <em><strong>A1</strong></em> for maximum point in circle.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) recognizing that <em>d</em> is the area under the curve <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {v(t)} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression in terms of <em>t</em>, with correct limits <em><strong>A2 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(d = \int_0^9 {(15\sqrt t } - 3t){\rm{d}}t\) , \(d = \int_0^9 v {\rm{d}}t\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(d = 148.5\) (m) (accept 149 to 3 sf) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph in part (a) was well done. It was pleasing to see many candidates considering the domain as they sketched their graph. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) (i) asked for an expression which bewildered a great many candidates. However, few had difficulty obtaining the correct answer in (b) (ii). </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first three terms of a geometric sequence are \({u_1} = 0.64,{\text{ }}{u_2} = 1.6\), and \({u_3} = 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \({S_6}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the least value of \(n\) such that \({S_n} > 75\,000\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{{u_1}}}{{{u_2}}},{\text{ }}\frac{4}{{1.6}},{\text{ }}1.6 = r(0.64)\)</p>
<p class="p1">\(r = 2.5\;\;\;\left( { = \frac{5}{2}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks] </em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into \({S_6}\) <strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{0.64({{2.5}^6} - 1)}}{{2.5 - 1}}\)</p>
<p class="p1">\({S_6} = 103.74\) (exact), \(104\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1 (analytic)</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{0.64({{2.5}^n} - 1)}}{{2.5 - 1}} > 75\,000,{\text{ }}\frac{{0.64({{2.5}^n} - 1)}}{{2.5 - 1}} = 75\,000\)</p>
<p class="p1">correct inequality (accept equation) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;n > 13.1803,{\text{ }}n = 13.2\)</p>
<p class="p1">\(n = 14\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong>METHOD 2 (table of values)</strong></p>
<p class="p1"><strong>both </strong>crossover values <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{S_{13}} = 63577.8,{\text{ }}{S_{14}} = 158945\)</p>
<p class="p1">\(n = 14\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following table shows a probability distribution for the random variable \(X\), where \({\text{E}}(X) = 1.2\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.18.09.png" alt="M17/5/MATME/SP2/ENG/TZ2/10"></p>
</div>
<div class="specification">
<p>A bag contains white and blue marbles, with at least three of each colour. Three marbles are drawn from the bag, without replacement. The number of blue marbles drawn is given by the random variable \(X\).</p>
</div>
<div class="specification">
<p>A game is played in which three marbles are drawn from the bag of ten marbles, without replacement. A player wins a prize if three white marbles are drawn.</p>
</div>
<div class="question">
<p>Jill plays the game nine times. Find the probability that she wins exactly two prizes.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{B}}(n,{\text{ }}p),{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){p^r}{q^{n - r}},{\text{ }}{(0.167)^2}{(0.833)^7},{\text{ }}\left( {\begin{array}{*{20}{c}} 9 \\ 2 \end{array}} \right)\)</p>
<p>0.279081</p>
<p>0.279 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were 280 taxis in the city. After <em>n</em> years the number of taxis, <em>T</em>, in the city is given by\[T = 280 \times {1.12^n} .\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find the number of taxis in the city at the end of 2005. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the year in which the number of taxis is double the number of taxis there were at the end of 2000.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">At the end of 2000 there were \(25600\) people in the city who used taxis. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">After <em>n</em> years the number of people, <em>P</em>, in the city who used taxis is given by\[P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1n}}}} .\](i) Find the value of <em>P</em> at the end of 2005, giving your answer to the nearest whole number. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) After seven complete years, will the value of <em>P</em> be double its value at the end of 2000? Justify your answer.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the ratio of the number of people using taxis in the city to the number of taxis. The city will reduce the number of taxis if \(R < 70\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of <em>R</em> at the end of 2000. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) After how many complete years will the city first reduce the number of taxis?</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(n = 5\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(T = 280 \times {1.12^5}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(T = 493\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of doubling <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 560</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting up equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(280 \times {1.12^n} = 560\), \({1.12^n} = 2\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 6.116 \ldots \) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">in the year 2007 <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1(5)}}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 39635.993 \ldots \) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 39636\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1(7)}}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 46806.997 \ldots \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">not doubled <em><strong>A1 N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reason for <strong>their</strong> answer <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(P < 51200\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) correct value <em><strong>A2 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{25600}}{{280}}\) , 91.4, \(640:7\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) setting up an inequality (accept an equation, or reversed inequality) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{P}{T} < 70\) , \(\frac{{2560000}}{{(10 + 90{{\rm{e}}^{ - 0.1n}})280 \times {{1.12}^n}}} < 70\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding the value \(9.31 \ldots \) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">after 10 years <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A number of candidates found this question very accessible. In part (a), many correctly solved </span><span style="font-family: times new roman,times; font-size: medium;">for <em>n</em>, but often incorrectly answered with the year 2006, thus misinterpreting that 6.12 years </span><span style="font-family: times new roman,times; font-size: medium;">after the end of 2000 is in the year 2007.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many found correct values in part (b) and often justified their result by simply noting the value after seven years is less than 51200. A common alternative was to divide 46807 by 25600 and note that this ratio is less than two. There were still a good number of candidates who failed to provide any justification as instructed.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Part (c) proved more challenging to candidates. Many found the correct ratio for <em>R</em>, however </span><span style="font-family: times new roman,times; font-size: medium;">few candidates then created a proper equation or inequality by dividing the function for <em>P</em> by </span><span style="font-family: times new roman,times; font-size: medium;">the function for <em>T</em> and setting this equal (or less) than 70. Such a function, although unfamiliar, </span><span style="font-family: times new roman,times; font-size: medium;">can be solved using the graphing or solving features of the GDC. Many candidates chose a </span><span style="font-family: times new roman,times; font-size: medium;">tabular approach but often only wrote down one value of the table, such as \(n = 10\) , \(R = 68.3\) . </span><span style="font-family: times new roman,times; font-size: medium;">What is essential is to include the two values between which the correct answer falls. </span><span style="font-family: times new roman,times; font-size: medium;">Sufficient evidence would include \(n = 9\) , \(R = 70.8\) so that it is clear the value of \(R = 70\) has </span><span style="font-family: times new roman,times; font-size: medium;">been surpassed.</span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}\sin 2x + 10\) , for \(0 \le x \le 4\) . Part of the graph of <em>f</em> is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/apple.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is an <em>x</em>-intercept at the point A, a local maximum point at M, where \(x = p\) and </span><span style="font-size: medium;"><span style="font-family: times new roman,times;">a local minimum point at N, where \(x = q\)</span><span style="font-family: times new roman,times;"> .</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-coordinate of A.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_p^q {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> . Explain why this is not the area of the shaded region.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(2.31\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 1.02 <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) 2.59 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\int_p^q {f(x){\rm{d}}x} = 9.96\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">split into two regions, make the area below the <em>x</em>-axis positive <em><strong>R1R1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well answered, the main problem being the accuracy.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well answered, the main problem being the accuracy.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many students lacked the calculator skills to successfully complete (6)(c) in that they could </span><span style="font-family: times new roman,times; font-size: medium;">not find the value of the definite integral. Some tried to find it by hand. </span><span style="font-family: times new roman,times; font-size: medium;">When trying to explain why the integral was not the area, most knew the region under the <em>x</em>-axis </span><span style="font-family: times new roman,times; font-size: medium;">was the cause of the integral not giving the total area, but the explanations were not </span><span style="font-family: times new roman,times; font-size: medium;">sufficiently clear. It was often stated that the area below the axis was negative rather than the </span><span style="font-family: times new roman,times; font-size: medium;">integral was negative.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The points A and B <span class="s1">lie on a line \(L\)</span>, and have position vectors \(\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ 2 \end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 1} \end{array}} \right)\) respectively. Let O <span class="s1">be the origin. This is shown on the following diagram.</span></p>
<p class="p1" style="text-align: center;"><span class="s1"><img src="images/Schermafbeelding_2017-02-01_om_15.56.14.png" alt="M16/5/MATME/SP2/ENG/TZ1/10"></span></p>
</div>
<div class="specification">
<p class="p1">The point C <span class="s1">also lies on \(L\)</span>, such that \(\overrightarrow {{\text{AC}}} = 2\overrightarrow {{\text{CB}}} \).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let \(\theta \) </span>be the angle between \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{OC}}} \).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let D be a point such that \(\overrightarrow {{\text{OD}}} = k\overrightarrow {{\text{OC}}} \)</span>, where \(k > 1\)<span class="s1">. Let E </span>be a point on \(L\) <span class="s1">such that \({\rm{C\hat ED}}\) </span>is a right angle. This is shown on the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_16.39.34.png" alt="M16/5/MATME/SP2/ENG/TZ1/10.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\overrightarrow {{\text{AB}}} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\).</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\theta \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\left| {\overrightarrow {{\text{DE}}} } \right| = (k - 1)\left| {\overrightarrow {{\text{OC}}} } \right|\sin \theta \).</p>
<p class="p2"><span class="s1">(ii) <span class="Apple-converted-space"> </span>The distance from D </span>to line \(L\) <span class="s1">is less than 3 </span>units. Find the possible values of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach (addition or subtraction) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({\text{AO}} + {\text{OB}},{\text{ B}} - {\text{A}}\)</p>
<p class="p1"><span class="Apple-converted-space">\(\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 9 \\ 6 \\ { - 3} \end{array}} \right)\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2">valid approach using \(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} {x + 3} \\ {y + 2} \\ {z - 2} \end{array}} \right),{\text{ }}\overrightarrow {{\text{CB}}} = \left( {\begin{array}{*{20}{c}} {6 - x} \\ {4 - y} \\ { - 1 - z} \end{array}} \right)\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {x + 3} \\ {y + 2} \\ {z - 2} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {12 - 2x} \\ {8 - 2y} \\ { - 2 - 2z} \end{array}} \right)\)</p>
<p class="p1">all three equations <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(x + 3 = 12 - 2x,{\text{ }}y + 2 = 8 - 2y,{\text{ }}z - 2 = - 2 - 2z\)<span class="s2">,</span></p>
<p class="p1"><span class="s2">\(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\) <span class="Apple-converted-space"> </span></span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p3"><strong>METHOD 2</strong></p>
<p class="p3">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(\overrightarrow {{\text{OC}}} - \overrightarrow {{\text{OA}}} = 2\left( {\overrightarrow {{\text{OB}}} - \overrightarrow {{\text{OC}}} } \right)\)</p>
<p class="p3">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(3\overrightarrow {{\text{OC}}} = 2\overrightarrow {{\text{OB}}} + \overrightarrow {{\text{OA}}} \)</p>
<p class="p4">correct substitution of \(\overrightarrow {{\text{OB}}} \) <span class="s3">and \(\overrightarrow {{\text{OA}}} \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(3\overrightarrow {{\text{OC}}} = 2\left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 1} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ 2 \end{array}} \right),{\text{ }}3\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 9 \\ 6 \\ 0 \end{array}} \right)\)</p>
<p class="p3"><span class="Apple-converted-space">\(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\) </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p3"><strong>METHOD 3</strong></p>
<p class="p3">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(\overrightarrow {{\text{AC}}} = \frac{2}{3}\overrightarrow {{\text{AB}}} \), diagram, \(\overrightarrow {{\text{CB}}} = \frac{1}{3}\overrightarrow {{\text{AB}}} \)</p>
<p class="p3"><img src="images/Schermafbeelding_2017-02-02_om_08.19.06.png" alt="M16/5/MATME/SP2/ENG/TZ1/10.b/M"></p>
<p class="p3">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 2} \end{array}} \right),{\text{ }}\overrightarrow {{\text{CB}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ { - 1} \end{array}} \right)\)</p>
<p class="p3">correct working involving \(\overrightarrow {{\text{OC}}} \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 2} \end{array}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 1} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ { - 1} \end{array}} \right)\)</p>
<p class="p3"><span class="Apple-converted-space">\(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\) </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">finding scalar product and magnitudes <span class="Apple-converted-space"> </span><strong><em>(A1)(A1)(A1)</em></strong></p>
<p class="p1">scalar product \( = (9 \times 3) + (6 \times 2) + ( - 3 \times 0){\text{ }}( = 39)\)</p>
<p class="p1">magnitudes \(\sqrt {81 + 36 + 9} {\text{ }}( = 11.22),{\text{ }}\sqrt {9 + 4} {\text{ }}( = 3.605)\)</p>
<p class="p1">substitution into formula <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta = \frac{{(9 \times 3) + 12}}{{\sqrt {126} \times \sqrt {13} }}\)</p>
<p class="p1">\(\theta = 0.270549{\text{ }}({\text{accept }}15.50135^\circ )\)</p>
<p class="p1"><span class="Apple-converted-space">\(\theta = 0.271{\text{ }}({\text{accept }}15.5^\circ )\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>attempt to use a trig ratio <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\sin \theta = \frac{{{\text{DE}}}}{{{\text{CD}}}},{\text{ }}\left| {\overrightarrow {{\text{CE}}} } \right| = \left| {\overrightarrow {{\text{CD}}} } \right|\cos \theta \)</p>
<p class="p1">attempt to express \(\overrightarrow {{\text{CD}}} \) in terms of \(\overrightarrow {{\text{OC}}} \) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\overrightarrow {{\text{OC}}} + \overrightarrow {{\text{CD}}} = \overrightarrow {{\text{OD}}} ,{\text{ OC}} + {\text{CD}} = {\text{OD}}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\left| {k\overrightarrow {{\text{OC}}} - \overrightarrow {{\text{OC}}} } \right|\sin \theta \)</p>
<p class="p1"><span class="s1">\(\left| {\overrightarrow {{\text{DE}}} } \right| = (k - 1)\left| {\overrightarrow {{\text{OC}}} } \right|\sin \theta \) <span class="Apple-converted-space"> </span></span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>valid approach involving the segment DE <span class="Apple-converted-space"> </span><span class="s2"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)</span>recognizing \(\left| {\overrightarrow {{\text{DE}}} } \right| < 3,{\text{ DE}} = 3\)</p>
<p class="p1">correct working (accept equation) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\((k - 1)(\sqrt {13} )\sin 0.271 < 3,{\text{ }}k - 1 = 3.11324\)</p>
<p class="p1"><span class="Apple-converted-space">\(1 < k < 4.11{\text{ }}({\text{accept }}k < 4.11{\text{ but not }}k = 4.11)\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority of candidates had little difficulty with parts (a) and (c). The most common error in both these parts were unforced arithmetic errors and occasional misreads of the vectors.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), candidates who were successful used a variety of different approaches, and it was pleasing to see the vast majority of these being well reasoned, however, there were numerous unsuccessful responses including those who attempted to use the given vector to work backwards. A lack of appropriate vector notation often meant that ideas were not always clearly communicated.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority of candidates had little difficulty with parts (a) and (c). The most common error in both these parts were unforced arithmetic errors and occasional misreads of the vectors.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">The majority of candidates struggled to make any progress in (d), with very few realizing that simple right-angled trigonometry could be used. Few were able to successfully express CD in terms of OC which was required to show the given result. Very few candidates attempted (d)(ii), with many unable to make the connection with results found in previous parts of the question.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2}\) and \(g(x) = 3\ln (x + 1)\), for \(x > - 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \(f(x) = g(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the area of the region enclosed by the graphs of \(f\) and \(g\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg </em>sketch</p>
<p class="p2">0, 1.73843</p>
<p class="p1"><span class="s1">\(x = 0,{\text{ }}x = 1.74{\text{ }}\left( {{\text{accept }}(0,{\text{ }}0){\text{ and }}(1.74,{\text{ }}3.02)} \right)\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">integrating and subtracting functions (in any order) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {g - f} \)</p>
<p class="p1">correct substitution of their limits <strong>or </strong><span class="s1">function (accept missing \({\text{d}}x\)</span>)</p>
<p class="p1"><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^{1.74} {g - f,{\text{ }}\int {3\ln (x + 1) - {x^2}} } \)</p>
<p class="p3"><span class="s2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award <strong><em>A1 </em></strong></span>if there is an error in the substitution.</p>
<p class="p3">1.30940</p>
<p class="p1"><span class="s1">1.31 <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates often did not make the connection between parts (a) and (b). The extraordinary number of failed analytical approaches in part (a) and correct use of the GDC to find the limits in part (b) suggests that candidates are equating the command term “solve” to mean use an algebraic approach to solve equations or inequalities, instead of their GDC. Many candidates appeared to interpret part (a) as something they should do by hand and often did not recognize that their answer to part (a) were the limits in part (b). Quite a few candidates failed to interpret a GDC solution of \(x = 5 \times {10^{ - 14}}\) correctly as \(x = 0\) and others found the solution \(x = 1.74\) as the only solution, ignoring the second intersection point until part (b).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Candidates often did not make the connection between parts (a) and (b). The extraordinary number of failed analytical approaches in part (a) and correct use of the GDC to find the limits in part (b) suggests that candidates are equating the command term “solve” to mean use an algebraic approach to solve equations or inequalities, instead of their GDC. Many candidates appeared to interpret part (a) as something they should do by hand and often did not recognize that their answer to part (a) were the limits in part (b). Quite a few candidates failed to interpret a GDC solution of \(x = 5 \times {10^{ - 14}}\) correctly as \(x = 0\) and others found the solution \(x = 1.74\) as the only solution, ignoring the second intersection point until part (b).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle’s displacement, in metres, is given by \(s(t) = 2t\cos t\) , for \(0 \le t \le 6\) , </span><span style="font-family: times new roman,times; font-size: medium;">where <em>t</em> is the time in seconds.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(s\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q7.jpg" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum velocity of the particle.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q7ms.jpg" alt> <em><strong>A1A1A1A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Note:</strong> Award <em><strong>A1</strong></em> for approximately correct shape (do not accept line segments). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this <em><strong>A1</strong></em> is awarded, award the following: </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for maximum and minimum within circles, </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for <em>x</em>-intercepts between 1 and 2 <strong>and</strong> between 4 and 5, </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for left endpoint at \((0{\text{, }}0)\) and right endpoint within circle. </span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. recognizing that \(v = s'\) , finding derivative, \(a = s''\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid method to find maximum <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(v\) , \(v'(t) = 0\) , \(t = 5.08698 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = 10.20025 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = 10.2\) \([10.2{\text{, }}10.3]\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates sketched an approximately correct shape for the displacement of a particle in the given domain, but many lost marks for carelessness in graphing the local extrema or the right endpoint. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), most candidates knew to differentiate displacement to find velocity, but few knew how to then find the maximum. Occasionally, a candidate would give the time value of the maximum. Others attempted to incorrectly set the first derivative equal to zero and solve analytically rather than take the maximum value from the graph of the velocity function. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = p\cos qx + r\) , for \( - 5 \le x \le 14\) , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/weather.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a minimum point at (0, −3) and a maximum point at (4, 7) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>r</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The equation \(y = k\) has exactly <strong>two</strong> solutions. Write down the value of <em>k</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of finding the amplitude <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{7 + 3}}{2}\) , amplitude \(= 5\) <br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = - 5\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) period \(= 8\) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(q = 0.785\) \(\left( { = \frac{{2\pi }}{8} = \frac{\pi }{4}} \right)\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(r = \frac{{7 - 3}}{2}\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 2\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = - 3\) (accept \(y = - 3\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not recognize that the value of <em>p</em> was negative. The value of <em>q</em> was often </span><span style="font-family: times new roman,times; font-size: medium;">interpreted incorrectly as the period but most candidates could find the value of <em>r</em>, the vertical </span><span style="font-family: times new roman,times; font-size: medium;">translation.</span></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), candidates either could not find a solution or found too many.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 2x + 3\) and \(g(x) = {x^3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \((f \circ g)(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the equation \((f \circ g)(x) = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to form composite (in any order) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f({x^3}),{\text{ }}{(2x + 3)^3}\)</p>
<p class="p1">\((f \circ g)(x) = 2{x^3} + 3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of appropriate approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2{x^3} = - 3\), sketch</p>
<p style="text-align: center;"><img src="image_1.html" alt></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{x^3} = \frac{{ - 3}}{2}\), sketch</p>
<p style="text-align: center;"><img src="image_2.html" alt></p>
<p>\( - 1.14471\)</p>
<p>\(x = \sqrt[3]{{\frac{{ - 3}}{2}}}\;\;\;{\text{(exact), }} - 1.14{\text{ }}[ - 1.15,{\text{ }} - 1.14]\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally well done, though there were some careless errors with the substitution into \(f\) in part (ai) and rearranging the equation in part (b). Although candidates understood that they were supposed to solve the equation \(2{x^3} + 3 = 0\), many wrote \(2{x^3} = 3\) or \(x = \sqrt {\frac{3}{2}} \). The majority of the candidates chose an algebraic method instead of using their GDC.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Generally well done, though there were some careless errors with the substitution into \(f\) in part (ai) and rearranging the equation in part (b). Although candidates understood that they were supposed to solve the equation \(2{x^3} + 3 = 0\), many wrote \(2{x^3} = 3\) or \(x = \sqrt {\frac{3}{2}} \). The majority of the candidates chose an algebraic method instead of using their GDC.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = - {x^4} + 2{x^3} - 1\), for \(0 \le x \le 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(f\) on the following grid.</p>
<p style="text-align: center;"><img src="image_3.html" alt></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \(f(x) = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360°\) about the <em>\(x\)</em>-axis.</p>
<p class="p1">Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="image_4.html" alt> <strong><em>A1A1A1 N3</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for both endpoints in circles,</p>
<p><strong><em>A1 </em></strong>for approximately correct shape (concave up to concave down).</p>
<p>Only if this <strong><em>A1 </em></strong>for shape is awarded, award <strong><em>A1 </em></strong>for maximum point in circle.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(x = 1\;\;\;x = 1.83928\)</p>
<p class="p1">\(x = 1{\text{ (exact)}}\;\;\;x = 1.84{\text{ }}[1.83,{\text{ }}1.84]\) <span class="Apple-converted-space"> </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute either (<strong><em>FT</em></strong> ) limits or function into formula with \({f^2}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(V = \pi \int_1^{1.84} {{f^2},{\text{ }}\int {{{( - {x^4} + 2{x^3} - 1)}^2}{\text{d}}x} } \)</p>
<p class="p1">\(0.636581\)</p>
<p class="p1">\(V = 0.637{\text{ }}[0.636,{\text{ }}0.637]\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [8 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Despite being a straightforward question, and although most candidates had a roughly correct shape for their graph, their sketches were either out of scale or missed one of the endpoints. In part (b), a few did not give both answers despite going on to use 1.84 in part (c).</p>
<p class="p1">Part (c) proved difficult for most candidates, as only a small number could write the correct expression for the volume: some included the correct limits but did not square the function, whilst others squared the function but did not write the correct limits in the integral. Many did not find a volume, or found an incorrect volume. The latter included finding the integral from 0 to 2, or dividing the region into three parts, showing a lack of understanding of “enclosed”.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Despite being a straightforward question, and although most candidates had a roughly correct shape for their graph, their sketches were either out of scale or missed one of the endpoints. In part (b), a few did not give both answers despite going on to use 1.84 in part (c).</p>
<p class="p1">Part (c) proved difficult for most candidates, as only a small number could write the correct expression for the volume: some included the correct limits but did not square the function, whilst others squared the function but did not write the correct limits in the integral. Many did not find a volume, or found an incorrect volume. The latter included finding the integral from 0 to 2, or dividing the region into three parts, showing a lack of understanding of “enclosed”.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Despite being a straightforward question, and although most candidates had a roughly correct shape for their graph, their sketches were either out of scale or missed one of the endpoints. In part (b), a few did not give both answers despite going on to use 1.84 in part (c).</p>
<p class="p1">Part (c) proved difficult for most candidates, as only a small number could write the correct expression for the volume: some included the correct limits but did not square the function, whilst others squared the function but did not write the correct limits in the integral. Many did not find a volume, or found an incorrect volume. The latter included finding the integral from 0 to 2, or dividing the region into three parts, showing a lack of understanding of “enclosed”.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos ({{\rm{e}}^x})\) , for \( - 2 \le x \le 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P2TZ2Q2.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = - {{\rm{e}}^x}\sin ({{\rm{e}}^x})\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/Jon.png" alt> <em><strong>A1A1A1A1 N4</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for shape that must have the correct domain (from \( - 2\) to \( + 2\) ) and correct range (from \( - 6\) to \(4\) ), <em><strong>A1</strong></em> for minimum in circle, <em><strong>A1</strong></em> for maximum in circle and <em><strong>A1</strong></em> for intercepts in circles. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many students failed in applying the chain rule to find the correct derivative, and some inappropriately used the product rule. However, many of those obtained full follow through marks in part (b) for the sketch of the function they found in part (a). </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many students failed in applying the chain rule to find the correct derivative, and some inappropriately used the product rule. However, many of those obtained full follow through marks in part (b) for the sketch of the function they found in part (a). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Most candidates sketched an approximately correct shape in the given domain, though there were some that did not realize they had to set their GDC to radians, producing a meaningless sketch.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> It is very important to stress to students that although they are asked to produce a sketch, it is still necessary to show its key features such as domain and range, stationary points and intercepts.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows a square \(ABCD\), and a sector \(OAB\) of a circle centre \(O\), radius \(r\). Part of the square is shaded and labelled \(R\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_15.28.33.png" alt></p>
<p>\[{\rm{A\hat OB}} = \theta {\text{, where }}0.5 \ \le \ \theta < \pi .\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the area of the square \(ABCD\) is \(2{r^2}(1 - \cos \theta )\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>When \(\theta = \alpha \), the area of the square \(ABCD\) is equal to the area of the sector \(OAB\).</p>
<p>(i) Write down the area of the sector when \(\theta = \alpha \).</p>
<p>(ii) Hence find \(\alpha \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>When \(\theta = \beta \), the area of \(R\) is more than twice the area of the sector.</p>
<p>Find all possible values of \(\beta \).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>area of \({\text{ABCD}} = {\text{A}}{{\text{B}}^2}\) (seen anywhere) <strong><em>(A1)</em></strong></p>
<p>choose cosine rule to find a side of the square <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{a^2} = {b^2} + {c^2} - 2bc\cos \theta \)</p>
<p>correct substitution (for triangle \(AOB\)) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;{r^2} + {r^2} - 2 \times r \times r\cos \theta ,{\text{ O}}{{\text{A}}^2} + {\text{O}}{{\text{B}}^2} - 2 \times {\text{OA}} \times {\text{OB}}\cos \theta \)</p>
<p>correct working for \({\text{A}}{{\text{B}}^2}\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;2{r^{\text{2}}} - 2{r^2}\cos \theta \)</p>
<p>\({\text{area}} = 2{r^2}(1 - \cos \theta )\) <strong><em>AG N0</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award no marks if the only working is \(2{r^2} - 2{r^2}\cos \theta \).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(\frac{1}{2}\alpha {r^2}\;\;\;\left( {{\text{accept }}2{r^2}(1 - \cos \alpha )} \right)\) <strong><em>A1 N1</em></strong></p>
<p>(ii) correct equation in one variable <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2(1 - \cos \alpha ) = \frac{1}{2}\alpha \)</p>
<p>\(\alpha = 0.511024\)</p>
<p>\(\alpha = 0.511\;\;\;({\text{accept }}\theta = 0.511)\) <strong><em>A2 N2</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for \(\alpha = 0.511\) and additional answers.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note: </strong>In this part, accept \(\theta \) instead of \(\beta \), and the use of equations instead of inequalities in the working.</p>
<p> </p>
<p>attempt to find \(R\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)subtraction of areas, \({\text{square}} - {\text{segment}}\)</p>
<p>correct expression for segment area <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{1}{2}\beta {r^2} - \frac{1}{2}{r^2}\sin \beta \)</p>
<p>correct expression for \(R\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2{r^2}(1 - \cos \beta ) - \left( {\frac{1}{2}\beta {r^2} - \frac{1}{2}{r^2}\sin \beta } \right)\)</p>
<p>correct inequality <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2{r^2}(1 - \cos \beta ) - \left( {\frac{1}{2}\beta {r^2} - \frac{1}{2}{r^2}\sin \beta } \right) > 2\left( {\frac{1}{2}\beta {r^2}} \right)\)</p>
<p>correct inequality in terms of angle only <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;2(1 - \cos \beta ) - \left( {\frac{1}{2}\beta - \frac{1}{2}\sin \beta } \right) > \beta \)</p>
<p>attempt to solve their inequality, must represent \(R > \) twice sector <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)sketch, one correct value</p>
<p> </p>
<p><strong>Note: </strong>Do not award the second <strong><em>(M1) </em></strong>unless the first <strong><em>(M1) </em></strong>for attempting to find \(R\) has been awarded.</p>
<p> </p>
<p><strong>both </strong>correct values \(1.30573\) and \(2.67369\) <strong><em>(A1)</em></strong></p>
<p>correct inequality \(1.31 < \beta < 2.67\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<p><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Those who attempted part (a) could in general show what was required by using the cosine rule. On rare occasions some more complicated approaches were seen using half of angle theta. In some cases, candidates did not show all the necessary steps and lost marks for not completely showing the given result.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">A number of candidates correctly answered part (bi) and created a correct equation in (bii), but did not solve the equation correctly, usually attempting an analytic method where the GDC would do. For many a major problem was to realize the need to reduce the equation to one variable before attempting to solve it. Occasionally, an answer would be written that was outside the given domain.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">When part (c) was attempted, many candidates did not recognize that the area in question requires the subtraction of a segment area, and often set the square area greater than twice the sector. Many candidates made mistakes when trying to eliminate brackets or just did not use them. Of those who created a correct inequality, few reached a fully correct conclusion.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1"><span class="s1">Let \(f(x) = k{x^2} + kx\) and \(g(x) = x - 0.8\)</span>. The graphs of \(f\) and \(g\) intersect at two distinct points.</p>
<p class="p1">Find the possible values of \(k\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to set up equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f = g,{\text{ }}k{x^2} + kx = x - 0.8\)</p>
<p>rearranging <strong>their </strong>equation to equal zero <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;k{x^2} + kx - x + 0.8 = 0,{\text{ }}k{x^2} + x(k - 1) + 0.8 = 0\)</p>
<p>evidence of discriminant (if seen explicitly, not just in quadratic formula) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{b^2} - 4ac,{\text{ }}\Delta = {(k - 1)^2} - 4k \times 0.8,{\text{ }}D = 0\)</p>
<p>correct discriminant <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{(k - 1)^2} - 4k \times 0.8,{\text{ }}{k^2} - 5.2k + 1\)</p>
<p>evidence of correct discriminant greater than zero <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\;\;\;{k^2} - 5.2k + 1 > 0,{\text{ }}{(k - 1)^2} - 4k \times 0.8 > 0\), correct answer</p>
<p>both correct values <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;0.2,{\text{ }}5\)</p>
<p>correct answer <strong><em>A2 N3</em></strong></p>
<p><em>eg</em>\(\;\;\;k < 0.2,{\text{ }}k \ne 0,{\text{ }}k > 5\)</p>
<p><strong><em>[8 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Many candidates knew to set the equations equal, and then some knew to manipulate the equation such that it is equal to zero. Those who recognized the discriminant in this equation earned further marks, although few set a correct discriminant greater than zero. Even in such cases, finding both inequalities proved elusive for most.</p>
<p class="p1">An alternative method was to graph each function and find where the line intersects the parabola in exactly one and in two places. Few could carry this approach to adequate completion, often neglecting a second inequality for \(k\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The velocity of a particle in ms<sup>−1</sup> is given by \(v = {{\rm{e}}^{\sin t}} - 1\) , for \(0 \le t \le 5\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(v\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ronan.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the total distance travelled by the particle in the first five seconds.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the positive \(t\)-intercept.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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" alt><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape crossing <em>x</em>-axis with \(3 < x < 3.5\) . </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this <em><strong>A1</strong></em> is awarded, award the following: </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for maximum in circle, <em><strong>A1</strong></em> for endpoints in circle. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = \pi \) (exact), \(3.14\) <strong><em>A1 N1</em> </strong></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[1 mark]</strong> </span></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing distance is area under velocity curve <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(s = \int v \) , shading on diagram, attempt to integrate </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach to find the total area <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\text{area A}} + {\text{area B}}\) , \(\int {v{\rm{d}}t - \int {v{\rm{d}}t} } \) , \(\int_0^{3.14} {v{\rm{d}}t + } \int_{3.14}^5 {v{\rm{d}}t} \) , \(\int {\left| v \right|} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working with integration and limits (accept \({\rm{d}}x\) or missing \({\rm{d}}t\) ) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\int_0^{3.14} {v{\rm{d}}t + } \int_5^{3.14} {v{\rm{d}}t} \) , \(3.067 \ldots + 0.878 \ldots \) , \(\int_0^5 {\left| {{{\rm{e}}^{\sin t}} - 1} \right|} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">distance \( = 3.95\) (m) <em><strong>A1 N3 </strong></em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[4 marks]</strong> </span></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a minor error on this question, where the units for velocity were given as ms<sup>-2</sup> rather than ms<sup>-1</sup> . Examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were considered at the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates continue to produce sloppy graphs resulting in loss of marks. Although the shape was often correctly drawn, students were careless when considering the domain and other key features such as the root and the location of the maximum point.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The fact that most candidates with poorly drawn graphs correctly found the root in (b)(i), clearly emphasized the disconnect between geometric and algebraic approaches to problems.</span></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (b)(ii), most appreciated that the definite integral would give the distance travelled but few could write a valid expression and normally just integrated from \(t = 0\) to \(t = 5\) without considering the part of the graph below the \(t\)-axis. Again, analytic approaches to evaluating their integral predominated over simpler GDC approaches and some candidates had their calculator set in degree mode rather than radian mode.</span></p>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Consider a geometric sequence where the first term is 768 and the second term is 576.</p>
<p>Find the least value of \(n\) such that the \(n\)th term of the sequence is less than 7.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to find \(r\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{576}}{{768}},{\text{ }}\frac{{768}}{{576}},{\text{ }}0.75\)</p>
<p>correct expression for \({u_n}\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(768{(0.75)^{n - 1}}\)</p>
<p><strong>EITHER (solving inequality)</strong></p>
<p>valid approach (accept equation) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({u_n} < 7\)</p>
<p>valid approach to find \(n\) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(768{(0.75)^{n - 1}} = 7,{\text{ }}n - 1 > {\log _{0.75}}\left( {\frac{7}{{768}}} \right)\), sketch</p>
<p>correct value</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(n = 17.3301\) <strong><em>(A1)</em></strong></p>
<p>\(n = 18\) (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>OR (table of values)</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({u_n} > 7\), one correct crossover value</p>
<p>both crossover values, \({u_{17}} = 7.69735\) <strong>and</strong> \({u_{18}} = 5.77301\) <strong><em>A2</em></strong></p>
<p>\(n = 18\) (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>OR (sketch of functions)</strong></p>
<p>valid approach <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)sketch of appropriate functions</p>
<p>valid approach <strong><em>(M1) </em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)finding intersections or roots (depending on function sketched)</p>
<p>correct value</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(n = 17.3301\) <strong><em>(A1)</em></strong></p>
<p>\(n = 18\) (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A rock falls off the top of a cliff. Let \(h\) be its height above ground in metres, </span><span style="font-family: times new roman,times; font-size: medium;">after \(t\) seconds.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The table below gives values of \(h\) and \(t\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="images/paige.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable </span><span style="font-family: times new roman,times; font-size: medium;">model for the data. Use Jane’s model to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) write down the height of the cliff;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) find the height of the rock after 4.5 seconds;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) find after how many seconds the height of the rock is \(30{\text{ m}}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Kevin thinks that the function \(g(t) = - 5.2{t^2} + 9.5t + 100\) is a better model for </span><span style="font-family: times new roman,times; font-size: medium;">the data. Use Kevin’s model to find when the rock hits the ground.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) On graph paper, using a scale of 1 cm to 1 second, and 1 cm to 10 m, </span><span style="font-family: times new roman,times; font-size: medium;">plot the data given in the table.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) By comparing the graphs of <em>f</em> and <em>g</em> with the plotted data, explain which </span><span style="font-family: times new roman,times; font-size: medium;">function is a better model for the height of the falling rock.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(106{\text{ m}}\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) substitute \(t = 4.5\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h = 44.9{\text{ m}}\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) set up suitable equation <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(t) = 30\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 4.91\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing that height is 0 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">set up suitable equation <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(g(t) = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 5.39{\text{ secs}}\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/grape.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A2 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for correct scales on axes, </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A2</em></strong> for 5 correct points, <em><strong>A1</strong></em> for 3 or 4 correct points.</span></p>
<p align="LEFT"><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) Jane’s function, with <strong>2</strong> valid reasons <em><strong>A1R1R1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. Jane’s passes very close to all the points, Kevin’s has the rock </span><span style="font-family: times new roman,times; font-size: medium;">clearly going up initially – not possible if rock falls</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Although Jane’s also goes up initially, it only goes up very slightly, and </span><span style="font-family: times new roman,times; font-size: medium;">so is the better model.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A farmer wishes to create a rectangular enclosure, ABCD, of area 525 m<sup>2</sup>, as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/friends.png" alt></span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The fencing used for side AB costs \(\$ 11\) per metre. The fencing for the other three sides </span><span style="font-family: times new roman,times; font-size: medium;">costs \(\$ 3\) per metre. The farmer creates an enclosure so that the cost is a minimum. Find this minimum cost.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for <strong>second</strong> side, using area = 525 <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. let \({\rm{AB}} = x\) , \({\rm{AD}} = \frac{{525}}{x}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up cost function using $3 for three sides and $11 for one side <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3({\rm{AD}} + {\rm{BC}} + {\rm{CD}}) + 11{\rm{AB}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for cost <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{525}}{x} \times 3 + \frac{{525}}{x} \times 3 + 11x + 3x\) , \(\frac{{525}}{{{\rm{AB}}}} \times 3 + \frac{{525}}{{{\rm{AB}}}} \times 3 + 11{\rm{AB}} + 3{\rm{AB}}\) , \(\frac{{3150}}{x} + 14x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>EITHER</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">sketch of cost function <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">identifying minimum point <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. marking point on graph, \(x = 15\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>OR</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivative (may be seen in equation below) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(C'(x) = \frac{{ - 1575}}{{{x^2}}} + \frac{{ - 1575}}{{{x^2}}} + 14\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting their derivative equal to 0 (seen anywhere) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 3150}}{{{x^2}}} + 14 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for <strong>second</strong> side, using area = 525 <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. let \({\rm{AD}} = x\) , \({\rm{AB}} = \frac{{525}}{x}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up cost function using \(\$ 3\) for three sides and \(\$ 11\) for one side <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3({\rm{AD}} + {\rm{BC}} + {\rm{CD}}) + 11{\rm{AB}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for cost <em><strong>A2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3\left( {x + x + \frac{{525}}{x}} \right) + \frac{{525}}{x} \times 11\) , \(3\left( {{\rm{AD}} + {\rm{AD}} + \frac{{525}}{{{\rm{AD}}}}} \right) + \frac{{525}}{{{\rm{AD}}}} \times 11\) , \(6x + \frac{{7350}}{x}\)</span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">sketch of cost function <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">identifying minimum point <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. marking point on graph, \(x = 35\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivative (may be seen in equation below) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(C'(x) = 6 - \frac{{7350}}{{{x^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting their derivative equal to 0 (seen anywhere) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6 - \frac{{7350}}{{{x^2}}} = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Although this question was a rather straight-forward optimisation question, the lack of structure caused many candidates difficulty. Some were able to calculate cost values but were unable to create an algebraic cost function. Those who were able to create a cost function in two variables often could not use the area relationship to obtain a function in a single variable and so could make no further progress. Of those few who created a correct cost function, most set the derivative to zero to find that the minimum cost occurred at \(x = 15\) , leading to \(\$ 420\). Although this is a correct approach earning full marks, candidates seem not to recognise that the result can be obtained from the GDC, without the use of calculus. </span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first two terms of a geometric sequence \({u_n}\) are \({u_1} = 4\) and \({u_2} = 4.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the common ratio.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find \({u_5}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Another sequence \({v_n}\) is defined by \({v_n} = a{n^k}\), where \(a,{\text{ }}k \in \mathbb{R}\), and \(n \in {\mathbb{Z}^ + }\), such that \({v_1} = 0.05\) and \({v_2} = 0.25\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(a\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(k\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the smallest value of \(n\) for which \({v_n} > {u_n}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(r = \frac{{{u_2}}}{{{u_1}}},{\text{ }}\frac{4}{{4.2}}\)</p>
<p class="p1">\(r = 1.05\;\;\;{\text{(exact)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>attempt to substitute into formula, with <strong>their </strong>\(r\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(4 \times {1.05^n},{\text{ }}4 \times 1.05 \times 1.05 \ldots \)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(4 \times {1.05^4},{\text{ }}4 \times 1.05 \times 1.05 \times 1.05 \times 1.05\)</p>
<p class="p1">\({u_5} = 4.862025{\text{ (exact), }}4.86{\text{ }}[4.86,{\text{ }}4.87]{\text{ }}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>attempt to substitute \(n = 1\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.05 = a \times {1^k}\)</p>
<p class="p1">\(a = 0.05\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct substitution of \(n = 2\) into \({v_2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.25 = a \times {2^k}\)</p>
<p class="p1">correct work <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)finding intersection point, \(k = {\log _2}\left( {\frac{{0.25}}{{0.05}}} \right),{\text{ }}\frac{{\log 5}}{{\log 2}}\)</p>
<p class="p1">\(2.32192\)</p>
<p class="p1">\(k = {\log _2}5\;\;\;{\text{(exact), }}2.32{\text{ }}[2.32,{\text{ }}2.33]\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct expression for \({u_n}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(4 \times {1.05^{n - 1}}\)</p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">correct substitution into inequality (accept equation) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.05 \times {n^k} > 4 \times {1.05^{n - 1}}\)</p>
<p class="p1">valid approach to solve inequality (accept equation) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)finding point of intersection, \(n = 7.57994{\text{ }}(7.59508{\text{ from 2.32)}}\)</p>
<p class="p1">\(n = 8\;\;\;\)(must be an integer) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">table of values</p>
<p class="p1">when \(n = 7,{\text{ }}{u_7} = 5.3604,{\text{ }}{v_7} = 4.5836\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">when \(n = 8,{\text{ }}{u_8} = 5.6284,{\text{ }}{v_8} = 6.2496\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(n = 8\;\;\;\)(must be an integer) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [14 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates answered part (a) correctly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">A surprising number assumed the second sequence to be geometric as well, and thus part (b) was confusing for many. It was quite common that students did not clearly show which work was relevant to part (i) and which to part (ii), thus often losing marks.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Few students successfully completed part (c) as tried to solve algebraically instead of graphically. Those who used the table of values did not always show two sets of values and consequently lost marks.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = {{\text{e}}^{2\,{\text{sin}}\left( {\frac{{\pi x}}{2}} \right)}}\), for <em>x</em> > 0.</p>
<p>The <em>k </em>th maximum point on the graph of <em>f</em> has <em>x</em>-coordinate <em>x<sub>k</sub></em> where \(k \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>x<sub>k</sub></em><sub> + 1</sub> = <em>x<sub>k</sub></em> + <em>a</em>, find <em>a</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <em>n</em> such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find maxima <em><strong>(M1)</strong></em></p>
<p><em>eg</em> one correct value of <em>x<sub>k</sub></em>, sketch of <em>f</em></p>
<p>any two correct consecutive values of <em>x<sub>k</sub></em> <em><strong>(A1)(A1)</strong></em></p>
<p><em>eg x</em><sub>1</sub> = 1, <em>x</em><sub>2</sub> = 5</p>
<p><em>a</em> = 4 <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing the sequence <em>x</em><sub>1,<sup> </sup></sub> <em>x</em><sub>2</sub><sub>,<sup> </sup></sub> <em>x</em><sub>3, …,</sub> <em>x</em><sub>n</sub> is arithmetic <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em>d</em> = 4</p>
<p>correct expression for sum<em> <strong>(A1)</strong><br></em></p>
<p><em>eg </em>\(\frac{n}{2}\left( {2\left( 1 \right) + 4\left( {n - 1} \right)} \right)\)</p>
<p>valid attempt to solve for <em>n</em> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> graph, 2<em>n</em><sup>2</sup> − <em>n</em> − 861 = 0</p>
<p><em>n</em> = 21 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{3x}}{2} + 1\) , \(g(x) = 4\cos \left( {\frac{x}{3}} \right) - 1\) . Let \(h(x) = (g \circ f)(x)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for \(h(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the period of \(h\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \(h\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to form any composition (even if order is reversed) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct composition \(h(x) = g\left( {\frac{{3x}}{2} + 1} \right)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(h(x) = 4\cos \left( {\frac{{\frac{{3x}}{2} + 1}}{3}} \right) - 1\) \(\left( {4\cos \left( {\frac{1}{2}x + \frac{1}{3}} \right) - 1,4\cos \left( {\frac{{3x + 2}}{6}} \right) - 1} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3 </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">period is \(4\pi (12.6)\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">range is \( - 5 \le h(x) \le 3\) \(\left( {\left[ { - 5,3} \right]} \right)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates handled the composition of the two given functions well. However, a large number of candidates had difficulties simplifying the result correctly. The period and range of the resulting trig function was not handled well. If candidates knew the definition of "range", they often did not express it correctly. Many candidates correctly used their GDCs to find the period and range, but this approach was not the most efficient. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right),{\text{ for }} - 4 \leqslant x \leqslant 4.\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of \(f\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of \(x\) where the function is decreasing.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(a\);</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The function \(f\) can also be written in the form \(f(x) = a\sin \left( {\frac{\pi }{4}(x + c)} \right)\), where \(a \in \mathbb{R}\), and \(0 \leqslant c \leqslant 2\). Find the value of \(c\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><br><img src="images/maths_9a_markscheme.png" alt> <strong><em> A1A1A1 N3</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="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;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for approximately correct sinusoidal shape.</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;"> <strong>Only</strong> if this <strong><em>A1</em></strong> is awarded, award the following:</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;"> <strong><em>A1</em></strong> for correct domain,</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;"> <strong><em>A1</em></strong> for approximately correct range.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="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;"><strong><em>[3 marks]</em></strong></span></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 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;">recognizes decreasing to the left of minimum or right of maximum,</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;"><em>eg</em> \(f'(x) < 0\) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>x-</em>values of minimum and maximum (may be seen on sketch in part (a)) <strong><em>(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(x = - 3,{\text{ (1, 1.4)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">two correct intervals <strong><em>A1A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \( - 4 < x < - 3,{\text{ }}1 \leqslant x \leqslant 4;{\text{ }}x < - 3,{\text{ }}x \geqslant 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 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;">recognizes that \(a\) is found from amplitude of wave <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>y-</em>value of minimum or maximum <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>(−3, −1.41) , (1, 1.41)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 1.41421\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = \sqrt 2 {\text{, (exact), 1.41,}}\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 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;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognize that shift for sine is found at <em>x</em>-intercept <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find <em>x</em>-intercept <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\cos \left( {\frac{\pi }{4}x} \right) + \sin \left( {\frac{\pi }{4}x} \right) = 0,{\text{ }}x = 3 + 4k,{\text{ }}k \in \mathbb{Z}\)</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;">\(x = - 1\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(c = 1\) <strong><em>A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to use a coordinate to make an equation <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\sqrt 2 \sin \left( {\frac{\pi }{4}c} \right) = 1,{\text{ }}\sqrt 2 \sin \left( {\frac{\pi }{4}(3 - c)} \right) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to solve resulting equation <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>sketch, \(x = 3 + 4k,{\text{ }}k \in \mathbb{Z}\)</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;">\(x = - 1\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(c = 1\) <strong><em>A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after \(t\) seconds is given by</p>
<p class="p1">\[v = 6t - 6,{\text{ for }}0 \leqslant t \leqslant 2.\]</p>
<p class="p1">After \(p\) <span class="s1">seconds, the particle is 2 m </span>from its initial position. Find the possible values of \(p\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">correct approach <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(s = \int {v,{\text{ }}\int_0^p {6t - 6{\text{d}}t} } \)</p>
<p class="p1">correct integration <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {6t - 6{\text{d}}t = 3{t^2} - 6t + C,{\text{ }}\left[ {3{t^2} - 6t} \right]_0^p} \)</p>
<p class="p1">recognizing that there are two possibilities <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)2 correct answers, \(s = \pm 2,{\text{ }}c \pm 2\)</p>
<p class="p1">two correct equations in \(p\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(3{p^2} - 6p = 2,{\text{ }}3{p^2} - 6p = - 2\)</p>
<p class="p1">0.42265, 1.57735</p>
<p class="p1"><span class="Apple-converted-space">\(p = 0.423{\text{ or }}p = 1.58\) </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Most candidates realized that they needed to calculate the integral of the velocity, and did it correctly. However, only a few realized that there were two possible positions for the particle, as it could move in two directions. In general, the only equation candidates wrote was \(3{p^2} - 6p = 2\), that gave solutions outside the given domain. Candidates failed to differentiate between displacement and distance travelled.</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the expansion of \({(x + 2)^{11}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the number of terms in this expansion.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the term containing \({x^2}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">12 terms <strong><em>A1 N1</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of binomial expansion <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( \begin{array}{l}<br>n\\<br>r<br>\end{array} \right){a^{n - r}}{b^r}\) </span><span style="font-family: times new roman,times; font-size: medium;">, an attempt to expand, Pascal’s triangle</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing correct term <strong><em> (A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 10th term , \(r = 9\) , \(\left( {\begin{array}{*{20}{c}}<br>{11}\\<br>9<br>\end{array}} \right)\) , \({(x)^2}{(2)^9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>{11}\\<br>9<br>\end{array}} \right){(x)^2}{(2)^9}\) , \(55 \times {2^9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(28160{x^2}\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates attempted this question, and many made good progress. A number of candidates spent time writing out Pascal’s triangle in full. Common errors included 11 for part (a) and not writing out the simplified form of the term for part (b). Another common error was adding instead of multiplying the parts of the term in part (b). </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates attempted this question, and many made good progress. A number of candidates spent time writing out Pascal’s triangle in full. Common errors included 11 for part (a) and not writing out the simplified form of the term for part (b). Another common error was adding instead of multiplying the parts of the term in part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves along a straight line such that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\), is given by \(v(t) = 10t{{\text{e}}^{ - 1.7t}}\), for \(t \geqslant 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On the grid below, sketch the graph of \(v\), for \(0 \leqslant t \leqslant 4\).</span></p>
<p style="font: normal normal normal 17px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_5a.png" alt></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the distance travelled by the particle in the first three seconds.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the velocity of the particle when its acceleration is zero.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><img src="images/maths_5a_markscheme.png" alt> <strong><em>A1A2 N3</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes: </strong>Award <strong><em>A1 </em></strong>for approximately correct domain \(0 \leqslant t \leqslant 4\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The shape must be approximately correct, with maximum skewed left. <strong>Only</strong> if the shape is approximately correct, award <strong><em>A2 </em></strong>for all the following approximately correct features, in circle of tolerance where drawn (accept seeing correct coordinates for the maximum, even if point outside circle):</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Maximum point, passes through origin, asymptotic to \(t\)-axis (but must not touch the axis).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">If only two of these features are correct, award <strong><em>A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach (including \(0\) and \(3\)) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_0^3 {10t{{\text{e}}^{ - 1.7t}}{\text{d}}t,{\text{ }}\int_0^3 {f(x)} } \), area from \(0\) to \(3\) (may be shaded in diagram)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{distance}} = 3.33{\text{ (m)}}\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">recognizing acceleration is derivative of velocity <strong><em>(R1)</em></strong></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;"><em>eg</em> \(a = \frac{{{\text{d}}v}}{{{\text{d}}t}}\), attempt to find \(\frac{{{\text{d}}v}}{{{\text{d}}t}}\), reference to maximum on the graph of \(v\)</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;">valid approach to find \(v\) when \(a = 0\) (may be seen on graph) <strong><em>(M1)</em></strong></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;"><em>eg</em> \(\frac{{{\text{d}}v}}{{{\text{d}}t}} = 0,{\text{ }}10{{\text{e}}^{ - 1.7t}} - 17t{{\text{e}}^{ - 1.7t}} = 0,{\text{ }}t = 0.588\)</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;">\({\text{velocity}} = 2.16{\text{ (m}}{{\text{s}}^{ - 1}})\) <strong><em>A1 N3</em></strong></span></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;"> </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;"><strong>Note: </strong>Award <strong><em>R1M1A0 </em></strong>for \((0.588, 216)\) if velocity is not identified as final answer</span></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;"> </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;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 2x + 4\) and \(g(x) = 7{x^2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({f^{ - 1}}(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \((f \circ g)(3.5)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">interchanging <em>x</em> and <em>y</em> (may be seen at any time) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct manipulation <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = 2y + 4\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}(x) = \frac{{x - 4}}{2}\) (accept \(y = \frac{{x - 4}}{2},\frac{{x - 4}}{2}\) ) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">attempt to form composite (in any order) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">e.g. \(f(7{x^2}){\text{, }}2(7{x^2}) + 4{\text{, }}7{(2x + 4)^2}\)</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">\((f \circ g)(x) = 14{x^2} + 4\) <em><strong>A1</strong></em> <em><strong>N2</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(7 \times {3.5^2}\) , \(14{(3.5)^2} + 4\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((f \circ g)(3.5) = 175.5\) (accept 176) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">All parts of this question were well answered by most of the candidates. Some misunderstood part (a) and found the derivative or the reciprocal, indicating they were not familiar with the notation for an inverse function. Occasionally, the composition symbol was mistaken for multiplication. Additionally, some candidates composed in the incorrect order. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">All parts of this question were well answered by most of the candidates. Some misunderstood part (a) and found the derivative or the reciprocal, indicating they were not familiar with the notation for an inverse function. Occasionally, the composition symbol was mistaken for multiplication. Additionally, some candidates composed in the incorrect order. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">All parts of this question were well answered by most of the candidates. Some misunderstood part (a) and found the derivative or the reciprocal, indicating they were not familiar with the notation for an inverse function. Occasionally, the composition symbol was mistaken for multiplication. Additionally, some candidates composed in the incorrect order. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a circle with centre O and radius \(r\) cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/scooby.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Points A and B are on the circumference of the circle and \({\rm{A}}\hat {\rm{O}}{\rm{B}} = 1.4\) radians .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point C is on [OA] such that \({\rm{B}}\hat {\rm{C}}{\rm{O}} = \frac{\pi }{2}\) radians</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \({\rm{OC}} = r\cos 1.4\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The area of the shaded region is \(25\) cm<sup>2</sup> . Find the value of \(r\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">use right triangle trigonometry <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\cos 1.4 = \frac{{{\rm{OC}}}}{r}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{OC}} = r\cos 1.4\) <em><strong>AG N0 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct value for BC </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\rm{BC}} = r\sin 1.4\) , \(\sqrt {{r^2} - {{(r\cos 1.4)}^2}} \) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of \(\Delta {\rm{OBC}} = \frac{1}{2}r\sin 1.4 \times r\cos 1.4\) \(\left( { = \frac{1}{2}{r^2}\sin 1.4 \times \cos 1.4} \right)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of sector \({{\rm{OAB}} = \frac{1}{2}{r^2} \times 1.4}\) <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to subtract in any order <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> sector – triangle, \({\frac{1}{2}{r^2}\sin 1.4 \times \cos 1.4 - 0.7{r^2}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({0.7{r^2} - \frac{1}{2}r\sin 1.4 \times r\cos 1.4 = 25}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve <em><strong>their</strong></em> equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> sketch, writing as quadratic, \(\frac{{25}}{{0.616 \ldots }}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 6.37\) <strong><em>A1 N4 </em></strong></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[7 marks]</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Exception to <em><strong>FT</strong></em> rule. Award <strong><em>A1FT</em></strong> for a correct <strong><em>FT</em></strong> answer from a quadratic equation involving two trigonometric functions. </span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">As to be expected, candidates found this problem challenging. In part (a), many were able to use right angle trigonometry to find the length of OC.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">As to be expected, candidates found this problem challenging. Those who used a systematic approach in part (b) were more successful than those whose work was scattered about the page. While a pleasing number of candidates successfully found the area of sector AOB, far fewer were able to find the area of triangle BOC. Candidates who took an analytic approach to solving the resulting equation were generally less successful than those who used their GDC. Candidates who converted the angle to degrees generally were not very successful.<br></span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows the graph of a function \(y = f(x)\), for \( - 6 \leqslant x \leqslant - 2\).</p>
<p>The points \(( - 6,{\text{ }}6)\) and \(( - 2,{\text{ }}6)\) lie on the graph of \(f\). There is a minimum point at \(( - 4,{\text{ }}0)\).</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p style="text-align: left;">Let \(g(x) = f(x - 5)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the grid above, sketch the graph of \(g\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the domain of \(g\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct interval <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0 \leqslant y \leqslant 6,{\text{ }}[0,{\text{ }}6]\), from 0 to 6</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-15_om_06.48.18.png" alt="M17/5/MATME/SP2/ENG/TZ2/03.b/M"> <strong><em>M1A1</em></strong> <strong><em>N2</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for a horizontal shift of the whole shape, 5 units to the left or right and <strong><em>A1 </em></strong>for the correct graph.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct interval <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 1 \leqslant x \leqslant 3,{\text{ }}[ - 1,{\text{ }}3]\), from \( - 1\) to 3</p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \({{\rm{e}}^x} = 4\sin x\) , for \(0 \le x \le 2\pi \) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of appropriate approach <em><strong>M1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. a sketch, writing \({{\rm{e}}^x} - 4\sin x = 0\) </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 0.371\) , \(x = 1.36\) <em><strong> A2A2 N2N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Although many students started with an analytical approach, many also realized they were not going further and successfully used their GDC to find the intercepts with the <em>x</em>-axis if they had set the equation equal to 0, or in other cases, they found the intersection of the two graphs. The better candidates drew a reasonable sketch and found the two values without difficulty. A good number of candidates did not provide a sketch, however, and they had more trouble earning the mark for showing method. Accuracy penalties were relatively common on this question.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Let <em>g</em>(<em>x</em>) = −(<em>x</em> − 1)<sup>2</sup> + 5.</p>
</div>
<div class="specification">
<p>Let <em>f</em>(<em>x</em>) = x<sup>2</sup>. The following diagram shows part of the graph of <em>f</em>.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The graph of <em>g</em> intersects the graph of <em>f</em> at <em>x</em> = −1 and <em>x</em> = 2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of the vertex of the graph of <em>g</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the grid above, sketch the graph of g for −2 ≤ <em>x</em> ≤ 4.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graphs of <em>f</em> and <em>g</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(1,5) (exact) <em><strong> A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong> A1A1A1 N3</strong></em></p>
<p><strong>Note:</strong> The shape must be a concave-down parabola.<br>Only if the shape is correct, award the following for points in circles:<br><em><strong>A1</strong></em> for vertex,<br><em><strong>A1 </strong></em>for correct intersection points,<br><em><strong>A1 </strong></em>for correct endpoints.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>integrating and subtracting functions (in any order) <em><strong>(M1)</strong></em><br><em>eg </em>\(\int {f - g} \)</p>
<p>correct substitution of limits or functions (accept missing d<em>x</em>, but do not accept any errors, including extra bits) <em><strong>(A1)</strong></em><br>eg \(\int_{ - 1}^2 {g - f,\,\,\int { - {{\left( {x - 1} \right)}^2}} } + 5 - {x^2}\)</p>
<p>area = 9 (exact) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a geometric series, \({u_1} = \frac{1}{{81}}\) and \({u_4} = \frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(r\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the smallest value of <em>n</em> for which \({S_n} > 40\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into formula for \(n\)th term of GP <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_4} = \frac{1}{{81}}{r^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up correct equation \(\frac{1}{{81}}{r^3} = \frac{1}{3}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(r = 3\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up an inequality (accept an equation) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\frac{1}{{81}}({3^n} - 1)}}{2} > 40\) , \(\frac{{\frac{1}{{81}}(1 - {3^n})}}{{ - 2}} > 40\) , \({3^n} > 6481\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of solving <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. graph, taking logs </span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(n > 7.9888 \ldots \) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>(A1)</strong> </span></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\therefore n = 8\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">if \(n = 7\) , sum \( = 13.49 \ldots \) ; if \(n = 8\) , sum \( = 40.49 \ldots \) <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(n = 8\) </span><span style="font-family: times new roman,times; font-size: medium;">(is the smallest value) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was well done. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b) a good number of candidates did not realize that they could use logs to solve the problem, nor did they make good use of their GDCs. Some students did use a trial and error approach to check various values however, in many cases, they only checked one of the "crossover" values. Most candidates had difficulty with notation, opting to set up an equation rather than an inequality. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {x^3} - 4x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Expand \({(x + h)^3}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the formula \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}\) </span><span style="font-family: times new roman,times; font-size: medium;">to show that </span><span style="font-family: times new roman,times; font-size: medium;">the derivative of \(f(x)\) is \(3{x^2} - 4\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} - 2)\) is parallel to the tangent at </span><span style="font-family: times new roman,times; font-size: medium;">a point Q. Find the coordinates of Q.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is decreasing for \(p < x < q\) . Find the value of <em>p</em> and of <em>q</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of values for the gradient of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to expand <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({(x + h)^3} = {x^3} + 3{x^2}h + 3x{h^2} + {h^3}\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(x + h\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{{{(x + h)}^3} - 4(x + h) + 1 - ({x^3} - 4x + 1)}}{h}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplifying <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{({x^3} + 3{x^2}h + 3x{h^2} + {h^3} - 4x - 4h + 1 - {x^3} + 4x - 1)}}{h}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">factoring out <em>h</em> <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{h(3{x^2} + 3xh + {h^2} - 4)}}{h}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f'(x) = 3{x^2} - 4\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>AG N0</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(1) = - 1\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up an appropriate equation <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{x^2} - 4 = - 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">at Q, \(x = - 1,y = 4\) (Q is \(( - 1{\text{, }}4)\)) <em><strong>A1 A1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that <em>f</em> is decreasing when \(f'(x) < 0\) <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct values for <em>p</em> and <em>q</em> (but do not accept \(p = 1.15{\text{, }}q = - 1.15\) ) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(p = - 1.15{\text{, }}q = 1.15\) ; \( \pm \frac{2}{{\sqrt 3 }}\) ; an interval such as \( - 1.15 \le x \le 1.15\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) \ge - 4\) , \(y \ge - 4\) , \(\left[ { - 4,\infty } \right[\) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (a), the basic expansion was not done well. Rather than use the binomial theorem, many candidates opted to expand by multiplication which resulted in algebraic errors. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), it was clear that many candidates had difficulty with differentiation from first principles. Those that successfully set the answer up, often got lost in the simplification. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) was poorly done with many candidates assuming that the tangents were horizontal and then incorrectly estimating the maximum of <em>f</em> as the required point. Many candidates unnecessarily found the equation of the tangent and could not make any further progress. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d) many correct solutions were seen but only a very few earned the reasoning mark. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was often not attempted and if it was, candidates were not clear on what was expected. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = 5 - {x^2}\). Part of the graph of \(f\)is shown in the following diagram.</span></p>
<p style="margin: 0px; font-style: normal; font-variant: normal; font-weight: normal; font-size: 21px; line-height: normal; font-family: 'Times New Roman'; text-align: center;"><img src="images/maths_2.png" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph crosses the \(x\)-axis at the points \(\rm{A}\) and \(\rm{B}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the \(x\)-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 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;">recognizing \(f(x) = 0\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(f = 0,{\text{ }}{x^2} = 5\)</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \pm 2.23606\)</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \pm \sqrt 5 {\text{ (exact), }}x = \pm 2.24\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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;">attempt to substitute either limits or the function into formula</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;">involving \({f^2}\) <strong><em>(M1)</em></strong></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;"><em>eg</em> \(\pi \int {{{\left( {5 - {x^2}} \right)}^2}{\text{d}}x,{\text{ }}\pi \int_{ - 2.24}^{2.24} {\left( {{x^4} - 10{x^2} + 25} \right)} ,{\text{ }}2\pi \int_0^{\sqrt 5 } {{f^2}} } \)</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;">\(187.328\)</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;">volume \(= 187\) <strong><em>A2 N3</em></strong></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;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br>