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</div><h2>SL Paper 1</h2><div class="specification">
<p>The following diagram shows the graph of \(f’\), the derivative of \(f\).</p>
<p><img src="images/Schermafbeelding_2017-08-11_om_08.50.59.png" alt="M17/5/MATME/SP1/ENG/TZ1/06"></p>
<p>The graph of \(f’\) has a local minimum at A, a local maximum at B and passes through \((4,{\text{ }} - 2)\).</p>
</div>
<div class="specification">
<p>The point \({\text{P}}(4,{\text{ }}3)\) lies on the graph of the function, \(f\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the gradient of the curve of \(f\) at P.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the normal to the curve of \(f\) at P.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the concavity of the graph of \(f\) when \(4 < x < 5\) <strong>and </strong>justify your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\( - 2\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></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>gradient of normal \( = \frac{1}{2}\) <strong><em>(A1)</em></strong></p>
<p>attempt to substitute their normal gradient and coordinates of P (in any order) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(y - 4 = \frac{1}{2}(x - 3),{\text{ }}3 = \frac{1}{2}(4) + b,{\text{ }}b = 1\)</p>
<p>\(y - 3 = \frac{1}{2}(x - 4),{\text{ }}y = \frac{1}{2}x + 1,{\text{ }}x - 2y + 2 = 0\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct answer <strong>and </strong>valid reasoning <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p>answer: <em>eg</em> graph of \(f\) is concave up, concavity is positive (between \(4 < x < 5\))</p>
<p>reason: <em>eg</em> slope of \(f’\) is positive, \(f’\) is increasing, \(f’’ > 0\),</p>
<p>sign chart (must clearly be for \(f’’\) and show A and B)</p>
<p><img src="images/Schermafbeelding_2017-08-11_om_10.53.43.png" alt="M17/5/MATME/SP1/ENG/TZ1/06.b/M"></p>
<p> </p>
<p><strong>Note:</strong> The reason given must refer to a specific function/graph. Referring to “the graph” or “it” is not sufficient.</p>
<p> </p>
<p><em><strong>[2 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.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.</div>
</div>
<br><hr><br><div class="specification">
<p>The values of the functions \(f\) and \(g\) and their derivatives for \(x = 1\) and \(x = 8\) are shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_16.42.43.png" alt="M17/5/MATME/SP1/ENG/TZ2/06"></p>
<p>Let \(h(x) = f(x)g(x)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(h(1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(h'(8)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>expressing \(h(1)\) as a product of \(f(1)\) and \(g(1)\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(1) \times g(1),{\text{ }}2(9)\)</p>
<p>\(h(1) = 18\) <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>attempt to use product rule (do <strong>not </strong>accept \(h’ = f' \times g’\)) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(h’ = fg' + gf',{\text{ }}h'(8) = f'(8)g(8) + g’(8)f(8)\)</p>
<p>correct substitution of values into product rule <strong><em>(A1) </em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(h’(8) = 4(5) + 2( - 3),{\text{ }} - 6 + 20\)</p>
<p>\(h’(8) = 14\) <strong><em>A1 N2</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;">The following diagram shows part of the graph of a quadratic function <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tent.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>x</em>-intercepts are at \(( - 4{\text{, }}0)\) and \((6{\text{, }}0)\) , and the <em>y</em>-intercept is at \((0{\text{, }}240)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(f(x)\) in the form \(f(x) = - 10(x - p)(x - q)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find another expression for \(f(x)\) in the form \(f(x) = - 10{(x - h)^2} + k\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f(x)\) can also be written in the form \(f(x) = 240 + 20x - 10{x^2}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\) , at time <em>t</em> seconds is </span><span style="font-family: times new roman,times; font-size: medium;">given by \(v = 240 + 20t - 10{t^2}\) , for \(0 \le t \le 6\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of<em> t</em> when the speed of the particle is greatest.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the acceleration of the particle when its speed is zero.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;"><span style="font-family: TimesNewRomanPS-ItalicMT; font-size: medium;">\(f(x) = - 10(x + 4)(x - 6)\)</span></span><span style="font-family: times new roman,times; font-size: medium;"> <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"><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;">attempting to find the <em>x</em>-coordinate of maximum point <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. averaging the <em>x</em>-intercepts, sketch, \(y' = 0\) , axis of symmetry</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to find the <em>y</em>-coordinate of maximum point <strong><em> (M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(k = - 10(1 + 4)(1 - 6)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = - 10{(x - 1)^2} + 250\) <em><strong>A1A1 N4</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;">attempt to expand \(f(x)\) <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 10({x^2} - 2x - 24)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to complete the square <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 10({(x - 1)^2} - 1 - 24)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = - 10{(x - 1)^2} + 250\) <em><strong>A1A1 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">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 simplify <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. distributive property, \( - 10(x - 1)(x - 1) + 250\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct simplification <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 10({x^2} - 6x + 4x - 24)\) , \( - 10({x^2} - 2x + 1) + 250\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 240 + 20x - 10{x^2}\) <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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) valid approach <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. vertex of parabola, \(v'(t) = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 1\) <strong><em>A1 N2</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) recognizing \(a(t) = v'(t)\) <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a(t) = 20 - 20t\) <strong><em>A1A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">speed is zero \( \Rightarrow t = 6\) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a(6) = - 100\) (\({\text{m}}{{\text{s}}^{ - 2}}\)) <em><strong>A1 N3</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 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;">Parts (a) and (c) of this question were very well done by most candidates. </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), many candidates attempted to use the method of completing the square, but were unsuccessful dealing with the coefficient of \( - 10\). Candidates who recognized that the <em>x</em>-coordinate of the vertex was 1, then substituted this value into the function from part (a), were generally able to earn full marks here. </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;">Parts (a) and (c) of this question were very well done by most candidates. </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), it was clear that many candidates were not familiar with the relationship between velocity and acceleration, and did not understand how those concepts were related to the graph which was given. A large number of candidates used time \(t = 1\) in part b(ii), rather than \(t = 6\) . To find the acceleration, some candidates tried to integrate the velocity function, rather than taking the derivative of velocity. Still others found the derivative in part b(i), but did not realize they needed to use it in part b(ii), as well.</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 \(g(x) = 2x\sin x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(g'(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the gradient of the graph of <em>g</em> at \(x = \pi \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<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. \(uv' + vu'\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivatives \(\cos x\) , 2 <em><strong>(A1)(A1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(g'(x) = 2x\cos x + 2\sin x\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1 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">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 into gradient function <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(g'(\pi )\)</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. \(2\pi \cos \pi + 2\sin \pi \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{gradient}} = - 2\pi \) <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 answered part (a) correctly, using the product rule to find the derivative, and earned full marks here. There were some who did not know to use the product rule, and of course did not find the correct derivative. </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), many candidates substituted correctly into their derivatives, but then used incorrect values for \(\sin x\) and \(\cos x\) , leading to the wrong gradient in their final answers. </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 \(f(x) = \frac{{ax}}{{{x^2} + 1}}\) , \( - 8 \le x \le 8\) , \(a \in \mathbb{R}\) .The graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bernie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The region between \(x = 3\) and \(x = 7\) is shaded.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f( - x) = - f(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(f''(x) = \frac{{2ax({x^2} - 3)}}{{{{({x^2} + 1)}^3}}}\) , find the coordinates of all points of inflexion.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">It is given that \(\int {f(x){\rm{d}}x = \frac{a}{2}} \ln ({x^2} + 1) + C\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the area of the shaded region, giving your answer in the form \(p\ln q\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of \(\int_4^8 {2f(x - 1){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<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;">evidence of substituting \( - x\) for \(x\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f( - x) = \frac{{a( - x)}}{{{{( - x)}^2} + 1}}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f( - x) = \frac{{ - ax}}{{{x^2} + 1}}\) \(( = - f(x))\) <em><strong>AG N0</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;">\(y = - f(x)\) is reflection of \(y = f(x)\) in <em>x</em> axis</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">and \(y = f( - x)\) is reflection of \(y = f(x)\) in <em>y</em> axis <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">sketch showing these are the same <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f( - x) = \frac{{ - ax}}{{{x^2} + 1}}\) \(( = - f(x))\) <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;">evidence of appropriate approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">to set the numerator equal to 0 <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2ax({x^2} - 3) = 0\) ; \(({x^2} - 3) = 0\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(0, 0) ,</span> <span style="font-family: Times New Roman; font-size: medium;">\(\left( {\sqrt 3 ,\frac{{a\sqrt 3 }}{4}} \right)\) , \(\left( { - \sqrt 3 , - \frac{{a\sqrt 3 }}{4}} \right)\) (accept \(x = 0\) , \(y = 0\) etc) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1A1A1A1A1 N5</strong> </span></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.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) correct expression <em><strong>A2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left[ {\frac{a}{2}\ln ({x^2} + 1)} \right]_3^7\) , \(\frac{a}{2}\ln 50 - \frac{a}{2}\ln 10\) , \(\frac{a}{2}(\ln 50 - \ln 10)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area = \(\frac{a}{2}\ln 5\) </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;">(ii) <strong>METHOD 1</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing the shift that does not change the area <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_4^8 {f(x - 1){\rm{d}}x} = \int_3^7 {f(x){\rm{d}}x} \) , \(\frac{a}{2}\ln 5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that the factor of 2 doubles the area <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_4^8 {2f(x - 1){\rm{d}}x = } 2\int_4^8 {f(x - 1){\rm{d}}x} \) \(\left( { = 2\int_3^7 {f(x){\rm{d}}x} } \right)\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\int_4^8 {2f(x - 1){\rm{d}}x = a\ln 5} \) (i.e. \(2 \times \) <strong>their</strong> answer to (c)(i)) <em><strong> </strong></em></span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></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;">changing variable</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">let \(w = x - 1\) , so \(\frac{{{\rm{d}}w}}{{{\rm{d}}x}} = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2\int {f(w){\rm{d}}w = } \frac{{2a}}{2}\ln ({w^2} + 1) + c\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting correct limits</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left[ {a\ln \left[ {{{(x - 1)}^2} + 1} \right]} \right]_4^8\) , \(\left[ {a\ln ({w^2} + 1)} \right]_3^7\) , \(a\ln 50 - a\ln 10\) </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; font-size: medium;">\(\int_4^8 {2f(x - 1){\rm{d}}x = a\ln 5} \) </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;"> [7 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 achieved by some candidates, although brackets around the \( - x\) were commonly neglected. Some attempted to show the relationship by substituting a specific value for <em>x</em> . This earned no marks as a general argument is required. </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;">Although many recognized the requirement to set the second derivative to zero in (b), a majority neglected to give their answers as ordered pairs, only writing the <em>x</em>-coordinates. Some did not consider the negative root. </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 those who found a correct expression in (c)(i), many finished by calculating \(\ln 50 - \ln 10 = \ln 40\) . Few recognized that the translation did not change the area, although some factored the 2 from the integrand, appreciating that the area is double that in (c)(i).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2} - x\), for \(x \in \mathbb{R}\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.25.10.png" alt="N17/5/MATME/SP1/ENG/TZ0/08"></p>
<p>The graph of \(f\) crosses the \(x\)-axis at the origin and at the point \({\text{P}}(1,{\text{ }}0)\).</p>
</div>
<div class="specification">
<p>The line <em>L</em> is the normal to the graph of <em>f</em> at P.</p>
</div>
<div class="specification">
<p>The line \(L\) intersects the graph of \(f\) at another point Q, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-11_om_09.27.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/08.c.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(f’(1) = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of \(L\) in the form \(y = ax + b\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(x\)-coordinate of Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of \(f\) and the line \(L\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(f’(x) = 2x - 1\) <strong><em>A1A1</em></strong></p>
<p>correct substitution <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2(1) - 1,{\text{ }}2 - 1\)</p>
<p>\(f’(1) = 1\) <strong><em>AG N0</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct approach to find the gradient of the normal <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{ - 1}}{{f'(1)}},{\text{ }}{m_1}{m_2} = - 1,{\text{ slope}} = - 1\)</p>
<p>attempt to substitute correct normal gradient and coordinates into equation of a line <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(y - 0 = - 1(x - 1),{\text{ }}0 = - 1 + b,{\text{ }}b = 1,{\text{ }}L = - x + 1\)</p>
<p>\(y = - x + 1\) <strong><em>A1 N2</em></strong></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>equating expressions <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(x) = L,{\text{ }} - x + 1 = {x^2} - x\)</p>
<p>correct working (must involve combining terms) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({x^2} - 1 = 0,{\text{ }}{x^2} = 1,{\text{ }}x = 1\)</p>
<p>\(x = - 1\,\,\,\,\,\left( {{\text{accept }}Q( - 1,{\text{ }}2)} \right)\) <strong><em>A2 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>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int {L - f,{\text{ }}\int_{ - 1}^1 {(1 - {x^2}){\text{d}}x} } \), splitting area into triangles and integrals</p>
<p>correct integration <strong><em>(A1)(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left[ {x - \frac{{{x^3}}}{3}} \right]_{ - 1}^1,{\text{ }} - \frac{{{x^3}}}{3} - \frac{{{x^2}}}{2} + \frac{{{x^2}}}{2} + x\)</p>
<p>substituting <strong>their</strong> limits into <strong>their</strong> integrated function and subtracting (in any order) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(1 - \frac{1}{3} - \left( { - 1 - \frac{{ - 1}}{3}} \right)\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>for substituting into original or differentiated function.</p>
<p> </p>
<p>area \( = \frac{4}{3}\) <strong><em>A2 N3</em></strong></p>
<p><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;">
[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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{\cos x}}{{\sin x}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , for \(\sin x \ne 0\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In the following table, \(f'\left( {\frac{\pi }{2}} \right) = p\) and \(f''\left( {\frac{\pi }{2}} \right) = q\) . The table also gives approximate values of \(f'(x)\) and \(f''(x)\) near \(x = \frac{\pi }{2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/batman.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the quotient rule to show that \(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f''(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>p</em> and of <em>q</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use information from the table to explain why there is a point of inflexion on the </span><span style="font-family: times new roman,times; font-size: medium;">graph of <em>f</em> where \(x = \frac{\pi }{2}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="color: #000000; font-family: Times New Roman; font-size: medium;"> <span style="color: #3f3f3f;">\(\frac{{\rm{d}}}{{{\rm{d}}x}}\sin x = \cos x\) </span></span><span style="font-family: times new roman,times; font-size: medium;">, \(\frac{{\rm{d}}}{{{\rm{d}}x}}\cos x = - \sin x\) (seen anywhere) <em><strong>(A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using the quotient rule <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. \(\frac{{\sin x( - \sin x) - \cos x(\cos x)}}{{{{\sin }^2}x}}\) , \(\frac{{ - {{\sin }^2}x - {{\cos }^2}x}}{{{{\sin }^2}x}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f'(x) = \frac{{ - ({{\sin }^2}x + {{\cos }^2}x)}}{{{{\sin }^2}x}}\) </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;">\(f'(x) = \frac{{ - 1}}{{{{\sin }^2}x}}\) </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;"> [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;"><strong>METHOD 1</strong> </span></p>
<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. \(f'(x) = - {(\sin x)^{ - 2}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f''(x) = 2({\sin ^{ - 3}}x)(\cos x)\) \(\left( { = \frac{{2\cos x}}{{{{\sin }^3}x}}} \right)\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1A1 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 \(2{\sin ^{ - 3}}x\) , <em><strong>A1</strong></em> for \(\cos x\) . </span></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;">derivative of \({\sin ^2}x = 2\sin x\cos x\) (seen anywhere) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing quotient rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(u = - 1\) , \(v = {\sin ^2}x\) , \(f'' = \frac{{{{\sin }^2}x \times 0 - ( - 1)2\sin x\cos x}}{{{{({{\sin }^2}x)}^2}}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f''(x) = \frac{{2\sin x\cos x}}{{{{({{\sin }^2}x)}^2}}}\) \(\left( { = \frac{{2\cos x}}{{{{\sin }^3}x}}} \right)\) </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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(\frac{\pi }{2}\) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 1}}{{{{\sin }^2}\frac{\pi }{2}}}\) , \(\frac{{2\cos \frac{\pi }{2}}}{{{{\sin }^3}\frac{\pi }{2}}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(p = - 1\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(q = 0\) <em><strong>A1A1 N1N1</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><span style="font-family: times new roman,times; font-size: medium;">second derivative is zero, second derivative changes sign <em><strong>R1R1 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">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 comfortably applied the quotient rule, although some did not completely </span><span style="font-family: times new roman,times; font-size: medium;">show that the Pythagorean identity achieves the numerator of the answer given. Whether </span><span style="font-family: times new roman,times; font-size: medium;">changing to \( - {(\sin x)^{ - 2}}\) , or applying the quotient rule a second time, most candidates </span><span style="font-family: times new roman,times; font-size: medium;">neglected the chain rule in finding the second derivative.</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;">Whether </span><span style="font-family: times new roman,times; font-size: medium;">changing to \( - {(\sin x)^{ - 2}}\) , or applying the quotient rule a second time, most candidates </span><span style="font-family: times new roman,times; font-size: medium;">neglected the chain rule in finding the second derivative.</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;">Those who knew the trigonometric </span><span style="font-family: times new roman,times; font-size: medium;">ratios at </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;">typically found the values of <em>p</em> and of <em>q</em>, sometimes in follow-through from an </span><span style="font-family: times new roman,times; font-size: medium;">incorrect \(f''(x)\) .</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;">Few candidates gave two reasons from the table that supported the existence </span><span style="font-family: times new roman,times; font-size: medium;">of a point of inflexion. Most stated that the second derivative is zero and neglected to consider </span><span style="font-family: times new roman,times; font-size: medium;">the sign change to the left and right of <em>q</em>. Some discussed a change of concavity, but without </span><span style="font-family: times new roman,times; font-size: medium;">supporting this statement by referencing the change of sign in \(f''(x)\) , so no marks were </span><span style="font-family: times new roman,times; font-size: medium;">earned.</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;">Consider \(f(x) = {x^2} + \frac{p}{x}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(x \ne 0\) , where <em>p</em> is a constant.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There is a minimum value of \(f(x)\) when \(x = - 2\) . Find the value of \(p\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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;">\(f'(x) = 2x - \frac{p}{{{x^2}}}\) <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 \(2x\) , <em><strong>A1</strong></em> for \( - \frac{p}{{{x^2}}}\) . </span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[2 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;">evidence of equating derivative to 0 (seen anywhere) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of finding \(f'( - 2)\) (seen anywhere) <em><strong>(M1)</strong> </em></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. \( - 4 - \frac{p}{4} = 0\) , \( - 16 - p = 0\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = - 16\) <em><strong>A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[4 marks]</span></strong></em></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;">Candidates did well on (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;">For (b), a great number of candidates substituted into the function instead of into the derivative. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The derivate of \({x^2}\) was calculated without difficulties, but there were numerous problems regarding the derivative of \(\frac{p}{x}\) . There were several candidates who considered both <em>p</em> and <em>x</em> as variables; some tried to use the quotient rule and had difficulties, others used negative exponents and were not successful. </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;">The following diagram shows the graphs of the <strong>displacement</strong>, <strong>velocity</strong> and </span><span style="font-family: times new roman,times; font-size: medium;"><strong>acceleration</strong> of a moving object as functions of time, <em>t</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/333.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Complete the following table by noting which graph A, B or C corresponds to </span><span style="font-family: times new roman,times; font-size: medium;">each function.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/444.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>t</em> when the velocity is greatest.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<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/towel.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A2A2 N4</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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = 3\) <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;">Many candidates answered this question completely and correctly, showing a good understanding of the graphical relationship between displacement, velocity and acceleration. When done incorrectly, many answered with the displacement as graph B and acceleration as graph 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;">Many candidates found the value of <em>t</em> which gave a maximum in the remaining graph, and were awarded follow through marks.</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 \(f(x) = \cos x + \sqrt 3 \sin x\) , \(0 \le x \le 2\pi \) . The following diagram shows the graph of \(f\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/movie.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The \(y\)-intercept is at (\(0\), \(1\)) , there is a minimum point at A (\(p\), \(q\)) and a maximum </span><span style="font-family: times new roman,times; font-size: medium;">point at B.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Hence</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) show that \(q = - 2\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) verify that A is a minimum point.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum value of \(f(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function \(f(x)\) can be written in the form \(r\cos (x - a)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> and of <em>a</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<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;">\(f'(x) = - \sin x + \sqrt 3 \cos x\) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) at A, \(f'(x) = 0\) <em><strong>R1</strong></em></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. \(\sin x = \sqrt 3 \cos x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\tan x = \sqrt 3 \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = \frac{\pi }{3}\) , \(\frac{{4\pi }}{3}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute <strong>their</strong> <em>x</em> into \(f(x)\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \left( {\frac{{4\pi }}{3}} \right) + \sqrt 3 \sin \left( {\frac{{4\pi }}{3}} \right)\)</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. \( - \frac{1}{2} + \sqrt 3 \left( { - \frac{{\sqrt 3 }}{2}} \right)\)<br></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"></span><span style="font-family: times new roman,times; font-size: medium;">correct working that clearly leads to \( - 2\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - \frac{1}{2} - \frac{3}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span><span style="font-family: times new roman,times; font-size: medium;">\(q = - 2\) <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct calculations to find \(f'(x)\) either side of \(x = \frac{{4\pi }}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(\pi ) = 0 - \sqrt 3 \) , \(f'(2\pi ) = 0 + \sqrt 3 \) </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x)\) changes sign from negative to positive <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so A is a minimum <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[10 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;">max when \(x = \frac{\pi }{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">R1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correctly substituting \(x = \frac{\pi }{3}\) into \(f(x)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2} + \sqrt 3 \left( {\frac{{\sqrt 3 }}{2}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">max value is 2 <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(r = 2\) , \(a = \frac{\pi }{3}\) <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">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) and (ii).</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>Consider a function \(f\). The line <em>L</em><sub>1</sub> with equation \(y = 3x + 1\) is a tangent to the graph of \(f\) when \(x = 2\)</p>
</div>
<div class="specification">
<p>Let \(g\left( x \right) = f\left( {{x^2} + 1} \right)\) and P be the point on the graph of \(g\) where \(x = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \(f'\left( 2 \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f\left( 2 \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the graph of <em>g</em> has a gradient of 6 at P.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <em>L</em><sub>2</sub> be the tangent to the graph of <em>g</em> at P. <em>L</em><sub>1</sub>Â intersects <em>L</em><sub>2</sub> at the point Q.</p>
<p>Find the y-coordinate of Q.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognize that \(f'\left( x \right)\)Â is the gradient of the tangent at \(x\)Â Â Â <strong><em>(M1)</em></strong></p>
<p><em>eg </em>  \(f'\left( x \right) = m\)</p>
<p>\(f'\left( 2 \right) = 3\)Â (accept <em>m</em> = 3)Â Â Â <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognize that \(f\left( 2 \right) = y\left( 2 \right)\)   <em><strong>(M1)</strong></em></p>
<p><em>eg</em>Â \(f\left( 2 \right) = 3 \times 2 + 1\)</p>
<p>\(f\left( 2 \right) = 7\)Â Â Â <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognize that the gradient of the graph of <em>g</em> is \(g'\left( x \right)\)   <em><strong>(M1)</strong></em></p>
<p>choosing chain rule to find \(g'\left( x \right)\)Â Â Â <em><strong>(M1)</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,u = {x^2} + 1,\,\,u' = 2x\)</p>
<p>\(g'\left( x \right) = f'\left( {{x^2} + 1} \right) \times 2x\)Â Â Â <em><strong>A2</strong></em></p>
<p>\(g'\left( 1 \right) = 3 \times 2\)Â Â Â <em><strong>A1</strong></em></p>
<p>\(g'\left( 1 \right) = 6\)Â Â Â <em><strong>AG N0 </strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<p>Â </p>
<p>Â </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Â at Q, <em>L</em><sub>1</sub> =Â <em>L</em><sub>2</sub>Â (seen anywhere)Â Â Â <em><strong> (M1)</strong></em></p>
<p>recognize that the gradient of <em>L</em><sub>2</sub>Â is <em>g'</em>(1)Â Â (seen anywhere)Â Â <em><strong> Â (M1)</strong></em><br><em>eg</em>Â <em>m</em> = 6</p>
<p>finding <em>g </em>(1)  (seen anywhere)   <em><strong>(A1)</strong></em><br><em>eg  </em>\(g\left( 1 \right) = f\left( 2 \right),\,\,g\left( 1 \right) = 7\)</p>
<p>attempt to substitute gradient and/or coordinates into equation of a straight line   <em><strong>M1</strong></em><br><em>eg  </em>\(y - g\left( 1 \right) = 6\left( {x - 1} \right),\,\,y - 1 = g'\left( 1 \right)\left( {x - 7} \right),\,\,7 = 6\left( 1 \right) + {\text{b}}\)</p>
<p>correct equation for <em>L</em><sub>2</sub> </p>
<p><em>eg  </em>\(y - 7 = 6\left( {x - 1} \right),\,\,y = 6x + 1\)   <em><strong>A1</strong></em></p>
<p>correct working to find Q    <em><strong>(A1)</strong></em><br><em>eg  </em>same <em>y</em>-intercept, \(3x = 0\)</p>
<p>\(y = 1\)Â Â Â <em><strong>A1 N2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<p>Â </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">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 \(f(x) = a\sin (b(x - c)) + d\) , for \(2 \le x \le 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/deanna.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum point at P(4, 12) and a minimum point at Q(8, −4) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the graph to write down the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>a</em> ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>c</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <em>d</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(b = \frac{\pi }{4}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">At a point R, the gradient is \( - 2\pi \) . Find the <em>x</em>-coordinate of R.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<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) \(a = 8\) <strong><em>A1 N1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(c = 2\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \(d = 4\) <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(i), (ii) and (iii).</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;">recognizing that period \( = 8\) <em><strong>(A1)</strong></em></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. \(8 = \frac{{2\pi }}{b}\) , \(b = \frac{{2\pi }}{8}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{\pi }{4}\) <em><strong>AG N0</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;">attempt to substitute <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(12 = 8\sin (b(4 - 2)) + 4\)</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. \(\sin 2b = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{\pi }{4}\) <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to differentiate or choosing chain rule <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \frac{\pi }{4}(x - 2)\) , \(\frac{\pi }{4} \times 8\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f'(x) = 2\pi \cos \left( {\frac{\pi }{4}(x - 2)} \right)\) (</span><span style="font-family: times new roman,times; font-size: medium;">accept \(2\pi \cos \frac{\pi }{4}(x - 2)\) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that gradient is \(f'(x)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = m\)</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. \( - 2\pi = 2\pi \cos \left( {\frac{\pi }{4}(x - 2)} \right)\) , \( - 1 = \cos \left( {\frac{\pi }{4}(x - 2)} \right)\)</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. \({\cos ^{ - 1}}( - 1) = \frac{\pi }{4}(x - 2)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">using \({\cos ^{ - 1}}( - 1) = \pi \) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi = \frac{\pi }{4}(x - 2)\)</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. \(4 = (x - 2)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 6\) <em><strong>A1 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><span style="font-family: times new roman,times; font-size: medium;">Part (a) of this question proved challenging for most candidates. </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;">Although a good number of candidates recognized that the period was 8 in part (b), there were some who did not seem to realize that this period could be found using the given coordinates of the maximum and minimum points. </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), not many candidates found the correct derivative using the chain rule. </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;">For part (d), a good number of candidates correctly set their expression equal to \( - 2\pi \) , but errors in their previous values kept most from correctly solving the equation. Most candidates who had the correct equation were able to gain full marks here. </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;">Given that \(f(x) = \frac{1}{x}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , answer the following.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the first four derivatives of \(f(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write an expression for \({f^{(n)}}(x)\) in terms of <em>x</em> and <em>n</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<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;">\(f'(x) = - {x^{ - 2}}\) (or \( - \frac{1}{{{x^2}}}\) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f''(x) = 2{x^{ - 3}}\) (or \(\frac{2}{{{x^3}}}\) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'''(x) = - 6{x^{ - 4}}\) (or \( - \frac{6}{{{x^4}}}\) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({f^{(4)}}(x) = 24{x^{ - 5}}\) (or \(\frac{{24}}{{{x^5}}}\) ) </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;">[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;">\({f^{(n)}}(x) = \frac{{{{( - 1)}^n}n!}}{{{x^{n + 1}}}}\) or \({( - 1)^n}n!({x^{ - (n + 1)}})\) <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">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 \(f(x) = 6 + 6\sin x\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/abba.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the curve of <em>f</em> , the <em>x</em>-axis, and the <em>y</em>-axis.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Solve for \(0 \le x < 2\pi \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(6 + 6\sin x = 6\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(6 + 6\sin x = 0\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the exact value of the <em>x</em>-intercept of <em>f</em> , for \(0 \le x < 2\pi \) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The area of the shaded region is <em>k</em> . Find the value of <em>k</em> , giving your answer in </span><span style="font-family: times new roman,times; font-size: medium;">terms of \(\pi \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Give a full geometric description of this transformation.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = 6 + 6\sin \left( {x - \frac{\pi }{2}} \right)\) . </span><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is transformed to the graph of <em>g</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_p^{p + \frac{{3\pi }}{2}} {g(x){\rm{d}}x} = k\) and \(0 \le p < 2\pi \) , write down the two values of <em>p</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<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) \(\sin x = 0\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 0\) , \(x = \pi \) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(\sin x = - 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = \frac{{3\pi }}{2}\) </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;">[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;">\(\frac{{3\pi }}{2}\) <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;">evidence of using anti-differentiation <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^{\frac{{3\pi }}{2}} {(6 + 6\sin x){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct integral \(6x - 6\cos x\) (seen anywhere) <em><strong>A1A1</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. \(6\left( {\frac{{3\pi }}{2}} \right) - 6\cos \left( {\frac{{3\pi }}{2}} \right) - ( - 6\cos 0)\) , \(9\pi - 0 + 6\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(k = 9\pi + 6\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1 N3</span></strong></em></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;">translation of \(\left( {\begin{array}{*{20}{c}}<br>{\frac{\pi }{2}}\\<br>0<br>\end{array}} \right)\) <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">d.</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 the area under <em>g</em> is the same as the shaded region in <em>f</em> <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{\pi }{2}\) , \(p = 0\) <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">e.</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 again had difficulty finding the common angles in the trigonometric </span><span style="font-family: times new roman,times; font-size: medium;">equations. In part (a), some did not show sufficient working in solving the equations. Others </span><span style="font-family: times new roman,times; font-size: medium;">obtained a single solution in (a)(i) and did not find another. Some candidates worked in </span><span style="font-family: times new roman,times; font-size: medium;">degrees; the majority worked in radians.</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;">While some candidates appeared to use their understanding of the graph of the original </span><span style="font-family: times new roman,times; font-size: medium;">function to find the <em>x</em>-intercept in part (b), most used their working from part (a)(ii) sometimes </span><span style="font-family: times new roman,times; font-size: medium;">with follow-through on an incorrect answer.</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;">Most candidates recognized the need for integration in part (c) but far fewer were able to see </span><span style="font-family: times new roman,times; font-size: medium;">the solution through correctly to the end. Some did not show the full substitution of the limits, </span><span style="font-family: times new roman,times; font-size: medium;">having incorrectly assumed that evaluating the integral at 0 would be 0; without this working, </span><span style="font-family: times new roman,times; font-size: medium;">the mark for evaluating at the limits could not be earned. Again, many candidates had trouble </span><span style="font-family: times new roman,times; font-size: medium;">working with the common trigonometric values.</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;">While there was an issue in the wording of the question with the given domains, this did not </span><span style="font-family: times new roman,times; font-size: medium;">appear to bother candidates in part (d). This part was often well completed with candidates </span><span style="font-family: times new roman,times; font-size: medium;">using a variety of language to describe the horizontal translation to the right by \(\frac{\pi }{2}\) .</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;">Most candidates who attempted part (e) realized that the integral was equal to the value that they had found in part (c), but a majority tried to integrate the function <em>g</em> without success. Some candidates used sketches to find one or both values for <em>p</em>. The problem in the wording of the question did not appear to have been noticed by candidates in this part either.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = {x^2}\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_17.08.23.png" alt="M17/5/MATME/SP1/ENG/TZ2/10"></p>
<p>The line \(L\) is the tangent to the graph of \(f\) at the point \({\text{A}}( - k,{\text{ }}{k^2})\), and intersects the \(x\)-axis at point B. The point C is \(( - k,{\text{ }}0)\).</p>
</div>
<div class="specification">
<p>The region \(R\) is enclosed by \(L\), the graph of \(f\), and the \(x\)-axis. This is shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_17.07.29.png" alt="M17/5/MATME/SP1/ENG/TZ2/10.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down \(f'(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the gradient of \(L\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the \(x\)-coordinate of B is \( - \frac{k}{2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of triangle ABC, giving your answer in terms of \(k\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the area of triangle ABC is \(p\) times the area of \(R\), find the value of \(p\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(f'(x) = 2x\) <em><strong>A1</strong></em> <em><strong>N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute \(x = - k\) into their derivative <strong><em>(M1)</em></strong></p>
<p>gradient of \(L\) is \( - 2k\) <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><strong>METHOD 1 </strong></p>
<p>attempt to substitute coordinates of A and their gradient into equation of a line <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({k^2} = - 2k( - k) + b\)</p>
<p>correct equation of \(L\) in any form <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(y - {k^2} = - 2k(x + k),{\text{ }}y = - 2kx - {k^2}\)</p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(y = 0\)</p>
<p>correct substitution into \(L\) equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - {k^2} = - 2kx - 2{k^2},{\text{ }}0 = - 2kx - {k^2}\)</p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2kx = - {k^2}\)</p>
<p>\(x = - \frac{k}{2}\) <strong><em>AG</em></strong> <strong><em>N0</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{gradient}} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}},{\text{ }} - 2k = \frac{{{\text{rise}}}}{{{\text{run}}}}\)</p>
<p>recognizing \(y = 0\) at B <strong><em>(A1)</em></strong></p>
<p>attempt to substitute coordinates of A and B into slope formula <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^2} - 0}}{{ - k - x}},{\text{ }}\frac{{ - {k^2}}}{{x + k}}\)</p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^2} - 0}}{{ - k - x}} = - 2k,{\text{ }}\frac{{ - {k^2}}}{{x + k}} = - 2k,{\text{ }} - {k^2} = - 2k(x + k)\)</p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2kx = - {k^2}\)</p>
<p>\(x = - \frac{k}{2}\) <strong><em>AG</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find area of triangle <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}({k^2})\left( {\frac{k}{2}} \right)\)</p>
<p>area of \({\text{ABC}} = \frac{{{k^3}}}{4}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (</strong>\(\int {f - {\text{triangle}}} \)<strong>)</strong></p>
<p>valid approach to find area from \( - k\) to 0 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_{ - k}^0 {{x^2}{\text{d}}x,{\text{ }}\int_0^{ - k} f } \)</p>
<p>correct integration (seen anywhere, even if <strong><em>M0 </em></strong>awarded) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{x^3}}}{3},{\text{ }}\left[ {\frac{1}{3}{x^3}} \right]_{ - k}^0\)</p>
<p>substituting <strong>their </strong>limits into <strong>their </strong>integrated function and subtracting <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0 - \frac{{{{( - k)}^3}}}{3}\), area from \( - k\) to 0 is \(\frac{{{k^3}}}{3}\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>for substituting into original or differentiated function.</p>
<p> </p>
<p>attempt to find area of \(R\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_{ - k}^0 {f(x){\text{d}}x - {\text{ triangle}}} \)</p>
<p>correct working for \(R\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^3}}}{3} - \frac{{{k^3}}}{4},{\text{ }}R = \frac{{{k^3}}}{{12}}\)</p>
<p>correct substitution into \({\text{triangle}} = pR\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^3}}}{4} = p\left( {\frac{{{k^3}}}{3} - \frac{{{k^3}}}{4}} \right),{\text{ }}\frac{{{k^3}}}{4} = p\left( {\frac{{{k^3}}}{{12}}} \right)\)</p>
<p>\(p = 3\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>METHOD 2 (</strong>\(\int {(f - L)} \)<strong>)</strong></p>
<p>valid approach to find area of \(R\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_{ - k}^{ - \frac{k}{2}} {{x^2} - ( - 2kx - {k^2}){\text{d}}x + \int_{ - \frac{k}{2}}^0 {{x^2}{\text{d}}x,{\text{ }}\int_{ - k}^{ - \frac{k}{2}} {(f - L) + \int_{ - \frac{k}{2}}^0 f } } } \)</p>
<p>correct integration (seen anywhere, even if <strong><em>M0 </em></strong>awarded) <strong><em>A2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{x^3}}}{3} + k{x^2} + {k^2}x,{\text{ }}\left[ {\frac{{{x^3}}}{3} + k{x^2} + {k^2}x} \right]_{ - k}^{ - \frac{k}{2}} + \left[ {\frac{{{x^3}}}{3}} \right]_{ - \frac{k}{2}}^0\)</p>
<p>substituting <strong>their </strong>limits into <strong>their </strong>integrated function and subtracting <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\frac{{{{\left( { - \frac{k}{2}} \right)}^3}}}{3} + k{{\left( { - \frac{k}{2}} \right)}^2} + {k^2}\left( { - \frac{k}{2}} \right)} \right) - \left( {\frac{{{{( - k)}^3}}}{3} + k{{( - k)}^2} + {k^2}( - k)} \right) + (0) - \left( {\frac{{{{\left( { - \frac{k}{2}} \right)}^3}}}{3}} \right)\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>for substituting into original or differentiated function.</p>
<p> </p>
<p>correct working for \(R\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^3}}}{{24}} + \frac{{{k^3}}}{{24}},{\text{ }} - \frac{{{k^3}}}{{24}} + \frac{{{k^3}}}{4} - \frac{{{k^3}}}{2} + \frac{{{k^3}}}{3} - {k^3} + {k^3} + \frac{{{k^3}}}{{24}},{\text{ }}R = \frac{{{k^3}}}{{12}}\)</p>
<p>correct substitution into \({\text{triangle}} = pR\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^3}}}{4} = p\left( {\frac{{{k^3}}}{{24}} + \frac{{{k^3}}}{{24}}} \right),{\text{ }}\frac{{{k^3}}}{4} = p\left( {\frac{{{k^3}}}{{12}}} \right)\)</p>
<p>\(p = 3\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[7 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.</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">Let \(f(x) = \sqrt {4x + 5} \), for \(x \geqslant - 1.25\).</p>
</div>
<div class="specification">
<p class="p1">Consider another function \(g\)<span class="s1">. Let R </span>be a point on the graph of \(g\). The \(x\)<span class="s1">-coordinate of R is 1. The equation of the tangent to the graph at R </span>is \(y = 3x + 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f'(1)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down \(g'(1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(g(1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(h(x) = f(x) \times g(x)\). Find the equation of the tangent to the graph of \(h\) at the point where \(x = 1\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">choosing chain rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},{\text{ }}u = 4x + 5,{\text{ }}u' = 4\)</p>
<p class="p1">correct derivative of \(f\) <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}{(4x + 5)^{ - \frac{1}{2}}} \times 4,{\text{ }}f'(x) = \frac{2}{{\sqrt {4x + 5} }}\)</p>
<p class="p1"><span class="Apple-converted-space">\(f'(1) = \frac{2}{3}\) </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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognize that \(g'(x)\) is the gradient of the tangent <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(g'(x) = m\)</p>
<p class="p1"><span class="Apple-converted-space">\(g'(1) = 3\) </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"><span class="s1">recognize that R </span>is on the tangent <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(g(1) = 3 \times 1 + 6\), sketch</p>
<p class="p1"><span class="Apple-converted-space">\(g(1) = 9\) </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">\(f(1) = \sqrt {4 + 5} {\text{ }}( = 3)\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(h(1) = 3 \times 9{\text{ }}( = 27)\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">choosing product rule to find \(h'(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(uv' + u'v\)</p>
<p class="p1">correct substitution to find \(h'(1)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(1) \times g'(1) + f'(1) \times g(1)\)</p>
<p class="p1">\(h'(1) = 3 \times 3 + \frac{2}{3} \times 9{\text{ }}( = 15)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">attempt to substitute coordinates (in any order) into the equation of a straight line <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(y - 27 = h'(1)(x - 1),{\text{ }}y - 1 = 15(x - 27)\)</p>
<p class="p1">\(y - 27 = 15(x - 1)\) <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">attempt to substitute coordinates (in any order) to find the \(y\)-intercept <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(27 = 15 \times 1 + b,{\text{ }}1 = 15 \times 27 + b\)</p>
<p class="p1">\(y = 15x + 12\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[7 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">Part a) was relatively well answered – the obvious errors seen were not using the chain rule correctly and simple fraction calculations being wrong.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In parts b) and c) it seemed that the students did not have a good conceptual understanding of what was actually happening in this question. There was lack of understanding of tangents, gradients and their relationship to the original function, \(g\). A working sketch may have been beneficial but few were seen and many did a lot more work than required.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In parts b) and c) it seemed that the students did not have a good conceptual understanding of what was actually happening in this question. There was lack of understanding of tangents, gradients and their relationship to the original function, \(g\). A working sketch may have been beneficial but few were seen and many did a lot more work than required.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part d) although candidates recognized \(h(x)\) as a product and may have correctly found \(h(1)\), they did not necessarily use the product rule to find \(h'(x)\), instead incorrectly using \(h'(x) = f'(x) \times g'(x)\). It was rare for a candidate to get as far as finding the equation of a straight line but those who did usually gained full marks.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \cos x\).</p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = {x^k}\), where \(k \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="specification">
<p>Let \(k = 21\) and \(h(x) = \left( {{f^{(19)}}(x) \times {g^{(19)}}(x)} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the first four derivatives of \(f(x)\)<span class="s1">.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find \({f^{(19)}}(x)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the first three derivatives of \(g(x)\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that \({g^{(19)}}(x) = \frac{{k!}}{{(k - p)!}}({x^{k - 19}})\), find \(p\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find \(h'(x)\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence, show that \(h'(\pi ) = \frac{{ - 21!}}{2}{\pi ^2}\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(f'(x) = - \sin x,{\text{ }}f''(x) = - \cos x,{\text{ }}{f^{(3)}}(x) = \sin x,{\text{ }}{f^{(4)}}(x) = \cos x\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach <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>\(\,\,\,\,\,\)</span>recognizing that 19 is one less than a multiple of 4<span class="s2">, \({f^{(19)}}(x) = {f^{(3)}}(x)\)</span></p>
<p class="p3"><span class="s2">\({f^{(19)}}(x) = \sin x\) <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>[4 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"> \(g'(x) = k{x^{k - 1}}\)</span></p>
<p class="p2"><span class="s1">\(g''(x) = k(k - 1){x^{k - 2}},{\text{ }}{g^{(3)}}(x) = k(k - 1)(k - 2){x^{k - 3}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p3">correct working that leads to the correct answer, involving the correct expression for the 19th <span class="s1">derivative <span class="Apple-converted-space"> </span></span><span class="s2"><strong><em>A2</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(k(k - 1)(k - 2) \ldots (k - 18) \times \frac{{(k - 19)!}}{{(k - 19)!}},{{\text{ }}_k}{P_{19}}\)</p>
<p class="p2"><span class="s3">\(p = 19\) (accept \(\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}\)</span><span class="s1">) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p3">correct working involving recognizing patterns in coefficients of first three derivatives (may be seen in part (b)(i)) leading to a general rule for 19th <span class="s1">coefficient <span class="Apple-converted-space"> </span></span><span class="s2"><strong><em>A2</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(g'' = 2!\left( {\begin{array}{*{20}{c}} k \\ 2 \end{array}} \right),{\text{ }}k(k - 1)(k - 2) = \frac{{k!}}{{(k - 3)!}},{\text{ }}{g^{(3)}}(x){ = _k}{P_3}({x^{k - 3}})\)</p>
<p class="p2">\({g^{(19)}}(x) = 19!\left( {\begin{array}{*{20}{c}} k \\ {19} \end{array}} \right),{\text{ }}19! \times \frac{{k!}}{{(k - 19)! \times 19!}},{{\text{ }}_k}{P_{19}}\)</p>
<p class="p2"><span class="s3">\(p = 19\) (accept \(\frac{{k!}}{{(k - 19)!}}{x^{k - 19}}\)</span><span class="s1">) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p2"><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">(i) <span class="Apple-converted-space"> </span>valid approach using product rule <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(uv' + vu',{\text{ }}{f^{(19)}}{g^{(20)}} + {f^{(20)}}{g^{(19)}}\)</p>
<p class="p1"><span class="s2">correct 20th </span>derivatives (must be seen in product rule) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)(A1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\({g^{(20)}}(x) = \frac{{21!}}{{(21 - 20)!}}x,{\text{ }}{f^{(20)}}(x) = \cos x\)</p>
<p class="p3"><span class="Apple-converted-space">\(h'(x) = \sin x(21!x) + \cos x\left( {\frac{{21!}}{2}{x^2}} \right){\text{ }}\left( {{\text{accept }}\sin x\left( {\frac{{21!}}{{1!}}x} \right) + \cos x\left( {\frac{{21!}}{{2!}}{x^2}} \right)} \right)\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>substituting \(x = \pi \) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({f^{(19)}}(\pi ){g^{(20)}}(\pi ) + {f^{(20)}}(\pi ){g^{(19)}}(\pi ),{\text{ }}\sin \pi \frac{{21!}}{{1!}}\pi + \cos \pi \frac{{21!}}{{2!}}{\pi ^2}\)</p>
<p class="p4">evidence of one correct value for \(\sin \pi \) or \(\cos \pi \) <span class="s3">(seen anywhere) <span class="Apple-converted-space"> </span></span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(\sin \pi = 0,{\text{ }}\cos \pi = - 1\)</p>
<p class="p1">evidence of correct values substituted into \(h'(\pi )\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(21!(\pi )\left( {0 - \frac{\pi }{{2!}}} \right),{\text{ }}21!(\pi )\left( { - \frac{\pi }{2}} \right),{\text{ }}0 + ( - 1)\frac{{21!}}{2}{\pi ^2}\)</p>
<p class="p5"> </p>
<p class="p6"><span class="s4"><strong>Note: </strong></span>If candidates write only the first line followed by the answer, award <span class="s1"><strong><em>A1A0A0</em></strong></span>.</p>
<p class="p7"> </p>
<p class="p3"><span class="s2">\(\frac{{ - 21!}}{2}{\pi ^2}\) <span class="Apple-converted-space"> </span></span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p3"><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.</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 diagram shows part of the graph of \(y = f'(x)\) . The <em>x</em>-intercepts are at points A and C. </span><span style="font-family: times new roman,times; font-size: medium;">There is a minimum at B, and a maximum at D.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/friday.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of \(f'(x)\) at C.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) </span><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong>, show that C corresponds to a minimum on the graph of <em>f</em> , i.e. it </span><span style="font-family: times new roman,times; font-size: medium;">has the same <em>x</em>-coordinate.</span></p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Which of the points A, B, D corresponds to a maximum on the graph of <em>f</em> ?</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that B corresponds to a point of inflexion on the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<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) \(f'(x) = 0\) <strong><em> A1 N1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(f'(x) < 0\) to the left of C, \(f'(x) > 0\) to the right of C <strong><em>R1R1 N2</em> </strong></span></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;"> \(f''(x) > 0\) <strong><em>R2 N2</em> </strong></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(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A <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 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;">\(f''(x) = 0\) <strong><em>R2</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">discussion of sign change of \(f''(x)\) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) < 0\) to the left of B and \(f''(x) > 0\) to the right of B; \(f''(x)\) changes </span><span style="font-family: times new roman,times; font-size: medium;">sign either side of B</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">B is a point of inflexion <em><strong>AG N0</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;">B is a minimum on the graph of the derivative \({f'}\) <strong><em>R2</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">discussion of sign change of \(f''(x)\) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) < 0\) to the left of B and \(f''(x) > 0\) to the right of B; \(f''(x)\) changes <span style="font-family: times new roman,times; font-size: medium;">sign either side of B</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">B is a point of inflexion <em><strong>AG N0</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">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 variation in successful and unsuccessful responses to this question was remarkable. Many candidates did not even attempt it. Candidates could often determine from the graph, the minimum and maximum values of the original function, but few could correctly use the graph to analyse and justify these results. Responses indicated that some candidates did not realize that they were looking at the graph of \({f'}\) and not the graph of \(f\) . </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;">The variation in successful and unsuccessful responses to this question was remarkable. Many candidates did not even attempt it. Candidates could often determine from the graph, the minimum and maximum values of the original function, but few could correctly use the graph to analyse and justify these results. Responses indicated that some candidates did not realize that they were looking at the graph of \({f'}\) and not the graph of \(f\) .</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), many candidates once more failed to respect the command term "show" and often provided an incomplete answer. Candidates should be encouraged to refer to the number of marks available for a particular part when deciding how much information should be given. </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 \(f(x) = {{\rm{e}}^{6x}}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(f'(x)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The tangent to the graph of <em>f</em> at the point \({\text{P}}(0{\text{, }}b)\) has gradient <em>m</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(m = 6\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find <em>b</em> .</span></p>
<p> </p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, write down the equation of this tangent.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<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;">\(f'(x) = 6{{\rm{e}}^{6x}}\) <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;">(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. \(f'(0)\) , \(6{{\rm{e}}^{6 \times 0}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct manipulation <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6{{\rm{e}}^0}\) , \(6 \times 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(m = 6\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of finding \(f(0)\) <em><strong> (M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = {{\rm{e}}^{6(0)}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = 1\) <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(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = 6x + 1\) <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;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the whole, candidates handled this question quite well with most candidates correctly applying the chain rule to an exponential function and successfully finding the equation of the tangent line. </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;">On the whole, candidates handled this question quite well with most candidates correctly applying the chain rule to an exponential function and successfully finding the equation of the tangent line. Some candidates lost a mark in (b)(i) for not showing sufficient working leading to the given answer. </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;">On the whole, candidates handled this question quite well. </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;">Consider the functions \(f(x)\) , \(g(x)\) and \(h(x)\) . The following table gives some values </span><span style="font-family: times new roman,times; font-size: medium;">associated with these functions.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/omt.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows parts of the graphs of \(h\) and \(h''\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/jls.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a point of inflexion on the graph of \(h\) at P, when \(x = 3\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(h(x) = f(x) \times g(x)\) ,<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(g(3)\) , of \(f'(3)\) , and of \(h''(2)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Explain why P is a point of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">find the \(y\)-coordinate of P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">find the equation of the normal to the graph of \(h\) at P.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<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;">\(g(3) = - 18\) , \(f'(3) = 1\) , \(h''(2) = - 6\) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(h''(3) = 0\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid reasoning <em><strong>R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({h''}\) changes sign at \(x = 3\) , change in concavity of \(h\) at \(x = 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so P is a point of inflexion <strong><em>AG N0 </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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">writing \(h(3)\) as a product of \(f(3)\) and \(g(3)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f(3) \times g(3)\) , \(3 \times ( - 18)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(h(3) = - 54\) <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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing need to find derivative of \(h\) <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({h'}\) , \(h'(3)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to use the product rule (do <strong>not</strong> accept \(h' = f' \times g'\) ) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(h' = fg' + gf'\) , \(h'(3) = f(3) \times g'(3) + g(3) \times f'(3)\)</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;"><em>eg </em> \(h'(3) = 3( - 3) + ( - 18) \times 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(h'(3) = - 27\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find the gradient of the normal <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \( - \frac{1}{m}\) , \( - \frac{1}{{27}}x\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute <strong>their</strong> coordinates and <strong>their</strong> normal gradient into the equation of a line <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \( - 54 = \frac{1}{{27}}(3) + b\) , \(0 = \frac{1}{{27}}(3) + b\) , \(y + 54 = 27(x - 3)\) , \(y - 54 = \frac{1}{{27}}(x + 3)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation in any form <em><strong>A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(y + 54 = \frac{1}{{27}}(x - 3)\) , \(y = \frac{1}{{27}}x - 54\frac{1}{9}\)</span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[7 marks]</strong> </span></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;">Nearly all candidates who attempted to answer parts (a) and (c) did so correctly, as these questions simply required them to understand the notation being used and to read the values from the given table.</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 majority of candidates earned one mark for stating that \(h''(x) = 0\) at point P. As this is not enough to determine a point of inflexion, very few candidates earned full marks on 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;">Nearly all candidates who attempted to answer parts (a) and (c) did so correctly, as these questions simply required them to understand the notation being used and to read the values from the given table.</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 to be quite challenging for even the strongest candidates, as almost none of them used the product rule to find \(h'(3)\). The most common error was to say \(h'(3) = f'(3) \times g'(3)\). Despite this error, many candidates were able to earn further method marks for their work in finding the equation of the normal. There were also a small number of candidates who were able to find the equation for \(h'(x)\) , and from that \(h''(x)\). These candidates were often successful in earning full marks, although this method was quite time-consuming.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\int {\frac{1}{{2x + 3}}} {\rm{d}}x\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \ln \sqrt P \) , find the value of <em>P</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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;">\(\int {\frac{1}{{2x + 3}}} {\rm{d}}x = \frac{1}{2}\ln (2x + 3) + C\) (accept \(\frac{1}{2}\ln |(2x + 3)| + C\) ) <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;">\(\int_0^3 {\frac{1}{{2x + 3}}} {\rm{d}}x = \left[ {\frac{1}{2}\ln (2x + 3)} \right]_0^3\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substitution of limits <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g.\(\frac{1}{2}\ln 9 - \frac{1}{2}\ln 3\) </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correctly using \(\ln a - \ln b = \ln \frac{a}{b}\)</span><span style="font-family: times new roman,times; font-size: medium;"> (seen anywhere) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{2}\ln 3\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correctly using \(a\ln b = \ln {b^a}\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln \sqrt {\frac{9}{3}} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 3\) (accept \(\ln \sqrt 3 \) ) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates were unable to correctly integrate but did recognize that the integral involved the natural log function; they most often missed the factor \(\frac{1}{2}\) or replaced it with 2.</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) proved difficult as many were unable to use the basic rules of logarithms.</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 following diagram shows the graph of a quadratic function <em>f</em> , for \(0 \le x \le 4\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q8.jpg" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph passes through the point P(0, 13) , and its vertex is the point V(2, 1) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function can be written in the form \(f(x) = a{(x - h)^2} + k\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>h</em> and of <em>k</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(a = 3\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f(x)\) , giving your answer in the form \(A{x^2} + Bx + C\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Calculate the area enclosed by the graph of <em>f</em> , the <em>x</em>-axis, and the lines \(x = 2\) and \(x = 4\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<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) \(h = 2\) , \(k = 1\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) attempt to substitute coordinates of any point (except the vertex) </span><span style="font-family: times new roman,times; font-size: medium;">on the graph into <em>f</em> <strong><em>M1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(13 = a{(0 - 2)^2} + 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working towards solution <em><strong> A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(13 = 4a + 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 3\) <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(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempting to expand <strong>their</strong> binomial <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = 3({x^2} - 2 \times 2x + 4) + 1\) , \({(x - 2)^2} = {x^2} - 4x + 4\)</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. \(f(x) = 3{x^2} - 12x + 12 + 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 3{x^2} - 12x + 13\) (accept \(A = 3\) , \(B = - 12\) , \(C = 13\) ) <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;"><strong>METHOD 1 </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">integral expression <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_2^4 {(3{x^2}} - 12x + 13)\) , \(\int {f{\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Area}} = [{x^3} - 6{x^2} + 13x]_2^4\) <em><strong>A1A1A1</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 \({x^3}\) , <em><strong>A1</strong></em> for \( - 6{x^2}\) , <em><strong>A1</strong></em> for \(13x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution of <strong>correct</strong> limits into <strong>their</strong> expression <em><strong>A1A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(({4^3} - 6 \times {4^2} + 13 \times 4) - ({2^3} - 6 \times {2^2} + 13 \times 2)\) , \(64 - 96 + 52 - (8 - 24 + 26)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for substituting 4, <em><strong>A1</strong></em> for substituting 2. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(64 - 96 + 52 - 8 + 24 - 26,20 - 10\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Area}} = 10\) <em><strong> A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[8 marks] </strong></em></span></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;">integral expression <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_2^4 {(3{{(x - 2)}^2}} + 1)\) , \(\int {f{\rm{d}}x} \)<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Area}} = [{(x - 2)^3} + x]_2^4\) <em><strong>A2A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A2</strong></em> for \({(x - 2)^3}\) , <em><strong>A1</strong></em> for \(x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span> </p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution of <strong>correct</strong> limits into <strong>their</strong> expression <em><strong>A1A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({(4 - 2)^3} + 4 - [{(2 - 2)^3} + 2]\) , \({2^3} + 4 - ({0^3} + 2)\) , \({2^3} + 4 - 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for substituting 4, <em><strong>A1</strong></em> for substituting 2. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(8 + 4 - 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Area}} = 10\) <em><strong>A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em> <strong>[8 marks]</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> METHOD 3 </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing area from 0 to 2 is same as area from 2 to 4 <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, \(\int_2^4 {f = \int_0^2 f } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">integral expression <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^2 {(3{x^2}} - 12x + 13)\) , \(\int {f{\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Area}} = [{x^3} - 6{x^2} + 13x]_0^2\) <em><strong>A1A1A1</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 \({x^3}\) , <strong><em>A1</em></strong> for \( - 6{x^2}\) , <em><strong>A1</strong></em> for \(13x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution of <strong>correct</strong> limits into <strong>their</strong> expression <em><strong>A1(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(({2^3} - 6 \times {2^2} + 13 \times 2) - ({0^3} - 6 \times {0^2} + 13 \times 0)\) , \(8 - 24 + 26\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for substituting 2, <em><strong>(A1)</strong></em> for substituting 0. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{Area}} = 10\) <em><strong>A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em> <strong>[8 marks]</strong> </em></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;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (a), nearly all the candidates recognized that <em>h</em> and <em>k</em> were the coordinates of the vertex of the parabola, and most were able to successfully show that \(a = 3\) . Unfortunately, a few candidates did not understand the "show that" command, and simply verified that \(a = 3\) would work, rather than showing how to find \(a = 3\) . </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), most candidates were able to find \(f(x)\) in the required form. For a few candidates, algebraic errors kept them from finding the correct function, even though they started with correct values for <em>a</em>, <em>h</em> and <em>k</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;">In part (c), nearly all candidates knew that they needed to integrate to find the area, but errors in integration, and algebraic and arithmetic errors prevented many from finding the correct area. </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;">In this question <em>s</em> represents displacement in metres and <em>t</em> represents time in seconds.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The velocity <em>v</em> m s<sup>–1</sup> of a moving body is given by \(v = 40 - at\) where <em>a</em> is a non-zero </span><span style="font-family: times new roman,times; font-size: medium;">constant.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Trains approaching a station start to slow down when they pass a point P. As a train </span><span style="font-family: times new roman,times; font-size: medium;">slows down, its velocity is given by \(v = 40 - at\) , where \(t = 0\) at P. The station is 500 m </span><span style="font-family: times new roman,times; font-size: medium;">from P.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) If \(s = 100\) when \(t = 0\) , find an expression for <em>s</em> in terms of <em>a</em> and <em>t</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) If \(s = 0\) when \(t = 0\) , write down an expression for <em>s</em> in terms of <em>a</em> and<em> t</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A train M slows down so that it comes to a stop at the station.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the time it takes train M to come to a stop, giving your answer in terms </span><span style="font-family: times new roman,times; font-size: medium;">of <em>a</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence show that \(a = \frac{8}{5}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">For a different train N, the value of <em>a</em> is 4.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that this train will stop <strong>before</strong> it reaches the station.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<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;"><strong>Note</strong>: In this question, do not penalize absence of units. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(s = \int {(40 - at){\rm{d}}t} \) <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 40t - \frac{1}{2}a{t^2} + c\) <em><strong>(A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(s = 100\) when \(t = 0\) (\(c = 100\) ) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 40t - \frac{1}{2}a{t^2} + 100\) <em><strong> A1 N5</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(s = 40t - \frac{1}{2}a{t^2}\) <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.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) stops at station, so \(v = 0\) <em><strong> (M1) </strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(t = \frac{{40}}{a}\) </span><span style="font-family: times new roman,times; font-size: medium;">(seconds) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of choosing formula for <em>s</em> from (a) (ii) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(t = \frac{{40}}{a}\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(40 \times \frac{{40}}{a} - \frac{1}{2}a \times \frac{{{{40}^2}}}{{{a^2}}}\)</span></p>
<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;">e.g. \(500 = s\) , \(500 = 40 \times \frac{{40}}{a} - \frac{1}{2}a \times \frac{{{{40}^2}}}{{{a^2}}}\) , \(500 = \frac{{1600}}{a} - \frac{{800}}{a}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of simplification to an expression which obviously leads to \(a = \frac{8}{5}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(500a = 800\) , \(5 = \frac{8}{a}\) , \(1000a = 3200 - 1600\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(a = \frac{8}{5}\) </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;"> [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;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = 40 - 4t\) , stops when \(v = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(40 - 4t = 0\) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = 10\) <em><strong> A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting into expression for <em>s</em> <em><strong>M1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 40 \times 10 - \frac{1}{2} \times 4 \times {10^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 200\) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since \(200 < 500\) (allow <em><strong>FT</strong></em> on their <em>s</em>, if \(s < 500\) ) <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">train stops before the station <em><strong>AG N0</strong> </em></span></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;">from (b) \(t = \frac{{40}}{4} = 10\) <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting into expression for <em><strong>s </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s = 40 \times 10 - \frac{1}{2} \times 4 \times {10^2}\) <strong><em>M1</em></strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 200\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since \(200 < 500\) <em><strong>R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">train stops before the station <em><strong>AG N0</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 3 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>a</em> is deceleration <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(4 > \frac{8}{5}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so stops in shorter time <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so less distance travelled <em><strong>R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so stops before station <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>
<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) proved accessible for most. </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), simple as it is, proved elusive as many candidates did not make the connection that \(v = 0\) when the train stops. Instead, many attempted to find the value of <em>t</em> using \(a = \frac{8}{5}\) . </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;">Few were successful in part (c). </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int {x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f(x)\), given that \(f’(x) = x{{\text{e}}^{{x^2} - 1}}\) and \(f( - 1) = 3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach to set up integration by substitution/inspection <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(u = {x^2} - 1,{\text{ d}}u = 2x,{\text{ }}\int {2x{{\text{e}}^{{x^2} - 1}}{\text{d}}x} \)</p>
<p>correct expression <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}\int {2x{{\text{e}}^{{x^2} - 1}}{\text{d}}x,{\text{ }}\frac{1}{2}\int {{{\text{e}}^u}{\text{d}}u} } \)</p>
<p>\(\frac{1}{2}{{\text{e}}^{{x^2} - 1}} + c\) <strong><em>A2</em></strong> <strong><em>N4</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>A1</em> </strong>if missing “\( + c\)”.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting \(x = - 1\) into <strong>their </strong>answer from (a) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}{{\text{e}}^0},{\text{ }}\frac{1}{2}{{\text{e}}^{1 - 1}} = 3\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2} + c = 3,{\text{ }}c = 2.5\)</p>
<p>\(f(x) = \frac{1}{2}{{\text{e}}^{{x^2} - 1}} + 2.5\) <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.</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="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = k{x^4}\) . The point \({\text{P}}(1{\text{, }}k)\) lies on the curve of <em>f</em> . At P, the normal to the curve </span><span style="font-family: times new roman,times; font-size: medium;">is parallel to \(y = - \frac{1}{8}x\) </span><span style="font-family: times new roman,times; font-size: medium;">. Find the value of <em>k</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">gradient of tangent \(= 8\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 4k{x^3}\) (seen anywhere) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing the gradient of the tangent is the derivative <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting the derivative equal to 8 <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4k{x^3} = 8\) , \(k{x^3} = 2\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting \(x = 1\) (seen anywhere) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = 2\) <em><strong>A1 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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Candidates‟ success with this question was mixed. Those who understood the relationship </span><span style="font-family: times new roman,times; font-size: medium;">between the derivative and the gradient of the normal line were not bothered by the lack of </span><span style="font-family: times new roman,times; font-size: medium;">structure in the question, solving clearly with only a few steps, earning full marks. Those who </span><span style="font-family: times new roman,times; font-size: medium;">were unclear often either gained a few marks for finding the derivative and substituting \(x = 1\) , </span><span style="font-family: times new roman,times; font-size: medium;">or no marks for working that did not employ the derivative. Misunderstandings included </span><span style="font-family: times new roman,times; font-size: medium;">simply finding the equation of the tangent or normal line, setting the derivative equal to the </span><span style="font-family: times new roman,times; font-size: medium;">gradient of the normal, and equating the function with the normal or tangent line equation. </span><span style="font-family: times new roman,times; font-size: medium;">Among the candidates who demonstrated greater understanding, more used the gradient of the </span><span style="font-family: times new roman,times; font-size: medium;">normal (the equation \( - \frac{1}{4}k = - \frac{1}{8}\) ) than the gradient of the tangent (\(4k = 8\) ) ; this led to </span><span style="font-family: times new roman,times; font-size: medium;">more algebraic errors in obtaining the final answer of \(k = 2\) . A number of unsuccessful </span><span style="font-family: times new roman,times; font-size: medium;">candidates wrote down a lot of irrelevant mathematics with no plan in mind and earned no </span><span style="font-family: times new roman,times; font-size: medium;">marks.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of the function \(y = f(x)\) passes through the point \(\left( {\frac{3}{2},4} \right)\) . The gradient function of <em>f</em> is given as \(f'(x) = \sin (2x - 3)\) . Find \(f(x)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of integration</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = \int {\sin (2x - 3){\rm{d}}x} \) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\( = - \frac{1}{2}\cos (2x - 3) + C\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting initial condition into <strong>their</strong> expression (even if <em>C</em> is missing) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4 = - \frac{1}{2}\cos 0 + C\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(C = 4.5\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f(x) = - \frac{1}{2}\cos (2x - 3) + 4.5\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N5</span></strong></em></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;">While most candidates realized they needed to integrate in this question, many did so unsuccessfully. Many did not account for the coefficient of <em>x</em>, and failed to multiply by \(\frac{1}{2}\). Some of the candidates who substituted the initial condition into their integral were not able to solve for "<em>c</em>", either because of arithmetic errors or because they did not know the correct value for \(\cos 0\) . </span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;"> The following diagram shows part of the graph of<span style="line-height: normal; background-color: #f7f7f7;"> \(y = f(x)\).</span></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_6.png" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph has a local maximum at \(A\), where \(x = - 2\), and a local minimum at \(B\), where \(x = 6\).</span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On the following axes, sketch the graph of \(y = f'(x)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the following in order from least to greatest: \(f(0),{\text{ }}f'(6),{\text{ }}f''( - 2)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="font-style: normal; font-variant: normal; font-weight: normal; font-stretch: normal; font-size: 21px; line-height: normal; font-family: 'Times New Roman'; min-height: 25px; margin: 0px; text-align: left;"><img src="images/maths_6a_markscheme.png" alt> <strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1A1A1A1 N4</em></strong></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> </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>Note: </strong>Award <strong><em>A1</em></strong> for <em>x</em>-intercept in circle at \(-2\), <strong><em>A1</em></strong> for <em>x</em>-intercept in circle at \(6\).</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;"> Award <strong><em>A1</em></strong> for approximately correct 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 <strong><em>A1</em></strong> for a negative <em>y</em>-intercept.</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>[4 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;">\(f''( - 2),{\text{ }}f'(6),{\text{ }}f(0)\) <strong><em>A2 N2</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>[2 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 class="p1">Let \(y = f(x)\)<span class="s1">, for \( - 0.5 \le \) x \( \le \) \(6.5\)</span>. The following diagram shows the graph of \(f'\), the derivative of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_13.43.06.png" alt></p>
<p class="p1">The graph of \(f'\) has a local maximum when \(x = 2\), a local minimum when \(x = 4\), and it crosses the <em>\(x\)-</em><span class="s1">axis at the point \((5,{\text{ }}0)\)</span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the graph of \(f\) has a local minimum when \(x = 5\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the set of values of \(x\) for which the graph of \(f\) is concave down.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The following diagram shows the shaded regions \(A\), \(B\) and \(C\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_13.46.59.png" alt></p>
<p class="p1">The regions are enclosed by the graph of \(f'\), the \(x\)-axis, the \(y\)-axis, and the line \(x = 6\).</p>
<p class="p1">The area of region \(A\) <span class="s1">is 12</span>, the area of region \(B\) <span class="s1">is 6.75 </span>and the area of region \(C\) <span class="s1">is 6.75</span>.</p>
<p class="p1">Given that \(f(0) = 14\), find \(f(6)\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The following diagram shows the shaded regions \(A\), \(B\) and \(C\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_13.46.59.png" alt></p>
<p class="p1">The regions are enclosed by the graph of \(f'\), the <em>x</em>-axis, the <em>y</em>-axis, and the line \(x = 6\).</p>
<p class="p1">The area of region \(A\) <span class="s1">is 12</span>, the area of region \(B\) <span class="s1">is 6.75 </span>and the area of region \(C\) <span class="s1">is 6.75</span>.</p>
<p class="p1">Let \(g(x) = {\left( {f(x)} \right)^2}\). Given that \(f'(6) = 16\), find the equation of the tangent to the graph of \(g\) at the point where \(x = 6\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<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">\(f'(5) = 0\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">valid reasoning including reference to the graph of \(f'\) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f'\) changes sign from negative to positive at \(x = 5\), labelled sign chart for \(f'\)</p>
<p class="p1">so \(f\) has a local minimum at \(x = 5\) <span class="Apple-converted-space"> </span><strong><em>AG N0<br></em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>It must be clear that any description is referring to the graph of \(f'\), simply giving the conditions for a minimum without relating them to \(f'\) does not gain the <strong><em>R1</em></strong>.</p>
<p class="p2"> </p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">\(f'(5) = 0\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">valid reasoning referring to second derivative <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f''(5) > 0\)</p>
<p class="p1">so \(f\) has a local minimum at \(x = 5\) <span class="Apple-converted-space"> </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</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">attempt to find relevant interval <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f'\) is decreasing, gradient of \(f'\) is negative, \(f'' < 0\)</p>
<p class="p1">\(2 < x < 4\;\;\;\)(accept “between 2 and 4”) <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>Notes: <span class="Apple-converted-space"> </span></strong>If no other working shown, award <strong><em>M1A0 </em></strong>for incorrect inequalities such as \(2 \le \) \(x\) \( \le \) 4, or “from 2 to 4”</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (one integral)</strong></p>
<p>correct application of Fundamental Theorem of Calculus <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_0^6 {f'(x){\text{d}}x = } f(6) - f(0),{\text{ }}f(6) = 14 + \int_0^6 {f'(x){\text{d}}x} \)</p>
<p>attempt to link definite integral with areas <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_0^6 {f'(x){\text{d}}x = - 12 - 6.75 + 6.75,{\text{ }}\int_0^6 {f'(x){\text{d}}x = {\text{Area }}A + {\text{Area }}B + {\text{ Area }}C} } \)</p>
<p>correct value for \(\int_0^6 {f'(x){\text{d}}x} \) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_0^6 {f'(x){\text{d}}x} = - 12\)</p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;f(6) - 14 = - 12,{\text{ }}f(6) = - 12 + f(0)\)</p>
<p>\(f(6) = 2\) <strong><em>A1 N3</em></strong></p>
<p><strong>METHOD 2 (more than one integral)</strong></p>
<p>correct application of Fundamental Theorem of Calculus <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_0^2 {f'(x){\text{d}}x} = f(2) - f(0),{\text{ }}f(2) = 14 + \int_0^2 {f'(x)} \)</p>
<p>attempt to link definite integrals with areas <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_0^2 {f'(x){\text{d}}x} = 12,{\text{ }}\int_2^5 {f'(x){\text{d}}x = - 6.75} ,{\text{ }}\int_0^6 {f'(x)} = 0\)</p>
<p>correct values for integrals <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_0^2 {f'(x){\text{d}}x} = - 12,{\text{ }}\int_5^2 {f'(x)} {\text{d}}x = 6.75,{\text{ }}f(6) - f(2) = 0\)</p>
<p>one correct intermediate value <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;f(2) = 2,{\text{ }}f(5) = - 4.75\)</p>
<p>\(f(6) = 2\) <strong><em>A1 N3</em></strong></p>
<p><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">correct calculation of \(g(6)\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{2^2},{\text{ }}g(6) = 4\)</p>
<p class="p1">choosing chain rule or product rule <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;g'\left( {f(x)} \right)f'(x),{\text{ }}\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},{\text{ }}f(x)f'(x) + f'(x)f(x)\)</p>
<p class="p1">correct derivative <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;g'(x) = 2f(x)f'(x),{\text{ }}f(x)f'(x) + f'(x)f(x)\)</p>
<p class="p1">correct calculation of \(g'(6)\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;2(2)(16),{\text{ }}g'(6) = 64\)</p>
<p class="p1">attempt to substitute <strong>their </strong>values of \(g'(6)\) and \(g(6)\) (in any order) into equation of a line <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{2^2} = (2 \times 2 \times 16)6 + b,{\text{ }}y - 6 = 64(x - 4)\)</p>
<p class="p1">correct equation in any form <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;y - 4 = 64(x - 6),{\text{ }}y = 64x - 380\)</p>
<p class="p1"><em><strong>[6 marks]</strong></em></p>
<p class="p1"><em><strong>[Total 15 marks]</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;">
[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><span style="font-family: times new roman,times; font-size: medium;">Let \(\int_1^5 {3f(x){\rm{d}}x = 12} \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\int_5^1 {f(x){\rm{d}}x = - 4} \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\int_1^2 {(x + f(x)){\rm{d}}x + } \int_2^5 {(x + f(x)){\rm{d}}x} \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<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 factorising 3/division by 3 <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_1^5 {3f(x){\rm{d}}x = 3\int_1^5 {f(x){\rm{d}}x} } \) , \(\frac{{12}}{3}\) , \(\int_1^5 {\frac{{3f(x){\rm{d}}x}}{3}} \) </span><span style="font-family: times new roman,times; font-size: medium;">(do not accept 4 as this is show that)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of stating that reversing the limits changes the sign <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_5^1 {f(x){\rm{d}}x = } - \int_1^5 {f(x){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int_5^1 {f(x){\rm{d}}x = } - 4\) </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;">[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 correctly combining the integrals (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(I = \int_1^2 {(x + f(x)){\rm{d}}x + } \int_2^5 {(x + f(x)){\rm{d}}x = } \int_1^5 {(x + f(x)){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correctly splitting the integrals (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(I = \int_1^5 {x{\rm{d}}x + } \int_1^5 {f(x){\rm{d}}x} \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int {x{\rm{d}}x = } \frac{{{x^2}}}{2}\) (seen anywhere) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(\int_1^5 {x{\rm{d}}x = } \left[ {\frac{{{x^2}}}{2}} \right]_1^5 = \frac{{25}}{2} - \frac{1}{2}\) \(\left( { = \frac{{24}}{2},12} \right)\) </span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(I =16\) <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>
<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 very poorly done. Very few candidates provided proper justification for part </span><span style="font-family: times new roman,times; font-size: medium;">(a), a common error being to write \(\int_1^5 {f(x){\rm{d}}x = } f(5) - f(1)\) . What was being looked for was that </span><span style="font-family: times new roman,times; font-size: medium;">\(\int_1^5 {3f(x){\rm{d}}x = } 3\int_1^5 {f(x){\rm{d}}x} \) and \(\int_5^1 {f(x){\rm{d}}x = } - \int_1^5 {f(x){\rm{d}}x} \) .</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) had similar problems with neither the combining of limits nor the splitting of integrals being done very often. A common error was to treat \(f(x)\) as 1 in order to make \(\int_1^5 {f(x){\rm{d}}x = 4} \) and then write \(\int_1^5 {(x + f(x)){\rm{d}}x = \left[ {x + 1} \right]} _1^5\) .</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A function \(f\) has its derivative given by \(f'(x) = 3{x^2} - 2kx - 9\), where \(k\) is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f''(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(f\) has a point of inflexion when \(x = 1\).</p>
<p class="p1">Show that \(k = 3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f'( - 2)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the equation of the tangent to the curve of \(f\) at \(( - 2,{\text{ }}1)\), giving your answer in the form \(y = ax + b\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(f'( - 1) = 0\), explain why the graph of \(f\) has a local maximum when \(x = - 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(f''(x) = 6x - 2k\) <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 class="p1">substituting \(x = 1\) into \(f''\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f''(1),{\text{ }}6(1) - 2k\)</p>
<p class="p1">recognizing \(f''(x) = 0\;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>M1</em></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>\(\;\;\;6 - 2k = 0\)</p>
<p class="p1">\(k = 3\) <span class="Apple-converted-space"> </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</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">correct substitution into \(f'(x)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;3{( - 2)^2} - 6( - 2) - 9\)</p>
<p class="p1">\(f'( - 2) = 15\) <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">recognizing gradient value (may be seen in equation) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;a = 15,{\text{ }}y = 15x + b\)</p>
<p class="p1">attempt to substitute \(( - 2,{\text{ }}1)\) into equation of a straight line <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;1 = 15( - 2) + b,{\text{ }}(y - 1) = m(x + 2),{\text{ }}(y + 2) = 15(x - 1)\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;31 = b,{\text{ }}y = 15x + 30 + 1\)</p>
<p class="p1">\(y = 15x + 31\) <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>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 </strong>(\({{\text{2}}^{{\text{nd}}}}\) derivative)</p>
<p>recognizing \(f'' < 0\;\;\;\)(seen anywhere) <strong><em>R1</em></strong></p>
<p>substituting \(x = - 1\) into \(f''\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f''( - 1),{\text{ }}6( - 1) - 6\)</p>
<p>\(f''( - 1) = - 12\) <strong><em>A1</em></strong></p>
<p>therefore the graph of \(f\) has a local maximum when \(x = - 1\) <strong><em>AG N0</em></strong></p>
<p><strong>METHOD 2 </strong>(\({{\text{1}}^{{\text{st}}}}\) derivative)</p>
<p>recognizing change of sign of \(f'(x)\;\;\;\)(seen anywhere) <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\;\;\;\)<em>sign chart</em>\(\;\;\;\)<img src="images/Schermafbeelding_2016-01-13_om_10.59.32.png" alt></p>
<p>correct value of \(f'\) for \( - 1 < x < 3\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;f'(0) = - 9\)</p>
<p>correct value of \(f'\) for \(x\) value to the left of \( - 1\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;f'( - 2) = 15\)</p>
<p>therefore the graph of \(f\) has a local maximum when \(x = - 1\) <strong><em>AG N0</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [14 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">Well answered and candidates coped well with \(k\) in the expression.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Mostly answered well with the common error being to substitute into \(f'\) instead of \(f''\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">A straightforward question that was typically answered correctly.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some candidates recalculated the gradient, not realising this had already been found in part c). Many understood they were finding a linear equation but were hampered by arithmetic errors.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Using change of sign of the first derivative was the most common approach used with a sign chart or written explanation. However, few candidates then supported their approach by calculating suitable values for \(f'(x)\). This was necessary because the question already identified a local maximum, hence candidates needed to explain why this was so. Some candidates did not mention the ‘first derivative’ just that ‘it’ was increasing/decreasing. Few candidates used the more efficient second derivative test.</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;">Consider \(f(x) = {x^2}\sin x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the gradient of the curve of \(f\) at \(x = \frac{\pi }{2}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<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 product rule <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(uv' + vu'\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivatives (must be seen in the product rule) \(\cos x\) , \(2x\) <strong><em>(A1)(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = {x^2}\cos x + 2x\sin x\) <strong><em>A1 N4 </em></strong></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;">substituting \(\frac{\pi }{2}\) into <strong>their</strong> \(f'(x)\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(f'\left( {\frac{\pi }{2}} \right)\) , \({\left( {\frac{\pi }{2}} \right)^2}\cos \left( {\frac{\pi }{2}} \right) + 2\left( {\frac{\pi }{2}} \right)\sin \left( {\frac{\pi }{2}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct values for <strong>both</strong> \(\sin \frac{\pi }{2}\) and \(\cos \frac{\pi }{2}\) seen in \(f'(x)\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(0 + 2\left( {\frac{\pi }{2}} \right) \times 1\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'\left( {\frac{\pi }{2}} \right) = \pi \) <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;">Many candidates correctly applied the product rule for the derivative, although a common error was to answer \(f'(x) = 2x\cos x\) .</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 generally understood that the gradient of the curve uses the derivative, although in some cases the substitution was made in the original function. Some candidates did not know the values of sine and cosine at \({\frac{\pi }{2}}\) .</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;">Consider \(f(x) = \frac{1}{3}{x^3} + 2{x^2} - 5x\) . Part of the graph of <em>f</em> is shown below. There is a maximum point at M, and a point of inflexion at N. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/abc.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of M.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of N.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The line <em>L</em> is the tangent to the curve of <em>f</em> at \((3{\text{, }}12)\). Find the equation of <em>L</em> in the form \(y = ax + b\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<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;">\(f'(x) = {x^2} + 4x - 5\) <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">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 attempting to solve \(f'(x) = 0\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of 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. \((x + 5)(x - 1)\) , \(\frac{{ - 4 \pm \sqrt {16 + 20} }}{2}\) , sketch</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = - 5\), \(x = 1\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so \(x = - 5\) <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">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;">\(f''(x) = 2x + 4\) (may be seen later) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of setting second derivative = 0 <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2x + 4 = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = - 2\) <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;">evidence of use of symmetry <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. midpoint of max/min, reference to shape of cubic</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct calculation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 5 + 1}}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = - 2\) <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>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to find the value of the derivative when \(x = 3\) <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(3) = 16\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach to finding the equation of a line <strong><em>M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y - 12 = 16(x - 3)\) , \(12 = 16 \times 3 + b\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 16x - 36\) <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">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 very well done with most candidates showing their work in an orderly </span><span style="font-family: times new roman,times; font-size: medium;">manner. There were a number of candidates, however, who were a bit sloppy in indicating </span><span style="font-family: times new roman,times; font-size: medium;">when a function was being equated to zero and they “solved” an expression rather than an </span><span style="font-family: times new roman,times; font-size: medium;">equation.</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 very well done with most candidates showing their work in an orderly </span><span style="font-family: times new roman,times; font-size: medium;">manner. There were a number of candidates, however, who were a bit sloppy in indicating </span><span style="font-family: times new roman,times; font-size: medium;">when a function was being equated to zero and they “solved” an expression rather than an </span><span style="font-family: times new roman,times; font-size: medium;">equation. Many candidates went through first and second derivative tests to verify that the </span><span style="font-family: times new roman,times; font-size: medium;">point they found was a maximum or an inflexion point; this was unnecessary since the graph </span><span style="font-family: times new roman,times; font-size: medium;">was given. Many also found the <em>y</em>-coordinate which was unnecessary and used up valuable </span><span style="font-family: times new roman,times; font-size: medium;">time on the exam.</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 was very well done with most candidates showing their work in an orderly </span><span style="font-family: times new roman,times; font-size: medium;">manner. There were a number of candidates, however, who were a bit sloppy in indicating </span><span style="font-family: times new roman,times; font-size: medium;">when a function was being equated to zero and they “solved” an expression rather than an </span><span style="font-family: times new roman,times; font-size: medium;">equation. Many candidates went through first and second derivative tests to verify that the </span><span style="font-family: times new roman,times; font-size: medium;">point they found was a maximum or an inflexion point; this was unnecessary since the graph </span><span style="font-family: times new roman,times; font-size: medium;">was given. Many also found the <em>y</em>-coordinate which was unnecessary and used up valuable </span><span style="font-family: times new roman,times; font-size: medium;">time on the exam.</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;">This question was very well done with most candidates showing their work in an orderly </span><span style="font-family: times new roman,times; font-size: medium;">manner. There were a number of candidates, however, who were a bit sloppy in indicating </span><span style="font-family: times new roman,times; font-size: medium;">when a function was being equated to zero and they “solved” an expression rather than an </span><span style="font-family: times new roman,times; font-size: medium;">equation. Many candidates went through first and second derivative tests to verify that the </span><span style="font-family: times new roman,times; font-size: medium;">point they found was a maximum or an inflexion point; this was unnecessary since the graph </span><span style="font-family: times new roman,times; font-size: medium;">was given. Many also found the <em>y</em>-coordinate which was unnecessary and used up valuable </span><span style="font-family: times new roman,times; font-size: medium;">time on the exam.</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 \(f(x) = \frac{1}{4}{x^2} + 2\) </span><span style="font-family: times new roman,times; font-size: medium;"> . The line <em>L</em> is the tangent to the curve of <em>f</em> at (4, 6) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \frac{{90}}{{3x + 4}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, for \(2 \le x \le 12\) . The following diagram shows the graph of <em>g</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/mickey.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the equation of <em>L</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the area of the region enclosed by the curve of <em>g</em> , the <em>x</em>-axis, and </span><span style="font-family: times new roman,times; font-size: medium;">the lines \(x = 2\) and \(x = 12\) . Give your answer in the form \(a\ln b\) , where \(a,b \in \mathbb{Z}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is reflected in the<em> x</em>-axis to give the graph of <em>h</em> . The area of the </span><span style="font-family: times new roman,times; font-size: medium;">region enclosed by the lines <em>L</em> , \(x = 2\) , \(x = 12\) and the <em>x</em>-axis is 120 \(120{\text{ c}}{{\text{m}}^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area enclosed by the lines<em> L</em> , \(x = 2\) , \(x = 12\) and the graph of <em>h</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<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;">finding \(f'(x) = \frac{1}{2}x\) </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;">attempt to find \(f'(4)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct value \(f'(4) = 2\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation in any form <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. </span><span style="font-family: times new roman,times; font-size: medium;">\(y - 6 = 2(x - 4)\) , \(y = 2x - 2\)</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;">\({\rm{area}} = \int_2^{12} {\frac{{90}}{{3x + 4}}} {\rm{d}}x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct integral <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30\ln (3x + 4)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting limits and subtracting <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30\ln (3 \times 12 + 4) - 30\ln (3 \times 2 + 4)\) , \(30\ln 40 - 30\ln 10\)</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. \(30(\ln 40 - \ln 10)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct application of \(\ln b - \ln a\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30\ln \frac{{40}}{{10}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 30\ln 4\) <em><strong>A1 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">b.</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 <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, area <em>h</em> = area <em>g</em> , 120 + <strong>their</strong> answer from (b)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 120 + 30\ln 4\) <em><strong>A2 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>
<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 most candidates answered part (a) correctly, finding the equation of the tangent, there were some who did not consider the value of their derivative when \(x = 4\) . </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 that they needed to integrate to find the area, but errors in integration, and misapplication of the rules of logarithms kept many from finding the correct area. </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), it was clear that a significant number of candidates understood the idea of the reflected function, and some recognized that the integral was the negative of the integral from part (b), but only a few recognized the relationship between the areas. Many thought the area between <em>h</em> and the <em>x</em>-axis was 120. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Fred makes an open metal container in the shape of a cuboid, as shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_09.43.42.png" alt="M16/5/MATME/SP1/ENG/TZ2/09"></p>
<p class="p1">The container has height \(x{\text{ m}}\), width \(x{\text{ m}}\) and length \(y{\text{ m}}\)<span class="s1">. The volume is \(36{\text{ }}{{\text{m}}^3}\)</span>.</p>
<p class="p1">Let \(A(x)\) be the outside surface area of the container.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(A(x) = \frac{{108}}{x} + 2{x^2}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(A'(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that the outside surface area is a minimum, find the height of the container.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Fred paints the outside of the container. A tin of paint covers a surface area of \({\text{10 }}{{\text{m}}^{\text{2}}}\) and costs $20<span class="s1">. Find the total cost of the tins needed to paint the container.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into the formula for volume <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(36 = y \times x \times x\)</p>
<p class="p1">valid approach to eliminate \(y\) (may be seen in formula/substitution) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(y = \frac{{36}}{{{x^2}}},{\text{ }}xy = \frac{{36}}{x}\)</p>
<p class="p1">correct expression for surface area <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(xy + xy + xy + {x^2} + {x^2},{\text{ area}} = 3xy + 2{x^2}\)</p>
<p class="p1">correct expression in terms of \(x\) only <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(3x\left( {\frac{{36}}{{{x^2}}}} \right) + 2{x^2},{\text{ }}{x^2} + {x^2} + \frac{{36}}{x} + \frac{{36}}{x} + \frac{{36}}{x},{\text{ }}2{x^2} + 3\left( {\frac{{36}}{x}} \right)\)</p>
<p class="p1"><span class="Apple-converted-space">\(A(x) = \frac{{108}}{x} + 2{x^2}\) </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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(A'(x) = - \frac{{108}}{{{x^2}}} + 4x,{\text{ }}4x - 108{x^{ - 2}}\) </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1 </em></strong>for each term.</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 that minimum is when \(A'(x) = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></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>\(\,\,\,\,\,\)\( - \frac{{108}}{{{x^2}}} + 4x = 0,{\text{ }}4x = \frac{{108}}{{{x^2}}}\)</p>
<p class="p1">correct simplification <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - 108 + 4{x^3} = 0,{\text{ }}4{x^3} = 108\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({x^3} = 27\)</p>
<p class="p1"><span class="Apple-converted-space">\({\text{height}} = 3{\text{ (m) }}({\text{accept }}x = 3)\) </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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to find area using <strong>their </strong>height <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{108}}{3} + 2{(3)^2},{\text{ }}9 + 9 + 12 + 12 + 12\)</p>
<p class="p1">minimum surface area \( = 54{\text{ }}{{\text{m}}^{\text{2}}}\) (may be seen in part (c)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">attempt to find the number of tins <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{54}}{{10}},{\text{ }}5.4\)</p>
<p class="p1"><span class="s1">6 </span>(tins) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1">$120 <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>[5 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">Many candidates answered part (a) of this question correctly, though some seemed to be working backwards from the given expression for area, which is not the intention of a "show that" question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), while many candidates found the correct derivative, some did so using cumbersome methods such as the quotient rule, rather than using the simpler power rule.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">It was disappointing to see the number of candidates who did not recognize that the derivative they had just found in part (b) would have to be equal to zero in order for the surface area to be a minimum. For the candidates who did set their derivative equal to zero, most were able to find the correct height.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (d) of this question, there were some arithmetic errors which kept candidates from finding the correct area. The most common error here, by far, was not considering that the number of tins purchased must be an integer.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of \(f(x) = 2x\sqrt {{a^2} - {x^2}} \)<span class="s1">, for \( - 1 \leqslant x \leqslant a\)</span>, where \(a > 1\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_10.04.36.png" alt="M16/5/MATME/SP1/ENG/TZ2/10"></p>
<p class="p1">The line \(L\) is the tangent to the graph of \(f\) <span class="s1">at the origin, O. The point \({\text{P}}(a,{\text{ }}b)\) </span>lies on \(L\).</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The point \({\text{Q}}(a,{\text{ }}0)\) </span>lies on the graph of \(f\). Let \(R\) be the region enclosed by the graph of \(f\) and the \(x\)-axis. This information is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-02_om_10.14.09.png" alt="M16/5/MATME/SP1/ENG/TZ2/10.b+c"></p>
<p class="p1">Let \({A_R}\) be the area of the region \(R\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Given that \(f'(x) = \frac{{2{a^2} - 4{x^2}}}{{\sqrt {{a^2} - {x^2}} }}\), for \( - 1 \leqslant x < a\), <span class="s1">find the equation of \(L\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find an expression for \(b\) in terms of \(a\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \({A_R} = \frac{2}{3}{a^3}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \({A_T}\) <span class="s1">be the area of the triangle OPQ</span>. Given that \({A_T} = k{A_R}\), find the value of \(k\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>recognizing the need to find the gradient when \(x = 0\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(f'(0)\)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2">\(f'(0) = \frac{{2{a^2} - 4(0)}}{{\sqrt {{a^2} - 0} }}\)</p>
<p class="p2"><span class="Apple-converted-space">\(f'(0) = 2a\) </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s2">correct equation with gradient 2\(a\) </span>(do not accept equations of the form \(L = 2ax\)) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(y = 2ax,{\text{ }}y - b = 2a(x - a),{\text{ }}y = 2ax - 2{a^2} + b\)</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1">attempt to substitute \(x = a\) into <strong>their </strong>equation of \(L\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(y = 2a \times a\)</p>
<p class="p1"><span class="s2">\(b = 2{a^2}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">equating gradients <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{b}{a} = 2a\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = 2{a^2}\) </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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">recognizing that area \( = \int_0^a {f(x){\text{d}}x} \) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">valid approach using substitution or inspection <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {2x\sqrt u {\text{d}}x,{\text{ }}u = {a^2} - {x^2},{\text{ d}}u = - 2x{\text{d}}x,{\text{ }}\frac{2}{3}{{({a^2} - {x^2})}^{\frac{3}{2}}}} \)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {2x\sqrt {{a^2} - {x^2}} {\text{d}}x = \int { - \sqrt u {\text{d}}u} } \)</p>
<p class="p1"><span class="Apple-converted-space">\(\int { - \sqrt u {\text{d}}u = - \frac{{{u^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \) </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\int {f(x){\text{d}}x = - \frac{2}{3}{{({a^2} - {x^2})}^{\frac{3}{2}}} + c} \) </span><strong><em>(A1)</em></strong></p>
<p class="p1">substituting limits and subtracting <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({A_R} = - \frac{2}{3}{({a^2} - {a^2})^{\frac{3}{2}}} + \frac{2}{3}{({a^2} - 0)^{\frac{3}{2}}},{\text{ }}\frac{2}{3}{({a^2})^{\frac{3}{2}}}\)</p>
<p class="p1"><span class="Apple-converted-space">\({A_R} = \frac{2}{3}{a^3}\) </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">recognizing that area \( = \int_0^a {f(x){\text{d}}x} \) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">valid approach using substitution or inspection <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {2x\sqrt u {\text{d}}x,{\text{ }}u = {a^2} - {x^2},{\text{ d}}u = - 2x{\text{d}}x,{\text{ }}\frac{2}{3}{{({a^2} - {x^2})}^{\frac{3}{2}}}} \)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {2x\sqrt {{a^2} - {x^2}} {\text{d}}x = \int { - \sqrt u {\text{d}}u} } \)</p>
<p class="p1"><span class="Apple-converted-space">\(\int { - \sqrt u {\text{d}}u = - \frac{{{u^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \) </span><strong><em>(A1)</em></strong></p>
<p class="p1">new limits for <span class="s1"><em>u </em></span>(even if integration is incorrect) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(u = 0{\text{ and }}u = {a^2},{\text{ }}\int_0^{{a^2}} {{u^{\frac{1}{2}}}{\text{d}}u,{\text{ }}\left[ { - \frac{2}{3}{u^{\frac{3}{2}}}} \right]} _{{a^2}}^0\)</p>
<p class="p1">substituting limits and subtracting <strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({A_R} = - \left( {0 - \frac{2}{3}{a^3}} \right),{\text{ }}\frac{2}{3}{({a^2})^{\frac{3}{2}}}\)</p>
<p class="p1"><span class="Apple-converted-space">\({A_R} = \frac{2}{3}{a^3}\) </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong><em>[6 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</strong></p>
<p class="p1">valid approach to find area of triangle <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2}({\text{OQ)(PQ), }}\frac{1}{2}ab\)</p>
<p class="p1">correct substitution into formula for \({A_T}\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({A_T} = \frac{1}{2} \times a \times 2{a^2},{\text{ }}{a^3}\)</p>
<p class="p1">valid attempt to find \(k\) (must be in terms of \(a\)) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({a^3} = k\frac{2}{3}{a^3},{\text{ }}k = \frac{{{a^3}}}{{\frac{2}{3}{a^3}}}\)</p>
<p class="p1">\(k = \frac{3}{2}\) <span class="Apple-converted-space"> </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">valid approach to find area of triangle <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^a {(2ax){\text{d}}x} \)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\([a{x^2}]_0^a,{\text{ }}{a^3}\)</p>
<p class="p1">valid attempt to find \(k\) (must be in terms of \(a\)) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({a^3} = k\frac{2}{3}{a^3},{\text{ }}k = \frac{{{a^3}}}{{\frac{2}{3}{a^3}}}\)</p>
<p class="p1">\(k = \frac{3}{2}\) <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>
<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">As is typically the case with question 10, this proved to be quite a challenging question for many candidates. In part (a), while many candidates seemed to recognize that there was some relationship between the given derivative and the gradient of the tangent line, most did not substitute zero for the \(x\)-value, and were unable to find the correct gradient of the line.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b), nearly every candidate understood that the area was equal to the integral of \(f\) from 0 to \(a\), very few were able to integrate correctly using either substitution or inspection. Many candidates did not even attempt to integrate, stopping after writing the integral expression.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (c), most candidates started with a correct expression for the area of the triangle such as \(\frac{{ab}}{2}\). However, very few were able to substitute their expression for \(b\) from part (a)(ii), and therefore did not find a value for \(k\).</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 \(f(x) = \frac{{2x}}{{{x^2} + 5}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use the quotient rule to show that \(f'(x) = \frac{{10 - 2{x^2}}}{{{{({x^2} + 5)}^2}}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {\frac{{2x}}{{{x^2} + 5}}{\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows part of the graph of \(f\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_10c.png" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The shaded region is enclosed by the graph of \(f\), the \(x\)-axis, and the lines \(x = \sqrt 5 \) and \(x = q\). This region has an area of \(\ln 7\). Find the value of \(q\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<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;">derivative of \(2x\) is \(2\) (must be seen in quotient rule) <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;">derivative of \({x^2} + 5\) is \(2x\) (must be seen in quotient rule) <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;">correct substitution into quotient rule <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> \(\frac{{({x^2} + 5)(2) - (2x)(2x)}}{{{{({x^2} + 5)}^2}}},{\text{ }}\frac{{2({x^2} + 5) - 4{x^2}}}{{{{({x^2} + 5)}^2}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working which clearly leads to given 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';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{2{x^2} + 10 - 4{x^2}}}{{{{({x^2} + 5)}^2}}},{\text{ }}\frac{{2{x^2} + 10 - 4{x^2}}}{{{x^4} + 10{x^2} + 25}}\)</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;">\(f'(x) = \frac{{10 - 2{x^2}}}{{{{({x^2} + 5)}^2}}}\) <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';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 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;">valid approach using substitution or inspection <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> \(u = {x^2} + 5,{\text{ d}}u = 2x{\text{d}}x,{\text{ }}\frac{1}{2}\ln ({x^2} + 5)\)</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;">\(\int {\frac{{2x}}{{{x^2} + 5}}{\text{d}}x = \int {\frac{1}{u}{\text{d}}u} } \) <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;">\(\int {\frac{1}{u}{\text{d}}u = \ln u + c} \) <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;">\(\ln ({x^2} + 5) + c\) <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">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;">correct expression for area <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> \(\left[ {\ln \left( {{x^2} + 5} \right)} \right]_{\sqrt 5 }^q,{\text{ }}\int\limits_{\sqrt 5 }^q {\frac{{2x}}{{{x^2} + 5}}{\text{d}}x} \)</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;">substituting limits into <strong>their </strong>integrated function and subtracting (in either order) <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> \(\ln ({q^2} + 5) - \ln \left( {{{\sqrt 5 }^2} + 5} \right)\)</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> \(\ln \left( {{q^2} + 5} \right) - \ln 10,{\text{ }}\ln \frac{{{q^2} + 5}}{{10}}\)</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;">equating <strong>their </strong>expression to \(\ln 7\) (seen anywhere) <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> \(\ln \left( {{q^2} + 5} \right) - \ln 10 = \ln 7,{\text{ }}\ln \frac{{{q^2} + 5}}{{10}} = \ln 7,{\text{ }}\ln ({q^2} + 5) = \ln 7 + \ln 10\)</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 equation without logs <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> \(\frac{{{q^2} + 5}}{{10}} = 7,{\text{ }}{q^2} + 5 = 70\)</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;">\({q^2} = 65\) <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;">\(q = \sqrt {65} \) <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;"> </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:<em> </em></strong>Award <strong style="font-style: normal;"><em>A0 </em></strong>for \(q = \pm \sqrt {65} \).</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;"> </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>[7 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="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\int_\pi ^a {\cos 2x{\text{d}}x} = \frac{1}{2}{\text{, where }}\pi < a < 2\pi \). Find the value of \(a\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<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 integration (ignore absence of limits and “\(+C\)”) <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> \(\frac{{\sin (2x)}}{2},{\text{ }}\int_\pi ^a {\cos 2x = \left[ {\frac{1}{2}\sin (2x)} \right]_\pi ^a} \)</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;">substituting limits into <strong>their </strong>integrated function and subtracting (in any order) <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> \(\frac{1}{2}\sin (2a) - \frac{1}{2}\sin (2\pi ),{\text{ }}\sin (2\pi ) - \sin (2a)\)</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;">\(\sin (2\pi ) = 0\) <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;">setting <strong>their </strong>result from an integrated function equal to \(\frac{1}{2}\) <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> \(\frac{1}{2}\sin 2a = \frac{1}{2},{\text{ }}\sin (2a) = 1\)</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;">recognizing \({\sin ^{ - 1}}1 = \frac{\pi }{2}\) <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> \(2a = \frac{\pi }{2},{\text{ }}a = \frac{\pi }{4}\)</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 value <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> \(\frac{\pi }{2} + 2\pi ,{\text{ }}2a = \frac{{5\pi }}{2},{\text{ }}a = \frac{\pi }{4} + \pi \)</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;">\(a = \frac{{5\pi }}{4}\) <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;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A rocket moving in a straight line has velocity \(v\) km s<sup>–1</sup> and displacement \(s\) km at </span><span style="font-family: times new roman,times; font-size: medium;">time \(t\) seconds. The velocity \(v\) is given by \(v(t) = 6{{\rm{e}}^{2t}} + t\) . When \(t = 0\) , \(s = 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for the displacement of the rocket in terms of \(t\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of anti-differentiation <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\int {(6{{\rm{e}}^{2t}} + t)} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 3{{\rm{e}}^{2t}} + \frac{{{t^2}}}{2} + C\) <strong><em>A2A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A2</strong></em> for \(3{{\rm{e}}^{2t}}\) , <em><strong>A1</strong></em> for \(\frac{{{t^2}}}{2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute (\(0\), \(10\)) into <strong>their</strong> integrated expression (even if \(C\) is missing) <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(10 = 3 + C\) , \(C = 7\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s = 3{{\rm{e}}^{2t}} + \frac{{{t^2}}}{2} + 7\) <em><strong>A1 N6</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Exception to the <em><strong>FT</strong></em> rule. If working shown, allow full <em><strong>FT</strong></em> on incorrect integration which must involve a power of \({\rm{e}}\). </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">
<div style="font-size: 13.28px; font-family: sans-serif; left: 630.533px; top: 632.893px; transform: scale(0.92, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="11.04000032901764"><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates earned full marks on this question, and many others were able to earn at least half of the available marks. Most candidates knew to integrate, but there were quite a few who tried to find the derivative instead. Many candidates integrated the term \(6{{\text{e}}^{2t}}\) incorrectly, but most were able to earn some further method marks for substituting into their integrated function. The majority of candidates who substituted (\(0\), \(10\)) into their integrated function knew that \({{\text{e}}^0} = 1\) . </span></div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph of a function \(f\). There is a local minimum point at \(A\), where \(x > 0\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_05.51.23.png" alt></p>
<p class="p1">The derivative of \(f\) is given by \(f'(x) = 3{x^2} - 8x - 3\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-coordinate of \(A\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The \(y\)-intercept of the graph is at (\(0,6\)). Find an expression for \(f(x)\).</p>
<p class="p1">The graph of a function \(g\) is obtained by reflecting the graph of \(f\) in the \(y\)-axis, followed by a translation of \(\left({\begin{array}{*{20}{c}}m\\n\end{array}}\right)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-coordinate of the local minimum point on the graph of \(g\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing that the local minimum occurs when \(f'(x) = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">valid attempt to solve \(3{x^2} - 8x - 3 = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)factorization, formula</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\((3x + 1)(x - 3),{\text{ }}x = \frac{{8 \pm \sqrt {64 + 36} }}{6}\)</p>
<p class="p1">\(x = 3\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A1 </em></strong>if both values \(x = \frac{{ - 1}}{3},{\text{ }}x = 3\) are given.</p>
<p class="p1"><em><strong>[5 marks]</strong></em></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">\(f(x) = \int {f'(x){\text{d}}x} \)</p>
<p class="p1">\(f(x) = {x^3} - 4{x^2} - 3x + c\;\;\;\)(do not penalize for missing “\( + c\)”) <span class="Apple-converted-space"> </span><strong><em>A1A1A1</em></strong></p>
<p class="p1">\(c = 6\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(f(x) = {x^3} - 4{x^2} - 3x + 6\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N6</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>applying reflection <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f( - x)\)</p>
<p><img src="image.html" alt></p>
<p>recognizing that the minimum is the image of \(A\) <strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;x = - 3\)</p>
<p class="p1">correct expression for \(x\) <strong><em>A1 N3</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\; - 3 + m,{\text{ }}\left( {\begin{array}{*{20}{c}} { - 3 + m} \\ { - 12 + n} \end{array}} \right),{\text{ }}(m - 3,{\text{ }}n - 12)\)</p>
<p class="p1"><em><strong>[3 marks]</strong></em></p>
<p class="p1"><em><strong>Total [14 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;">
<p class="p1">The majority of candidates approached part (a) correctly, and most recognized that only one solution was possible within the given domain.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Nearly all candidates answered part (b) correctly, earning all the available marks for integrating the polynomial and solving for \(C\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) proved to be much more difficult for candidates, who either did not know how to apply the transformations correctly, or who engaged in lengthy and unnecessary manipulations of the function, rather than simply finding the image of the local minimum point \(A\).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 1 + {{\text{e}}^{ - x}}\) and \(g(x) = 2x + b\), for \(x \in \mathbb{R}\), where \(b\) is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \((g \circ f)(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3\), find the value of \(b\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to form composite <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(g(1 + {{\text{e}}^{ - x}})\)</p>
<p>correct function <strong><em>A1 N2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\((g \circ f)(x) = 2 + b + 2{{\text{e}}^{ - x}},{\text{ }}2(1 + {{\text{e}}^{ - x}}) + b\)</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>evidence of \(\mathop {\lim }\limits_{x \to \infty } (2 + b + 2{{\text{e}}^{ - x}}) = 2 + b + \mathop {\lim }\limits_{x \to \infty } (2{{\text{e}}^{ - x}})\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2 + b + 2{{\text{e}}^{ - \infty }}\), graph with horizontal asymptote when \(x \to \infty \)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if candidate clearly has incorrect limit, such as \(x \to 0,{\text{ }}{{\text{e}}^\infty },{\text{ }}2{{\text{e}}^0}\).</p>
<p> </p>
<p>evidence that \({{\text{e}}^{ - x}} \to 0\) (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\mathop {\lim }\limits_{x \to \infty } ({{\text{e}}^{ - x}}) = 0,{\text{ }}1 + {{\text{e}}^{ - x}} \to 1,{\text{ }}2(1) + b = - 3,{\text{ }}{{\text{e}}^{{\text{large negative number}}}} \to 0\), graph of \(y = {{\text{e}}^{ - x}}\) or</p>
<p>\(y = 2{{\text{e}}^{ - x}}\) with asymptote \(y = 0\), graph of composite function with asymptote \(y = - 3\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2 + b = - 3\)</p>
<p>\(b = - 5\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[4 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="question">
<p>Let \(f’(x) = \frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}\). Given that \(f(0) = 1\), find \(f(x)\).</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>\(\,\,\,\,\,\)\(\int {f'{\text{d}}x,{\text{ }}\int {\frac{{3{x^2}}}{{{{({x^3} + 1)}^5}}}{\text{d}}x} } \)</p>
<p>correct integration by substitution/inspection <strong><em>A2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(x) = - \frac{1}{4}{({x^3} + 1)^{ - 4}} + c,{\text{ }}\frac{{ - 1}}{{4{{({x^3} + 1)}^4}}}\)</p>
<p>correct substitution into <strong>their </strong>integrated function (must include \(c\)) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(1 = \frac{{ - 1}}{{4{{({0^3} + 1)}^4}}} + c,{\text{ }} - \frac{1}{4} + c = 1\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if candidates substitute into \(f’\) or \(f’’\).</p>
<p> </p>
<p>\(c = \frac{5}{4}\) <strong><em>(A1)</em></strong></p>
<p>\(f(x) = - \frac{1}{4}{({x^3} + 1)^{ - 4}} + \frac{5}{4}{\text{ }}\left( { = \frac{{ - 1}}{{4{{({x^3} + 1)}^4}}} + \frac{5}{4},{\text{ }}\frac{{5{{({x^3} + 1)}^4} - 1}}{{4{{({x^3} + 1)}^4}}}} \right)\) <strong><em>A1</em></strong> <strong><em>N4</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><span style="font-family: times new roman,times; font-size: medium;">A function <em>f</em> has its first derivative given by \(f'(x) = {(x - 3)^3}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the second derivative.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(3)\) and \(f''(3)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point P on the graph of <em>f</em> has <em>x</em>-coordinate \(3\). Explain why P is not a point </span><span style="font-family: times new roman,times; font-size: medium;">of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<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;">\(f''(x) = 3{(x - 3)^2}\) <em><strong>A2 N2</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;">attempt to expand \({(x - 3)^3}\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = {x^3} - 9{x^2} + 27x - 27\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f''(x) = 3{x^2} - 18x + 27\) <strong><em>A1 N2</em></strong> </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;">\(f'(3) = 0\) , \(f''(3) = 0\) <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><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;">\({f''}\) does not change sign at P <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence for this <em><strong>R1 N0</strong> </em></span></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;">\({f'}\) </span><span style="font-family: times new roman,times; font-size: medium;">changes sign at P so P is a maximum/minimum (i.e. not inflexion) <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence for this <em><strong>R1 N0</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 3 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(f(x) = \frac{1}{4}{(x - 3)^4} + c\) and sketching this function <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">indicating minimum at \(x = 3\) <em><strong>R1 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">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 completed parts (a) and (b) successfully. </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 completed parts (a) and (b) successfully. </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 rare few earned any marks in part (c) - most justifying the point of inflexion with the zero answers in part (b), not thinking that there is more to consider. </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 \(f(x) = \frac{{6x}}{{x + 1}}\) , for \(x > 0\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\) , for \(x > 0\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(g'(x) = \frac{1}{{x(x + 1)}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{1}{{x(x + 1)}}\) . The area enclosed by the graph of <em>h</em> , the <em>x</em>-axis and </span><span style="font-family: times new roman,times; font-size: medium;">the lines \(x = \frac{1}{5}\) </span><span style="font-family: times new roman,times; font-size: medium;"> and \(x = k\) is \(\ln 4\) . Given that \(k > \frac{1}{5}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , find the value of <em>k</em> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<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;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing quotient rule <em><strong> (M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{u'v - uv'}}{{{v^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct differentiation (must be seen in quotient rule) <em><strong>(A1)(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\rm{d}}}{{{\rm{d}}x}}(6x) = 6\) , \(\frac{{\rm{d}}}{{{\rm{d}}x}}(x + 1) = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into quotient rule <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{(x + 1)6 - 6x}}{{{{(x + 1)}^2}}}\) , \(\frac{{6x + 6 - 6x}}{{{{(x + 1)}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{6}{{{{(x + 1)}^2}}}\) <em><strong>A1 N4</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[5 marks]</strong> </span></em></p>
<p> </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;">evidence of choosing product rule <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6x{(x + 1)^{ - 1}}\) , \(uv' + vu'\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct differentiation (must be seen in product rule) <em><strong> (A1)(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\rm{d}}}{{{\rm{d}}x}}(6x) = 6\) , \(\frac{{\rm{d}}}{{{\rm{d}}x}}{(x + 1)^{ - 1}} = - 1{(x + 1)^{ - 2}} \times 1\)</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. \(6x \times - {(x + 1)^{ - 2}} + {(x + 1)^{ - 1}} \times 6\) , \(\frac{{ - 6x + 6(x + 1)}}{{{{(x + 1)}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{6}{{{{(x + 1)}^2}}}\) <em><strong>A1 N4</strong> </em></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">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;">evidence of choosing chain rule <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. formula, \(\frac{1}{{\left( {\frac{{6x}}{{x + 1}}} \right)}} \times \left( {\frac{{6x}}{{x + 1}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct reciprocal of \(\frac{1}{{\left( {\frac{{6x}}{{x + 1}}} \right)}}\) is \(\frac{{x + 1}}{{6x}}\) (seen anywhere) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into chain rule <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{{\left( {\frac{{6x}}{{x + 1}}} \right)}} \times \frac{6}{{{{(x + 1)}^2}}}\) , \(\left( {\frac{6}{{{{(x + 1)}^2}}}} \right)\left( {\frac{{x + 1}}{{6x}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working that clearly leads to the answer <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\frac{6}{{(x + 1)}}} \right)\left( {\frac{1}{{6x}}} \right)\) , \(\left( {\frac{1}{{{{(x + 1)}^2}}}} \right)\left( {\frac{{x + 1}}{x}} \right)\) , \(\frac{{6(x + 1)}}{{6x{{(x + 1)}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{1}{{x(x + 1)}}\) <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[4 marks]</strong> </span></em></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 subtract logs <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln a - \ln b\) , \(\ln 6x - \ln (x + 1)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivatives (must be seen in correct expression) <em><strong>A1A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{6}{{6x}} - \frac{1}{{x + 1}}\) , \(\frac{1}{x} - \frac{1}{{x + 1}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working that clearly leads to the answer <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{x + 1 - x}}{{x(x + 1)}}\) , \(\frac{{6x + 6 - 6x}}{{6x(x + 1)}}\) , \(\frac{{6(x + 1 - x)}}{{6x(x + 1)}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{1}{{x(x + 1)}}\) <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid method using integral of <em>h</em>(<em>x</em>) (accept missing/incorrect limits or missing \({\text{d}}x\) ) <strong><em> (M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\rm{area}} = \int_{\frac{1}{5}}^k {h(x){\rm{d}}x} \) , \(\int{\left( {\frac{1}{{x(x + 1)}}} \right)} \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that integral of derivative will give original function <em><strong>(R1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int{\left( {\frac{1}{{x(x + 1)}}} \right)} {\rm{d}}x = \ln \left( {\frac{{6x}}{{x + 1}}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution and subtraction <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln \left( {\frac{{6k}}{{k + 1}}} \right) - \ln \left( {\frac{{6 \times \frac{1}{5}}}{{\frac{1}{5} + 1}}} \right)\) , \(\ln \left( {\frac{{6k}}{{k + 1}}} \right) - \ln (1)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting <strong>their</strong> expression equal to \(\ln 4\) <em><strong>(M1)</strong></em><strong> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln \left( {\frac{{6k}}{{k + 1}}} \right) - \ln (1) = \ln 4\) , \(\ln \left( {\frac{{6k}}{{k + 1}}} \right) = \ln 4\) , \(\int_{\frac{1}{5}}^k {h(x){\rm{d}}x = \ln 4} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation without logs <strong><em> A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g.\(\frac{{6k}}{{k + 1}} = 4\) , \(6k = 4(k + 1)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <strong><em> (A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6k = 4k + 4\) , \(2k = 4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 2\) <strong><em>A1 N4</em> </strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[7 marks] </span></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><span style="font-family: times new roman,times; font-size: medium;">In part (a), most candidates recognized the need to apply the quotient rule to find the derivative, and many were successful in earning full marks here. </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), many candidates struggled with the chain rule, or did not realize the chain rule was necessary to find the derivative. Again, some candidates attempted to work backward from the given answer, which is not allowed in a "show that" question. A few clever candidates simplified the situation by applying properties of logarithms before finding their derivative. </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 part (c), many candidates recognized the need to integrate the function, and that their integral would equal \(\ln 4\) . However, many did not recognize that the integral of <em>h</em> is <em>g</em> . Those candidates who made this link between the parts (b) and (c) often carried on correctly to find the value of <em>k</em> , with a few candidates having errors in working with logarithms. </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;">A toy car travels with velocity <em>v</em> ms<sup>−1</sup> for six seconds. This is shown in the </span><span style="font-family: times new roman,times; font-size: medium;">graph below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/kbells.png" alt></span></p>
</div>
<div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows the graph of \(y = f(x)\), for </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\( - 4 \le x \le 5\).</span></span></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><img src="images/maths_3.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the car’s velocity at \(t = 3\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(f( - 3)\);</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the car’s acceleration at \(t = 1.5\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the total distance travelled.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<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;">\(4{\text{ (m}}{{\text{s}}^{ - 1}}{\text{)}}\) <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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f( - 3) = - 1\) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a(i).</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 acceleration is the gradient <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a(1.5) = \frac{{4 - 0}}{{2 - 0}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 2\) \({\text{(m}}{{\text{s}}^{ - 2}}{\text{)}}\) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing area under curve <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. trapezium, triangles, integration</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. \(\frac{1}{2}(3 + 6)4\) , \(\int_0^6 {\left| {v(t)} \right|} {\rm{d}}t\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">distance 18 (m) <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;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<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">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 \({L_x}\) be a family of lines with equation given by \(r\) \( = \left( {\begin{array}{*{20}{c}} x \\ {\frac{2}{x}} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} {{x^2}} \\ { - 2} \end{array}} \right)\), where \(x > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the equation of \({L_1}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A line \({L_a}\) crosses the \(y\)-axis at a point \(P\).</p>
<p class="p1">Show that \(P\) has coordinates \(\left( {0,{\text{ }}\frac{4}{a}} \right)\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The line \({L_a}\) crosses the \(x\)-axis at \({\text{Q}}(2a,{\text{ }}0)\). Let \(d = {\text{P}}{{\text{Q}}^2}\).</p>
<p class="p1">Show that \(d = 4{a^2} + \frac{{16}}{{{a^2}}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">There is a minimum value for \(d\). Find the value of \(a\) that gives this minimum value.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute \(x = 1\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)<strong><em>r </em></strong>\( = \left( {\begin{array}{*{20}{c}} 1 \\ {\frac{2}{1}} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} {{1^2}} \\ { - 2} \end{array}} \right),{\text{ }}{L_1} = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \end{array}} \right)\)</p>
<p>correct equation (vector or Cartesian, but do not accept “\({L_1}\)”)</p>
<p><em>eg</em>\(\;\;\;\)<strong><em>r </em></strong>\( = \left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 1 \\ { - 2} \end{array}} \right),{\text{ }}y = - 2x + 4\;\;\;\)(must be an equation) <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 class="p1">appropriate approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} 0 \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} a \\ {\frac{2}{a}} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} {{a^2}} \\ { - 2} \end{array}} \right)\)</p>
<p class="p1">correct equation for \(x\)-coordinate <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;0 = a + t{a^2}\)</p>
<p class="p1">\(t = \frac{{ - 1}}{a}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">substituting <strong>their </strong>parameter to find \(y\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;y = \frac{2}{a} - 2\left( {\frac{{ - 1}}{a}} \right),{\text{ }}\left( {\begin{array}{*{20}{c}} a \\ {\frac{2}{a}} \end{array}} \right) - \frac{1}{a}\left( {\begin{array}{*{20}{c}} {{a^2}} \\ { - 2} \end{array}} \right)\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;y = \frac{2}{a} + \frac{2}{a},{\text{ }}\left( {\begin{array}{*{20}{c}} a \\ {\frac{2}{a}} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} a \\ { - \frac{2}{a}} \end{array}} \right)\)</p>
<p class="p1">finding correct expression for \(y\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;y = \frac{4}{a},{\text{ }}\left( {\begin{array}{*{20}{c}} 0 \\ {\frac{4}{a}} \end{array}} \right)\) \({\text{P}}\left( {0,{\text{ }}\frac{4}{a}} \right)\) <span class="Apple-converted-space"> </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;\)distance formula, Pythagorean Theorem, \(\overrightarrow {{\text{PQ}}} = \left( {\begin{array}{*{20}{c}} {2a} \\ { - \frac{4}{a}} \end{array}} \right)\)</p>
<p>correct simplification <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;{(2a)^2} + {\left( {\frac{4}{a}} \right)^2}\)</p>
<p>\(d = 4{a^2} + \frac{{16}}{{{a^2}}}\) <strong><em>AG N0</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing need to find derivative <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;d',{\text{ }}d'(a)\)</p>
<p>correct derivative <strong><em>A2</em></strong></p>
<p><em>eg</em>\(\;\;\;8a - \frac{{32}}{{{a^3}}},{\text{ }}8x - \frac{{32}}{{{x^3}}}\)</p>
<p>setting <strong>their </strong>derivative equal to \(0\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;8a - \frac{{32}}{{{a^3}}} = 0\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;8a = \frac{{32}}{{{a^3}}},{\text{ }}8{a^4} - 32 = 0\)</p>
<p>working towards solution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{a^4} = 4,{\text{ }}{a^2} = 2,{\text{ }}a = \pm \sqrt 2 \)</p>
<p>\(a = \sqrt[4]{4}\;\;\;(a = \sqrt 2 )\;\;\;({\text{do not accept }} \pm \sqrt 2 )\)<strong><em> A1 N3</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
<p><strong><em>Total [17 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">In part (a), most candidates correctly substituted 1 for \(x\), although many of them did not earn full marks for their work here, as they wrote their vector equation using \({L_1} = \), not understanding that \({L_1}\) is the name of the line, and not a vector.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Very few candidates answered parts (b) and (c) correctly, often working backwards from the given answer, which is not appropriate in "show that" questions. In these types of questions, candidates are required to clearly show their working and reasoning, which will hopefully lead them to the given answer.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Very few candidates answered parts (b) and (c) correctly, often working backwards from the given answer, which is not appropriate in "show that" questions. In these types of questions, candidates are required to clearly show their working and reasoning, which will hopefully lead them to the given answer.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Fortunately, a good number of candidates recognized the need to find the derivative of the given expression for \(d\) in part (d) of the question, and so were able to earn at least some of the available marks in the final part.</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 a semicircle centre O, diameter [AB], with radius 2.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let P be a point on the circumference, with \({\rm{P}}\widehat {\rm{O}}{\rm{B}} = \theta \) radians.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/three_mins.png" alt></span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>S</em> be the total area of the two segments shaded in the diagram below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/flo.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the triangle OPB, in terms of \(\theta \) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Explain why the area of triangle OPA is the same as the area triangle OPB.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(S = 2(\pi - 2\sin \theta )\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(\theta \) when <em>S</em> is a local minimum, justifying that it is a minimum.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find a value of \(\theta \) for which <em>S</em> has its greatest value.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<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 area of a triangle <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. </span><span style="font-family: times new roman,times; font-size: medium;">\(A = \frac{1}{2} \times 2 \times 2 \times \sin \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 2\sin \theta \) <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"><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;">\({\rm{P}}\widehat {\rm{O}}{\rm{A = }}\pi - \theta \) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{area }}\Delta {\rm{OPA}} = \frac{1}{2}2 \times 2 \times \sin (\pi - \theta )\) \(( = 2\sin (\pi - \theta ))\) </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;">since \(\sin (\pi - \theta ) = \sin \theta \) <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">then both triangles have the same area <em><strong> AG N0</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;">triangle OPA has the same height and the same base as triangle OPB <em><strong>R3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">then both triangles have the same area <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">area semicircle \( = \frac{1}{2} \times \pi {(2)^2}\) \(( = 2\pi )\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({\text{area }}\Delta {\rm{APB}} = 2\sin \theta + 2\sin \theta \) \(( = 4\sin \theta )\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(S{\text{ = area of semicircle}} - {\text{area }}\Delta {\rm{APB}}\) \(( = 2\pi - 4\sin \theta )\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(S = 2(\pi - 2\sin \theta )\) <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">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;">attempt to differentiate <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{d}}S}}{{{\rm{d}}\theta }} = - 4\cos \theta \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting derivative equal to 0 <em><strong> (M1)</strong></em></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. \( - 4\cos \theta = 0\) , \(\cos \theta = 0\) , \(4\cos \theta = 0\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\theta = \frac{\pi }{2}\) </span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3</span></strong></em></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;">evidence of using second derivative <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(S''(\theta ) = 4\sin \theta \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(S''\left( {\frac{\pi }{2}} \right) = 4\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">it is a minimum because \(S''\left( {\frac{\pi }{2}} \right) > 0\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">R1 N0</span></strong></em></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;">evidence of using first derivative <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> for \(\theta < \frac{\pi }{2},S'(\theta ) < 0\) (may use diagram) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">for \(\theta > \frac{\pi }{2},S'(\theta ) > 0\) (may use diagram) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">it is a minimum since the derivative goes from negative to positive <em><strong>R1 N0</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;">\(2\pi - 4\sin \theta \) is minimum when \(4\sin \theta \) is a maximum <em><strong>R3</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(4\sin \theta \) is a maximum when \(\sin \theta = 1\) <em><strong>(A2)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\theta = \frac{\pi }{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A3 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">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><em>S</em> is greatest when \(4\sin \theta \) is smallest (or equivalent) <em><strong>(R1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 0\) (or \(\pi \) ) <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">e.</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;">Most candidates could obtain the area of triangle OPB as equal to \(2\sin \theta \) , though \(2\theta \) was </span><span style="font-family: times new roman,times; font-size: medium;">given quite often as the area.</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;">A minority recognized the equality of the sines of supplementary angles and </span><span style="font-family: times new roman,times; font-size: medium;">the term complementary was frequently used instead of supplementary. Only a handful of </span><span style="font-family: times new roman,times; font-size: medium;">candidates used the simple equal base and altitude argument.</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;">Many candidates seemed to </span><span style="font-family: times new roman,times; font-size: medium;">see why \(S = 2(\pi - 2\sin \theta )\) but the arguments presented for showing why this result was true </span><span style="font-family: times new roman,times; font-size: medium;">were not very convincing in many cases. Explicit evidence of why the area of the semicircle </span><span style="font-family: times new roman,times; font-size: medium;">was \(2\pi \) was often missing as was an explanation for \(2(2\sin \theta )\) </span><span style="font-family: times new roman,times; font-size: medium;">and for subtraction.</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;">Only a small number of candidates recognized the fact <em>S</em> would be minimum when sin was </span><span style="font-family: times new roman,times; font-size: medium;">maximum, leading to a simple non-calculus solution. Those who chose the calculus route </span><span style="font-family: times new roman,times; font-size: medium;">often had difficulty finding the derivative of <em>S</em>, failing in a significant number of cases to </span><span style="font-family: times new roman,times; font-size: medium;">recognize that the derivative of a constant is 0, and also going through painstaking application </span><span style="font-family: times new roman,times; font-size: medium;">of the product rule to find the simple derivative. When it came to justify a minimum, there was </span><span style="font-family: times new roman,times; font-size: medium;">evidence in some cases of using some form of valid test, but explanation of the test being </span><span style="font-family: times new roman,times; font-size: medium;">used was generally poor.</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;">Candidates who answered part (d) correctly generally did well in </span><span style="font-family: times new roman,times; font-size: medium;">part (e) as well, though answers outside the domain of \(\theta \) were frequently seen.</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;">Consider a function \(f(x)\) such that \(\int_1^6 {f(x){\text{d}}x = 8} \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_1^6 {2f(x){\text{d}}x} \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_1^6 {\left( {f(x) + 2} \right){\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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;">appropriate 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> \(2\int {f(x),{\text{ }}2(8)} \)</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;">\(\int_1^6 {2f(x){\text{d}}x = 16} \) <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">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;">appropriate 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> \(\int {f(x) + \int {2,{\text{ }}8 + \int 2 } } \)</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;">\(\int {2{\text{d}}x = 2x} \) (seen anywhere) <strong><em>(A1)</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;">substituting limits into <strong>their </strong>integrated function and subtracting (in any order) <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> \(2(6) - 2(1),{\text{ }}8 + 12 - 2\)</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;">\(\int_1^6 {\left( {f(x) + 2} \right){\text{d}}x = 18} \) <strong><em>A1 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>[4 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;">A rectangle is inscribed in a circle of radius 3 cm and centre O, as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bike.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The point P(<em>x</em> , <em>y</em>) is a vertex of the rectangle and also lies on the circle. The angle </span><span style="font-family: times new roman,times; font-size: medium;">between (OP) and the <em>x</em>-axis is \(\theta \) radians, where \(0 \le \theta \le \frac{\pi }{2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down an expression in terms of \(\theta \) for</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(x\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(y\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let the area of the rectangle be <em>A</em>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(A = 18\sin 2\theta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, find the exact value of \(\theta \) which maximizes the area of </span><span style="font-family: times new roman,times; font-size: medium;">the rectangle.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) Use the second derivative to justify that this value of \(\theta \) does give </span><span style="font-family: times new roman,times; font-size: medium;">a maximum.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<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) \(x = 3\cos \theta \) <em><strong>A1 N1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(y = 3\sin \theta \) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">finding area <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A = 2x \times 2y\) , \(A = 8 \times \frac{1}{2}bh\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(A = 4 \times 3\sin \theta \times 3\cos \theta \) , \(8 \times \frac{1}{2} \times 3\cos \theta \times 3\sin \theta \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(A = 18(2\sin \theta \cos \theta )\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(A = 18\sin 2\theta \) <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }} = 36\cos 2\theta \) <em><strong>A2 N2 </strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) for setting derivative equal to 0 <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(36\cos 2\theta = 0\) , \(\frac{{{\rm{d}}A}}{{{\rm{d}}\theta }} = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2\theta = \frac{\pi }{2}\) <em><strong> (A1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = \frac{\pi }{4}\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) valid reason (seen anywhere) <em><strong>R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. at \(\frac{\pi }{4}\), \(\frac{{{{\rm{d}}^2}A}}{{{\rm{d}}{\theta ^2}}} < 0\) ; maximum when \(f''(x) < 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding second derivative \(\frac{{{{\rm{d}}^2}A}}{{{\rm{d}}{\theta ^2}}} = - 72\sin 2\theta \) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(\frac{\pi }{4}\) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 72\sin \left( {2 \times \frac{\pi }{4}} \right)\) , \( - 72\sin \left( {\frac{\pi }{2}} \right)\) , \( - 72\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\theta = \frac{\pi }{4}\) produces the maximum area <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[8 marks]</strong> </span></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 familiar with the circular nature of sine and cosine found part (a) accessible. However, a good number of candidates left this part blank, which suggests that there was difficulty interpreting the meaning of the<em> x</em> and <em>y</em> in the diagram. </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;">Those with answers from (a) could begin part (b), but many worked backwards and thus earned no marks. In a "show that" question, a solution cannot begin with the answer given. The area of the rectangle could be found by using \(2x \times 2y\) , or by using the eight small triangles, but it was essential that the substitution of the double-angle formula was shown before writing the given answer. </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;">As the area function was given in part (b), many candidates correctly found the derivative in (c) and knew to set this derivative to zero for a maximum value. Many gave answers in degrees, however, despite the given domain in radians. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Although some candidates found the second derivative function correctly, few stated that the second derivative must be negative at a maximum value. Simply calculating a negative value is not sufficient for a justification. </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 \(f(x) = \frac{{{{(\ln x)}^2}}}{2}\), for \(x > 0\).</span></p>
</div>
<div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(g(x) = \frac{1}{x}\). The following diagram shows parts of the graphs of \(f'\) and <em>g</em>.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; min-height: 25px; text-align: center; margin: 0px;"><img src="images/maths_10b.png" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(f'\) has an <em>x</em>-intercept at \(x = p\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(f'(x) = \frac{{\ln x}}{x}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">There is a minimum on the graph of \(f\). Find the \(x\)-coordinate of this minimum.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \(p\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(g\) intersects the graph of \(f'\) when \(x = q\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(q\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(g\) intersects the graph of \(f'\) when \(x = q\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(R\) be the region enclosed by the graph of \(f'\), the graph of \(g\) and the line \(x = p\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the area of \(R\) is \(\frac{1}{2}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<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;"><strong>METHOD 1</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;">correct use of chain rule <strong><em>A1A1</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> \(\frac{{2\ln x}}{2} \times \frac{1}{x},{\text{ }}\frac{{2\ln x}}{{2x}}\)</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>Note: </strong>Award <strong><em>A1 </em></strong>for \(\frac{{2\ln x}}{{2x}}\), <strong><em>A1 </em></strong>for \( \times \frac{1}{x}\).</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;">\(f'(x) = \frac{{\ln x}}{x}\) <strong><em>AG N0</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>
<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>METHOD 2</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;">correct substitution into quotient rule, with derivatives seen <strong><em>A1</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> \(\frac{{2 \times 2\ln x \times \frac{1}{x} - 0 \times {{(\ln x)}^2}}}{4}\)</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;">correct working <strong><em>A1</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> \(\frac{{4\ln x \times \frac{1}{x}}}{4}\)</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;">\(f'(x) = \frac{{\ln x}}{x}\) <strong><em>AG N0</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">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;">setting derivative \( = 0\) <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> \(f'(x) = 0,{\text{ }}\frac{{\ln x}}{x} = 0\)</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> \(\ln x = 0,{\text{ }}x = {{\text{e}}^0}\)</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;">\(x = 1\) <strong><em>A1 N2</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>
<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;">intercept when \(f'(x) = 0\) <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;">\(p = 1\) <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">c.</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;">equating functions <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> \(f' = g,{\text{ }}\frac{{\ln x}}{x} = \frac{1}{x}\)</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;">correct working <strong><em>(A1)</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> \(\ln x = 1\)</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;">\(q = {\text{e (accept }}x = {\text{e)}}\) <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';"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></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: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of integrating and subtracting functions (in any order, seen anywhere) <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> \(\int_q^e {\left( {\frac{1}{x} - \frac{{\ln x}}{x}} \right){\text{d}}x{\text{, }}\int {f' - g} } \)</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;">correct integration \(\ln x - \frac{{{{(\ln x)}^2}}}{2}\) <strong><em>A2</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;">substituting limits into <strong>their </strong>integrated function and subtracting (in any order) <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> \((\ln {\text{e}} - \ln 1) - \left( {\frac{{{{(\ln {\text{e}})}^2}}}{2} - \frac{{{{(\ln 1)}^2}}}{2}} \right)\)</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>Note: </strong>Do not award <strong><em>M1 </em></strong>if the integrated function has only one term.</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;">correct working <strong><em>A1</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> \((1 - 0) - \left( {\frac{1}{2} - 0} \right),{\text{ }}1 - \frac{1}{2}\)</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;">\({\text{area}} = \frac{1}{2}\) <strong><em>AG N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px '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>Candidates may work with two separate integrals, and only combine them at the end. Award marks in line with the markscheme.</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>[5 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><span style="font-family: times new roman,times; font-size: medium;">The following is the graph of a function \(f\) , for \(0 \le x \le 6\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="images/charlie.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The first part of the graph is a quarter circle of radius \(2\) with centre at the origin.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Find \(\int_0^2 {f(x){\rm{d}}x} \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the graph of \(f\) , the \(x\)-axis, the \(y\)-axis and the </span><span style="font-family: times new roman,times; font-size: medium;">line \(x = 6\) . The area of this region is \(3\pi \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_2^6 {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_0^2 {f(x){\rm{d}}x} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The shaded region is enclosed by the graph of \(f\) , the \(x\)-axis, the \(y\)-axis and the </span><span style="font-family: times new roman,times; font-size: medium;">line \(x = 6\) . The area of this region is \(3\pi \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_2^6 {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<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) attempt to find quarter circle area <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\frac{1}{4}(4\pi )\) , \(\frac{{\pi {r^2}}}{4}\) , \(\int_0^2 {\sqrt {4 - {x^2}{\rm{d}}x} } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of region \( = \pi \) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int_0^2 {f(x){\rm{d}}x = - \pi } \) <strong><em>A2 N3 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">attempted set up with both regions <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\text{shaded area}} - {\text{quarter circle}}\) , \(3\pi - \pi \) , \(3\pi - \int_0^2 {f = \int_2^6 f } \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int_2^6 {f(x){\rm{d}}x = 2\pi } \) <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>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [7 marks]<br></span></strong></em></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;">attempt to find quarter circle area <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\frac{1}{4}(4\pi )\) , \(\frac{{\pi {r^2}}}{4}\) , \(\int_0^2 {\sqrt {4 - {x^2}{\rm{d}}x} } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of region \( = \pi \) <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int_0^2 {f(x){\rm{d}}x = - \pi } \) <strong><em>A2 N3 </em></strong></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;">attempted set up with both regions <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\text{shaded area}} - {\text{quarter circle}}\) , \(3\pi - \pi \) , \(3\pi - \int_0^2 {f = \int_2^6 f } \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int_2^6 {f(x){\rm{d}}x = 2\pi } \) <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>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [7 marks]<br></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 dir="ltr" style="font-size: 13.28px; font-family: sans-serif; left: 96.032px; top: 828.133px; transform: scale(1.05935, 1); transform-origin: 0% 0% 0px;" data-font-name="Helvetica" data-canvas-width="604.8907380271482"><span style="font-family: times new roman,times; font-size: medium;">There was a minor error on the diagram, where the point on the \(y\)-axis was labelled \(2\) (to indicate the length of the radius), rather than \( - 2\). Examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p dir="ltr" style="font-size: 13.28px; font-family: sans-serif; left: 309.667px; top: 844.133px; transform: scale(1.01373, 1); transform-origin: 0% 0% 0px;" data-font-name="Helvetica" data-canvas-width="329.4635298187781"><span style="font-family: times new roman,times; font-size: medium;">While most candidates were able to correctly find the area of the quarter circle in part (a), very few considered that the value of the definite integral is negative for the part of the function below the \(x\)-axis. In part (b), most went on to earn full marks by subtracting the area of the quarter circle from \(3 \pi\).</span></p>
<p dir="ltr" style="font-size: 13.28px; font-family: sans-serif; left: 309.667px; top: 844.133px; transform: scale(1.01373, 1); transform-origin: 0% 0% 0px;" data-font-name="Helvetica" data-canvas-width="329.4635298187781"><span style="font-family: times new roman,times; font-size: medium;">Candidates who did not understand the connection between area and the value of the integral often tried to find a function to integrate. These candidates were not successful using this method.</span></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-size: 13.28px; font-family: sans-serif; left: 96.032px; top: 828.133px; transform: scale(1.05935, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="604.8907380271482"><span style="font-family: times new roman,times; font-size: medium;">There was a minor error on the diagram, where the point on the \(y\)-axis was labelled \(2\) (to indicate the length of the radius), rather than \( - 2\). Examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p style="font-size: 13.28px; font-family: sans-serif; left: 309.667px; top: 844.133px; transform: scale(1.01373, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="329.4635298187781"><span style="font-family: times new roman,times; font-size: medium;">While most candidates were able to correctly find the area of the quarter circle in part (a), very few considered that the value of the definite integral is negative for the part of the function below the \(x\)-axis. In part (b), most went on to earn full marks by subtracting the area of the quarter circle from \(3 \pi\).</span></p>
<p style="font-size: 13.28px; font-family: sans-serif; left: 309.667px; top: 844.133px; transform: scale(1.01373, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="329.4635298187781"><span style="font-family: times new roman,times; font-size: medium;">Candidates who did not understand the connection between area and the value of the integral often tried to find a function to integrate. These candidates were not successful using this method.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-size: 13.28px; font-family: sans-serif; left: 96.032px; top: 828.133px; transform: scale(1.05935, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="604.8907380271482"><span style="font-family: times new roman,times; font-size: medium;">There was a minor error on the diagram, where the point on the \(y\)-axis was labelled \(2\) (to indicate the length of the radius), rather than \( - 2\). Examiners were instructed to notify the IB assessment centre of any candidates adversely affected. Candidate scripts did not indicate any adverse effect.</span></p>
<p style="font-size: 13.28px; font-family: sans-serif; left: 309.667px; top: 844.133px; transform: scale(1.01373, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="329.4635298187781"><span style="font-family: times new roman,times; font-size: medium;">While most candidates were able to correctly find the area of the quarter circle in part (a), very few considered that the value of the definite integral is negative for the part of the function below the \(x\)-axis. In part (b), most went on to earn full marks by subtracting the area of the quarter circle from \(3 \pi\).</span></p>
<p style="font-size: 13.28px; font-family: sans-serif; left: 309.667px; top: 844.133px; transform: scale(1.01373, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="329.4635298187781"><span style="font-family: times new roman,times; font-size: medium;">Candidates who did not understand the connection between area and the value of the integral often tried to find a function to integrate. These candidates were not successful using this method.</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;">Consider \(f(x) = \ln ({x^4} + 1)\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The second derivative is given by \(f''(x) = \frac{{4{x^2}(3 - {x^4})}}{{{{({x^4} + 1)}^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The equation \(f''(x) = 0\) has only three solutions, when \(x = 0\) , \( \pm \sqrt[4]{3}\) \(( \pm 1.316 \ldots )\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(f(0)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the set of values of \(x\) for which \(f\) is increasing.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f''(1)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong>, show that there is no point of inflexion on the graph of \(f\) at \(x = 0\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is a point of inflexion on the graph of \(f\) at \(x = \sqrt[4]{3}\) \((x = 1.316 \ldots )\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(f\) , for \(x \ge 0\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<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;">substitute \(0\) into \(f\)<em><strong> </strong></em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\ln (0 + 1)\) , \(\ln 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(0) = 0\) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{1}{{{x^4} + 1}} \times 4{x^3}\) (seen anywhere) <strong><em>A1A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \(\frac{1}{{{x^4} + 1}}\) and <strong><em>A1</em></strong> for \(4{x^3}\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing \(f\) increasing where \(f'(x) > 0\) (seen anywhere) <strong><em>R1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(f'(x) > 0\) , diagram of signs </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve \(f'(x) > 0\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(4{x^3} = 0\) , \({x^3} > 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f\) increasing for \(x > 0\) (accept \(x \ge 0\) ) <strong><em>A1 N1 </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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) substituting \(x = 1\) into \(f''\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\frac{{4(3 - 1)}}{{{{(1 + 1)}^2}}}\) , \(\frac{{4 \times 2}}{4}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f''(1) = 2\) <strong><em> A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) valid interpretation of point of inflexion (seen anywhere) <strong><em>R1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> no change of sign in \(f''(x)\) , no change in concavity, </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'\) increasing both sides of zero </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \(f''(x)\) for \(x < 0\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(f''( - 1)\) , \(\frac{{4{{( - 1)}^2}(3 - {{( - 1)}^4})}}{{{{({{( - 1)}^4} + 1)}^2}}}\) , diagram of signs </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working leading to positive value <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(f''( - 1) = 2\) , discussing signs of numerator <strong>and</strong> denominator </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">there is no point of inflexion at \(x = 0\) <em><strong>AG N0 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </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><span style="font-family: times new roman,times; font-size: medium;"><img src="images/jcp.png" alt></span><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>Notes</strong>: Award <em><strong>A1</strong></em> for shape concave up left of POI and concave down right of POI. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> Only if this <em><strong>A1</strong></em> is awarded, then 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 curve through (\(0\), \(0\)) , <strong><em>A1</em></strong> for increasing throughout. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> Sketch need not be drawn to scale. Only essential features need to be clear. </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;">Many candidates left their answer to part (a) as \(\ln 1\). While this shows an understanding for substituting a value into a function, it leaves an unfinished answer that should be expressed as an integer.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Candidates who attempted to consider where \(f\) is increasing generally understood the derivative is needed. However, a number of candidates did not apply the chain rule, which commonly led to answers such as “increasing for all \(x\)”. Many set their derivative equal to zero, while neglecting to indicate in their working that \(f'(x) > 0\) for an increasing function. Some created a diagram of signs, which provides appropriate evidence as long as it is clear that the signs represent \(f’\).</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;">Finding \(f''(1)\) proved no challenge, however, using this value to <strong>show that</strong> no point of inflexion exists proved elusive for many. Some candidates recognized the signs must not change in the second derivative. Few candidates presented evidence in the form of a calculation, which follows from the “hence” command of the question. In this case, a sign diagram without numerical evidence was not sufficient.</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;">Few candidates created a correct graph from the information given or found in the question. This included the point (\(0\), \(0\)), the fact that the function is always increasing for \(x > 0\) , the concavity at \(x = 1\) and the change in concavity at the given point of inflexion. Many incorrect attempts showed a graph concave down to the right of \(x = 0\) , changing to concave up.</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;">The acceleration, \(a{\text{ m}}{{\text{s}}^{ - 2}}\), of a particle at time <em>t</em> seconds is given by \(a = 2t + \cos t\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the acceleration of the particle at \(t = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the velocity, <em>v</em>, at time<em> t</em>, given that the initial velocity of the particle </span><span style="font-family: times new roman,times; font-size: medium;">is \({\text{m}}{{\text{s}}^{ - 1}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_0^3 {v{\rm{d}}t} \)</span><span style="font-family: times new roman,times; font-size: medium;"> , giving your answer in the form \(p - q\cos 3\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">What information does the answer to part (c) give about the motion of </span><span style="font-family: times new roman,times; font-size: medium;">the particle?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<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 \(t = 0\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a(0) = 0 + \cos 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a(0) = 1\) <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 integrating the acceleration function <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {(2t + \cos t){\text{d}}t} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression \({t^2} + \sin t + c\) <strong><em>A1A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> If "\( + c\)" is omitted, award no further marks.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting (2,0) into indefinite integral <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2 = 0 + \sin 0 + c\) , \(c = 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v(t) = {t^2} + \sin t + 2\) <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;">\(\int {({t^2} + \sin t + 2)} {\rm{d}}t = \frac{{{t^3}}}{3} - \cos t + 2t\) <em><strong>A1A1A1</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 each correct term. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(v(3) - v(0)\) <strong><em> (M1)</em> </strong></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. \((9 - \cos 3 + 6) - (0 - \cos 0 + 0)\) , \((15 - \cos 3) - ( - 1)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(16 - \cos 3\) (accept \(p = 16\) , \(q = - 1\) ) <em><strong>A1A1 N3</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 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;">reference to motion, reference to first 3 seconds <em><strong>R1R1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. displacement in 3 seconds, distance travelled in 3 seconds </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">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;">Parts (a) and (b) of this question were generally 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;">Parts (a) and (b) of this question were generally well done. </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;">Problems arose in part (c) with many candidates not substituting \(s(3) - s(0)\) correctly, leading to only a partially correct final answer. There were also a notable few who were not aware that \(\cos 0 = 1\) in both parts (a) and (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;">There were a variety of interesting answers about the motion of the particle, few being able to give both parts of the answer correctly. </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 \(f:x \mapsto {\sin ^3}x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the range of the function <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Consider \(f(x) = 1\) , \(0 \le x \le 2\pi \) . Write down the number of solutions to </span><span style="font-family: times new roman,times; font-size: medium;">this equation. Justify your answer.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where \(a{\text{, }}p{\text{, }}q \in \mathbb{Z}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">. Find the volume generated when the </span><span style="font-family: times new roman,times; font-size: medium;">curve of <em>g</em> is revolved through \(2\pi \) about the <em>x</em>-axis.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<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) range of <em>f</em> is \([ - 1{\text{, }}1]\) , \(( - 1 \le f(x) \le 1)\) <em><strong>A2 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\sin ^3}x \Rightarrow 1 \Rightarrow \sin x = 1\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">justification for one solution on \([0{\text{, }}2\pi ]\) <em><strong>R1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, unit circle, sketch of \(\sin x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">1 solution (seen anywhere) <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.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 3{\sin ^2}x\cos x\) <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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">using \(V = \int_a^b {\pi {y^2}{\rm{d}}x} \) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(V = \int_0^{\frac{\pi }{2}} {\pi (\sqrt 3 } \sin x{\cos ^{\frac{1}{2}}}x{)^2}{\rm{d}}x\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \pi \int_0^{\frac{\pi }{2}} {3{{\sin }^2}x\cos x{\rm{d}}x} \) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(V = \pi \left[ {{{\sin }^3}x} \right]_0^{\frac{\pi }{2}}\) \(\left( { = \pi \left( {{{\sin }^3}\left( {\frac{\pi }{2}} \right) - {{\sin }^3}0} \right)} \right)\) <em><strong>A2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(\sin \frac{\pi }{2} = 1\) </span><span style="font-family: times new roman,times; font-size: medium;">and \(\sin 0 = 0\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi \left( {1 - 0} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = \pi \) <em><strong>A1 N1</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 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 not done well by most candidates. No more than one-third of them could correctly give the range of \(f(x) = {\sin ^3}x\) and few could provide adequate justification for there being exactly one solution to \(f(x) = 1\) in the interval \([0{\text{, }}2\pi ]\) .</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 not done well by most candidates.</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;">This question was not done well by most candidates. No more than one-third of them could correctly give the range of \(f(x) = {\sin ^3}x\) and few could provide adequate justification for there being exactly one solution to \(f(x) = 1\) in the interval \([0{\text{, }}2\pi ]\) . Finding the derivative of this function also presented major problems, thus making part (c) of the question much more difficult. In spite of the formula for volume of revolution being given in the Information Booklet, fewer than half of the candidates could correctly put the necessary function and limits into \(\pi \int_a^b {{y^2}{\rm{d}}x} \) and fewer still could square \(\sqrt 3 \sin x{\cos ^{\frac{1}{2}}}x\) correctly. From those who did square correctly, the correct antiderivative was not often recognized. All manner of antiderivatives were suggested instead.</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;">A function <em>f</em> is defined for \( - 4 \le x \le 3\) . The graph of <em>f</em> is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/poo.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph has a local maximum when \(x = 0\) , and local minima when \(x = - 3\) , \(x = 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-intercepts of the graph of the <strong>derivative</strong> function, \(f'\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down all values of <em>x</em> for which \(f'(x)\) is positive.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">At point D on the graph of <em>f</em> , the <em>x</em>-coordinate is \( - 0.5\). Explain why \(f''(x) < 0\) </span><span style="font-family: times new roman,times; font-size: medium;">at D.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<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;"><em>x</em>-intercepts at \( - 3\), 0, 2 <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 3 < x < 0\) , \(2 < x < 3\) <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;">correct reasoning <em><strong>R2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. the graph of <em>f</em> is <strong>concave-down</strong> (accept convex), the first derivative is decreasing</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">therefore the second derivative is negative <em><strong>AG</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;">Candidates had mixed success with parts (a) and (b). Weaker candidates either incorrectly </span><span style="font-family: times new roman,times; font-size: medium;">used the <em>x</em>-intercepts of <em>f</em> or left this question blank. Some wrote down only two of the three </span><span style="font-family: times new roman,times; font-size: medium;">values in part (a). Candidates who answered part (a) correctly often had trouble writing the set </span><span style="font-family: times new roman,times; font-size: medium;">of values in part (b); difficulties included poor notation and incorrectly including the </span><span style="font-family: times new roman,times; font-size: medium;">endpoints. Other candidates listed individual <em>x</em>-values here rather than a range of values.</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;">Candidates had mixed success with parts (a) and (b). Weaker candidates either incorrectly </span><span style="font-family: times new roman,times; font-size: medium;">used the <em>x</em>-intercepts of <em>f</em> or left this question blank. Some wrote down only two of the three </span><span style="font-family: times new roman,times; font-size: medium;">values in part (a). Candidates who answered part (a) correctly often had trouble writing the set </span><span style="font-family: times new roman,times; font-size: medium;">of values in part (b); difficulties included poor notation and incorrectly including the </span><span style="font-family: times new roman,times; font-size: medium;">endpoints. Other candidates listed individual <em>x</em>-values here rather than a range of values.</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;">Many candidates had difficulty explaining why the second derivative is negative in part (c). A </span><span style="font-family: times new roman,times; font-size: medium;">number claimed that since the point D was “close” to a maximum value, the second derivative </span><span style="font-family: times new roman,times; font-size: medium;">must be negative; this incorrect appeal to the second derivative test indicates a lack of </span><span style="font-family: times new roman,times; font-size: medium;">understanding of how the test works and the relative concept of closeness. Some candidates </span><span style="font-family: times new roman,times; font-size: medium;">claimed D was a point of inflexion, again demonstrating poor understanding of the second </span><span style="font-family: times new roman,times; font-size: medium;">derivative. Among candidates who answered part (c) correctly, some stated that <em>f</em> was </span><span style="font-family: times new roman,times; font-size: medium;">concave down while others gave well-formed arguments for why the first derivative was </span><span style="font-family: times new roman,times; font-size: medium;">decreasing. A few candidates provided nicely sketched graphs of \(f'\) and \(f''\) and used them </span><span style="font-family: times new roman,times; font-size: medium;">in their explanations.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph of a function <em>h </em>passes through the point \(\left( {\frac{\pi }{{12}}, 5} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(h'(x) = 4\cos 2x\), find \(h(x)\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of anti-differentiation <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> \(\int {h'(x), \int {4\cos 2x{\text{d}}x} } \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct integration <strong><em>(A2)</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> \(h(x) = 2\sin 2x + c, \frac{{4\sin 2x}}{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute \(\left( {\frac{\pi }{{12}},5} \right)\) into their 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> \(2\sin \left( {2 \times \frac{\pi }{{12}}} \right) + c = 5,{\text{ }}2\sin \left( {\frac{\pi }{6}} \right) = 5\)</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;">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> \(2\left( {\frac{1}{2}} \right) + c = 5,{\text{ }}c = 4\)</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;">\(h(x) = 2\sin 2x + 4\) <strong><em>A1 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;"><strong><em>[6 marks]</em></strong></span></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;">Let \(f(x) = {{\rm{e}}^{ - 3x}}\) and \(g(x) = \sin \left( {x - \frac{\pi }{3}} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(g'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = {{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right)\) . Find the exact value of \(h'\left( {\frac{\pi }{3}} \right)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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) \( - 3{{\rm{e}}^{ - 3x}}\) <em><strong> A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(\cos \left( {x - \frac{\pi }{3}} \right)\) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing product rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(uv' + vu'\)</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. \( - 3{{\rm{e}}^{ - 3x}}\sin \left( {x - \frac{\pi }{3}} \right) + {{\rm{e}}^{ - 3x}}\cos \left( {x - \frac{\pi }{3}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">complete correct substitution of \(x = \frac{\pi }{3}\) <em><strong>(A1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 3{{\rm{e}}^{ - 3\frac{\pi }{3}}}\sin \left( {\frac{\pi }{3} - \frac{\pi }{3}} \right) + {{\rm{e}}^{ - 3\frac{\pi }{3}}}\cos \left( {\frac{\pi }{3} - \frac{\pi }{3}} \right)\)ï€ ïƒ§        </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(h'\left( {\frac{\pi }{3}} \right) = {{\rm{e}}^{ - \pi }}\) <em><strong>A1 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">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;">A good number of candidates found the correct derivative expressions in (a). Many applied the product rule, although with mixed success. </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;">Often the substitution of \({\frac{\pi }{3}}\) was incomplete or not done at all. </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;">The velocity <em>v</em> ms<sup>−1</sup> of a particle at time <em>t</em> seconds, is given by \(v = 2t + \cos 2t\) , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le t \le 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the velocity of the particle when \(t = 0\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t = k\) , the acceleration is zero.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \(k = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the exact velocity when \(t = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">When \(t < \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> and when \(t > \frac{\pi }{4}\) , \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} > 0\) </span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch a graph of <em>v</em> against <em>t</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>d</em> be the distance travelled by the particle for \(0 \le t \le 1\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for <em>d</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Represent <em>d</em> on your sketch.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<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;">\(v = 1\) <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;">(i) \(\frac{{\rm{d}}}{{{\rm{d}}t}}(2t) = 2\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\frac{{\rm{d}}}{{{\rm{d}}t}}(\cos 2t) = - 2\sin 2t\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1</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 coefficient 2 and <em><strong>A1</strong></em> for <span lang="EN-US">\( - \sin 2t\)</span> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of considering acceleration = 0 <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{\rm{d}}v}}{{{\rm{d}}t}} = 0\) , \(2 - 2\sin 2t = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct manipulation <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sin 2k = 1\) , \(\sin 2t = 1\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(2k = \frac{\pi }{2}\) (accept \(2t = \frac{\pi }{2}\) ) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1</span></em></strong></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(k = \frac{\pi }{4}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG N0</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) attempt to substitute \(t = \frac{\pi }{4}\) into <em>v<strong> </strong></em></span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\left( {\frac{\pi }{4}} \right) + \cos \left( {\frac{{2\pi }}{4}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = \frac{\pi }{2}\) <em><strong>A1 N2</strong></em></span></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(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;"><br><img src="images/ray.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A2 N4</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Award <em><strong>A1</strong></em> for <em>y</em>-intercept at \((0{\text{, }}1)\) , <strong><em>A1</em></strong> for curve having zero gradient at \(t = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A2</strong></em> for shape that is concave down to the left of \(\frac{\pi }{4}\) </span><span style="font-family: times new roman,times; font-size: medium;">and concave up to the right </span><span style="font-family: times new roman,times; font-size: medium;">of \(\frac{\pi }{4}\) </span><span style="font-family: times new roman,times; font-size: medium;">. If a correct curve is drawn without indicating \(t = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , do not award the </span><span style="font-family: times new roman,times; font-size: medium;">second <em><strong>A1</strong></em> for the zero gradient, but award the final <em><strong>A2</strong></em> if appropriate. Sketch need not be drawn to scale. Only essential features 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">c.</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 expression <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^1 {(2t + \cos 2t){\rm{d}}t} \) , \(\left[ {{t^2} + \frac{{\sin 2t}}{2}} \right]_0^1\) , \(1 + \frac{{\sin 2}}{2}\) , \(\int_0^1 {v{\rm{d}}t} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/deb.png" alt></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;"><strong>Note</strong>: The line at \(t = 1\) needs to be clearly after \(t = \frac{\pi }{4}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</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(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 gave a correct initial velocity, although a substantial number of candidates answered that \(0 + \cos 0 = 0\) .</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 (b), students commonly applied the chain rule correctly to achieve the derivative, and many recognized that the acceleration must be zero. Occasionally a student would use a double-angle identity on the velocity function before differentiating. This is not incorrect, but it usually caused problems when trying to show \(k = \frac{\pi }{4}\) . At times students would reach the equation \(\sin 2k = 1\) and then substitute the \(\frac{\pi }{4}\) , which does not satisfy the “show that” instruction. </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;">The challenge in this question is sketching the graph using the information achieved and provided. This requires students to make graphical interpretations, and as typical in section B, to link the early parts of the question with later parts. Part (a) provides the <em>y</em>-intercept, and part (b) gives a point with a horizontal tangent. Plotting these points first was a helpful strategy. Few understood either the notation or the concept that the function had to be increasing on either side of the \(\frac{\pi }{4}\) , with most thinking that the point was either a max or min. It was the astute student who recognized that the derivatives being positive on either side of \(\frac{\pi }{4}\) creates a point of inflexion. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Additionally, important points should be labelled in a sketch. Indicating the \(\frac{\pi }{4}\) on the <em>x</em>-axis is a requirement of a clear graph. Although students were not penalized for not labelling the \(\frac{\pi }{2}\) on the <em>y</em>-axis, there should be a recognition that the point is higher than the <em>y</em>-intercept. </span></p>
<p> </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;">While some candidates recognized that the distance is the area under the velocity graph, surprisingly few included neither the limits of integration in their expression, nor the “d<em>t</em>”. Most unnecessarily attempted to integrate the function, often giving an answer with “+C”, and only earned marks if the limits were included with their result. Few recognized that a shaded area is an adequate representation of distance on the sketch, with most fruitlessly attempting to graph a new curve.</span></p>
<div class="question_part_label">d(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In this question, you are given that \(\cos \frac{\pi }{3} = \frac{1}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , and \(\sin \frac{\pi }{3} = \frac{{\sqrt 3 }}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The displacement of an object from a fixed point, O is given by \(s(t) = t - \sin 2t\) for</span><span style="font-family: times new roman,times; font-size: medium;"> \(0 \le t \le \pi \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(s'(t)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In this interval, there are only two values of <em>t</em> for which the object is not moving. </span><span style="font-family: times new roman,times; font-size: medium;">One value is \(t = \frac{\pi }{6}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the other value.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(s'(t) > 0\) between these two values of <em>t</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the distance travelled between these two values of <em>t</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<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;">\(s'(t) = 1 - 2\cos 2t\) <em><strong>A1A2 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 1, <em><strong>A2</strong></em> for \(- 2\cos 2t\) . </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;">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. setting \(s'(t) = 0\)</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. \(2\cos 2t = 1\) , \(\cos 2t = \frac{1}{2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2t = \frac{\pi }{3}\) , \(\frac{{5\pi }}{3}\) , \(\ldots \) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = \frac{{5\pi }}{6}\) <em><strong>A1 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">b.</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. choosing a value in the interval \(\frac{\pi }{6} < t < \frac{{5\pi }}{6}\) </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. \(s'\left( {\frac{\pi }{2}} \right) = 1 - 2\cos \pi \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s'\left( {\frac{\pi }{2}} \right) = 3\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(s'(t) > 0\) <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[3 marks]</strong> </span></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 approach using <em>s</em> or integral of \(s'\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {s'(t){\rm{d}}t} \) ; \(s\left( {\frac{{5\pi }}{6}} \right)\) , \(s\left( {\frac{\pi }{6}} \right)\) ; \(\left[ {t - \sin 2t} \right]_{\frac{\pi }{6}}^{\frac{{5\pi }}{6}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting values and subtracting <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s\left( {\frac{{5\pi }}{6}} \right) - s\left( {\frac{\pi }{6}} \right)\) , \(\left( {\frac{\pi }{6} - \frac{{\sqrt 3 }}{2}} \right) - \left( {\frac{{5\pi }}{6} - \left( { - \frac{{\sqrt 3 }}{2}} \right)} \right)\)</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. \(\frac{{5\pi }}{6} - \sin \frac{{5\pi }}{3} - \left[ {\frac{\pi }{6} - \sin \frac{\pi }{3}} \right]\) , \(\left( {\frac{{5\pi }}{6} - \left( { - \frac{{\sqrt 3 }}{2}} \right)} \right) - \left( {\frac{\pi }{6} - \frac{{\sqrt 3 }}{2}} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">distance is \(\frac{{2\pi }}{3} + \sqrt 3 \) <em><strong>A1A1 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 \(\frac{{2\pi }}{3}\) , <em><strong>A1</strong></em> for \(\sqrt 3 \) . </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;">The derivative in part (a) was reasonably well done, but errors here often caused trouble in later parts. Candidates occasionally attempted to use the double angle identity for \(\sin 2t\) before differentiating, but they rarely were successful in then applying the product rule. </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 understood that they needed to set their derivative equal to zero, but fewer were able to take the next step to solve the resulting double angle equation. Again, some candidates over-complicated the equation by using the double angle identity. Few ended up with the correct answer \(\frac{{5\pi }}{6}\) . </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), many candidates knew they needed to test a value between \(\pi /6\) and their value from part (b), but fewer were able to successfully complete that calculation. Some candidates simply tested their boundary values while others unsuccessfully attempted to make use of the second derivative. </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;">Although many candidates did not attempt part (d), those who did often demonstrated a good understanding of how to use the displacement function <em>s</em> or the integral of their derivative from part (a). Candidates who had made an error in part (b) often could not finish, as \(\sin (2t)\) could not be evaluated at their value without a calculator. Of those who had successfully found the other boundary of \(5\pi /6\) , a common error was giving the incorrect sign of the value of \(\sin (5\pi /3)\) . Again, this part was a good discriminator between the grade 6 and 7 candidates.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The following diagram shows the graph of \(f(x) = \frac{x}{{{x^2} + 1}}\), for \(0 \le x \le 4\), and the line \(x = 4\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-13_om_17.54.57.png" alt></p>
<p class="p1">Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis and the line \(x = 4\).</p>
<p class="p1">Find the area of \(R\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">substitution of limits or function <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;A = \int_0^4 {f(x),{\text{ }}\int {\frac{x}{{{x^2} + 1}}{\text{d}}x} } \)</p>
<p class="p1">correct integration by substitution/inspection <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1">\(\frac{1}{2}\ln ({x^2} + 1)\)</p>
<p class="p1">substituting limits into <strong>their </strong>integrated function and subtracting (in any order) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{2}\left( {\ln ({4^2} + 1) - \ln ({0^2} + 1)} \right)\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{1}{2}\left( {\ln ({4^2} + 1) - \ln ({0^2} + 1)} \right),{\text{ }}\frac{1}{2}\left( {\ln (17) - \ln (1)} \right),{\text{ }}\frac{1}{2}\ln 17 - 0\)</p>
<p class="p1">\(A = \frac{1}{2}\ln (17)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Exception to <strong><em>FT </em></strong>rule. Allow full <strong><em>FT </em></strong>on incorrect integration involving a \(\ln \) function.</p>
<p class="p3"> </p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Very few candidates earned full marks in this question. While most candidates knew to integrate, many seemed unfamiliar with integrating using substitution or inspection. This topic is part of the syllabus, but it did not occur to many candidates to use a substitution method. A large number of them tried to integrate the individual terms in the numerator and denominator as though this were a polynomial function. While there were some candidates who knew the integral would involve a natural log function and substituted 4 and 0 into their function, many ended up with undefined values such as or did not know what to do with expressions containing \(\ln 1\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \sqrt x \) . Line <em>L</em> is the normal to the graph of <em>f</em> at the point (4, 2) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the diagram below, the shaded region <em>R</em> is bounded by the <em>x</em>-axis, the graph of <em>f</em> and </span><span style="font-family: times new roman,times; font-size: medium;">the line <em>L</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ring.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that the equation of <em>L</em> is \(y = - 4x + 18\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Point A is the <em>x</em>-intercept of <em>L</em> . Find the <em>x</em>-coordinate of A.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">Find an expression for the area of <em>R</em></span> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region <em>R</em> is rotated \(360^\circ \) about the <em>x</em>-axis. Find the volume of the solid formed, </span><span style="font-family: times new roman,times; font-size: medium;">giving your answer in terms of \(\pi \) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<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;">finding derivative <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = \frac{1}{2}{x^{\frac{1}{2}}},\frac{{1}}{{2\sqrt x }}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct value of derivative or its negative reciprocal (seen anywhere) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{{2\sqrt 4 }}\) , \(\frac{1}{4}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">gradient of normal = \(\frac{1}{{{\text{gradient of tangent}}}}\) (seen anywhere) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - \frac{1}{{f'(4)}} = - 4\) , \( - 2\sqrt x \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting into equation of line (for normal) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y - 2 = - 4(x - 4)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = - 4x + 18\) <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;">recognition that \(y = 0\) at A <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 4x + 18 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = \frac{{18}}{4}\) \(\left( { = \frac{9}{2}} \right)\) <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;">splitting into two appropriate parts (areas and/or integrals) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for area of <em>R</em> <em><strong>A2 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. area of <em>R</em> = \(\int_0^4 {\sqrt x } {\rm{d}}x + \int_4^{4.5} {( - 4x + 18){\rm{d}}x} \) , \(\int_0^4 {\sqrt x } {\rm{d}}x + \frac{1}{2} \times 0.5 \times 2\) (triangle) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> if d<em>x</em> is missing. </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;">correct expression for the volume from \(x = 0\) to \(x = 4\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \int_0^4 {\pi \left[ {f{{(x)}^2}} \right]} {\rm{d}}x\) , \({\int_0^4 {\pi \sqrt x } ^2}{\rm{d}}x\) , \(\int_0^4 {\pi x{\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = \left[ {\frac{1}{2}\pi {x^2}} \right]_0^4\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = \pi \left( {\frac{1}{2} \times 16 - \frac{1}{2} \times 0} \right)\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(V = 8\pi \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding the volume from \(x = 4\) to \(x = 4.5\)</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;">recognizing a cone <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \frac{1}{3}\pi {r^2}h\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(V = \frac{1}{3}\pi {(2)^2} \times \frac{1}{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\( = \frac{{2\pi }}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">total volume is \(8\pi + \frac{2}{3}\pi \) \(\left( { = \frac{{26}}{3}\pi } \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N4</span></strong></em></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;"> \(V = \pi \int_4^{4.5} {{{( - 4x + 18)}^2}{\rm{d}}x} \) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\( = \int_4^{4.5} {\pi (16{x^2} - 144x + 324){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \pi \left[ {\frac{{16}}{3}{x^3} - 72{x^2} + 324x} \right]_4^{4.5}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\( = \frac{{2\pi }}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">total volume is \(8\pi + \frac{2}{3}\pi \) \(\left( { = \frac{{26}}{3}\pi } \right)\) <em><strong>A1 N4</strong></em></span></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">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;">Parts (a) and (b) were well done by most candidates. </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 well done by most candidates. </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;">While quite a few candidates understood that both functions must be used to find the area in part (c), very few were actually able to write a correct expression for this area and this was due to candidates not knowing that they needed to integrate from \(0\) to \(4\) and then from \(4\) to \(4.5\). </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;">On part (d), some candidates were able to earn follow through marks by setting up a volume expression, but most of these expressions were incorrect. If they did not get the expression for the area correct, there was little chance for them to get part (d) correct. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">For those candidates who used their expression in part (c) for (d), there was a surprising amount of them who incorrectly applied distributive law of the exponent with respect to the addition or subtraction. </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;"><span>Let </span><span>\(f(x) = {x^3}\)</span><span>. The following diagram shows part of the graph of </span><span><em>f</em> </span><span>.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/gone.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The point \({\rm{P}}(a,f(a))\) , where \(a > 0\) , lies on the graph of <em>f</em> . The tangent at P crosses the <em>x</em>-axis at the point \({\rm{Q}}\left( {\frac{2}{3},0} \right)\) . This tangent intersects the graph of <em>f</em> at the point R(−2, −8) .</span></p>
<p> </p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The equation of the tangent at P is \(y = 3x - 2\) . Let <em>T</em> be the region enclosed by </span><span style="font-family: times new roman,times; font-size: medium;">the graph of <em>f</em> , the tangent [PR] and the line \(x = k\) , between \(x = - 2\) and \(x = k\) </span><span style="font-family: times new roman,times; font-size: medium;">where \( - 2 < k < 1\) . This is shown in the diagram below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chad.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Show that the gradient of [PQ] is \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \(f'(a)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (iii) Hence show that \(a = 1\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that the area of <em>T</em> is \(2k + 4\) , show that <em>k</em> satisfies the equation </span><span style="font-family: times new roman,times; font-size: medium;">\({k^4} - 6{k^2} + 8 = 0\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<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) substitute into gradient \( = \frac{{{y_1} - {y_2}}}{{{x_1} - {x_2}}}\) </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. \(\frac{{f(a) - 0}}{{a - \frac{2}{3}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(f(a) = {a^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{a^3} - 0}}{{a - \frac{2}{3}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">gradient \(\frac{{{a^3}}}{{a - \frac{2}{3}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG N0</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct answer <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{a^2}\) , \(f'(a) = 3\) , \(f'(a) = \frac{{{a^3}}}{{a - \frac{2}{3}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) <strong>METHOD 1</strong></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. \(f'(a) = {\rm{gradient}}\) , \(3{a^2} = \frac{{{a^3}}}{{a - \frac{2}{3}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplify <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{a^2}\left( {a - \frac{2}{3}} \right) = {a^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">rearrange <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{a^3} - 2{a^2} = {a^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of solving <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> e.g. \(2{a^3} - 2{a^2} = 2{a^2}(a - 1) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 1\) <em><strong>AG N0</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;">gradient RQ \( = \frac{{ - 8}}{{ - 2 - \frac{2}{3}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplify <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 8}}{{ - \frac{8}{3}}},3\)</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. \(f'(a) = {\rm{gradient}}\) , \(3{a^2} = \frac{{ - 8}}{{ - 2 - \frac{2}{3}}}\) , \(\frac{{{a^3}}}{{a - \frac{2}{3}}} = 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplify <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{a^2} = 3\) , \({a^2} = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 1\) <em><strong>AG N0</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 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;">approach to find area of <em>T</em> involving subtraction and integrals <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {f - (3x - 2){\rm{d}}x} \) , \(\int_{ - 2}^k {(3x - 2) - \int_{ - 2}^k {{x^3}} } \) , \(\int {({x^3} - 3x + 2)} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct integration with correct signs <em><strong>A1A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{4}{x^4} - \frac{3}{2}{x^2} + 2x\) , \(\frac{3}{2}{x^2} - 2x - \frac{1}{4}{x^4}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct limits \( - 2\) and <em>k</em> (seen anywhere) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{ - 2}^k {({x^3} - 3x + 2){\rm{d}}x} \) , \(\left[ {\frac{1}{4}{x^4} - \frac{3}{2}{x^2} + 2x} \right]_{ - 2}^k\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute <em>k</em> and <span lang="EN-US">\( - 2\) </span> <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into <strong>their</strong> integral if 2 or more terms <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\frac{1}{4}{k^4} - \frac{3}{2}{k^2} + 2k} \right) - (4 - 6 - 4)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting <strong>their</strong> integral expression equal to \(2k + 4\) (seen anywhere) <em><strong>(M1)</strong></em></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. \(\frac{1}{4}{k^4} - \frac{3}{2}{k^2} + 2 = 0\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({k^4} - 6{k^2} + 8 = 0\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG N0</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [9 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) seemed to be well-understood by many candidates, and most were able to earn at least partial marks here. Part (ai) was a "show that" question, and unfortunately there were some candidates who did not show how they arrived at the given expression. </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), the concept seemed to be well-understood. Most candidates saw the necessity of using definite integrals and subtracting the two functions, and the integration was generally done correctly. However, there were a number of algebraic and arithmetic errors which prevented candidates from correctly showing the desired final result. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A quadratic function \(f\) can be written in the form \(f(x) = a(x - p)(x - 3)\). The graph of \(f\) has axis of symmetry \(x = 2.5\) and \(y\)-intercept at \((0,{\text{ }} - 6)\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line \(y = kx - 5\) is a tangent to the curve of \(f\). Find the values of \(k\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using <em>x</em>-intercept)</strong></p>
<p>determining that 3 is an \(x\)-intercept <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(x - 3 = 0\), <img src="images/Schermafbeelding_2017-08-11_om_13.55.43.png" alt="M17/5/MATME/SP1/ENG/TZ1/09.a/M"></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3 - 2.5,{\text{ }}\frac{{p + 3}}{2} = 2.5\)</p>
<p>\(p = 2\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>METHOD 2 (expanding <em>f </em>(<em>x</em>)) </strong></p>
<p>correct expansion (accept absence of \(a\)) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(a{x^2} - a(3 + p)x + 3ap,{\text{ }}{x^2} - (3 + p)x + 3p\)</p>
<p>valid approach involving equation of axis of symmetry <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{ - b}}{{2a}} = 2.5,{\text{ }}\frac{{a(3 + p)}}{{2a}} = \frac{5}{2},{\text{ }}\frac{{3 + p}}{2} = \frac{5}{2}\)</p>
<p>\(p = 2\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>METHOD 3 (using derivative)</strong></p>
<p>correct derivative (accept absence of \(a\)) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(a(2x - 3 - p),{\text{ }}2x - 3 - p\)</p>
<p>valid approach <strong>(<em>M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f’(2.5) = 0\)</p>
<p>\(p = 2\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute \((0,{\text{ }} - 6)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 6 = a(0 - 2)(0 - 3),{\text{ }}0 = a( - 8)( - 9),{\text{ }}a{(0)^2} - 5a(0) + 6a = - 6\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 6 = 6a\)</p>
<p>\(a = - 1\) <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.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using discriminant)</strong></p>
<p>recognizing tangent intersects curve once <strong><em>(M1)</em></strong></p>
<p>recognizing one solution when discriminant = 0 <strong><em>M1</em></strong></p>
<p>attempt to set up equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(g = f,{\text{ }}kx - 5 = - {x^2} + 5x - 6\)</p>
<p>rearranging their equation to equal zero <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({x^2} - 5x + kx + 1 = 0\)</p>
<p>correct discriminant (if seen explicitly, not just in quadratic formula) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({(k - 5)^2} - 4,{\text{ }}25 - 10k + {k^2} - 4\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(k - 5 = \pm 2,{\text{ }}(k - 3)(k - 7) = 0,{\text{ }}\frac{{10 \pm \sqrt {100 - 4 \times 21} }}{2}\)</p>
<p>\(k = 3,{\text{ }}7\) <strong><em>A1A1</em></strong> <strong><em>N0</em></strong></p>
<p><strong>METHOD 2 (using derivatives)</strong></p>
<p>attempt to set up equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(g = f,{\text{ }}kx - 5 = - {x^2} + 5x - 6\)</p>
<p>recognizing derivative/slope are equal <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f’ = {m_T},{\text{ }}f' = k\)</p>
<p>correct derivative of \(f\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 2x + 5\)</p>
<p>attempt to set up equation in terms of either \(x\) or \(k\) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(( - 2x + 5)x - 5 = - {x^2} + 5x - 6,{\text{ }}k\left( {\frac{{5 - k}}{2}} \right) - 5 = - {\left( {\frac{{5 - k}}{2}} \right)^2} + 5\left( {\frac{{5 - k}}{2}} \right) - 6\)</p>
<p>rearranging their equation to equal zero <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({x^2} - 1 = 0,{\text{ }}{k^2} - 10k + 21 = 0\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(x = \pm 1,{\text{ }}(k - 3)(k - 7) = 0,{\text{ }}\frac{{10 \pm \sqrt {100 - 4 \times 21} }}{2}\)</p>
<p>\(k = 3,{\text{ }}7\) <strong><em>A1A1</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[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;">
[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;">Part of the graph of \(f(x) = a{x^3} - 6{x^2}\) is shown below.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/1.jpg" alt></span></p>
<p> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">The point P lies on the graph of </span><span style="font-family: 'times new roman', times; font-size: medium;">\(f\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> . At P, <em>x</em> = 1.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The graph of \(f\) has a gradient of \(3\) at the point P. Find the value of \(a\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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;">\(f'(x) = 3a{x^2} - 12x\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> <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 each correct term.</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;">setting their derivative equal to 3 (seen anywhere) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. </span><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = 3\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute \(x = 1\) into \(f'(x)\) </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>(M1)</strong></em></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. </span><span style="font-family: 'times new roman', times; font-size: medium;">\(3a{(1)^2} - 12(1)\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into \(f'(x)\) </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>(A1)</strong></em></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. \(3a - 12\) , \(3a = 15\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 5\) </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>A1 N2</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">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;">A majority of candidates answered part (a) correctly, and a good number earned full marks on both </span><span style="font-family: 'times new roman', times; font-size: medium;">parts of this question. In part (b), some common errors included setting the derivative equal to zero, or </span><span style="font-family: 'times new roman', times; font-size: medium;">substituting 3 for <em>x</em> in their derivative. There were also a few candidates who incorrectly tried to work </span><span style="font-family: 'times new roman', times; font-size: medium;">with \(f(x)\) , rather than \(f'(x)\) , 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;">A majority of candidates answered part (a) correctly, and a good number earned full marks on both </span><span style="font-family: 'times new roman', times; font-size: medium;">parts of this question. In part (b), some common errors included setting the derivative equal to zero, or </span><span style="font-family: 'times new roman', times; font-size: medium;">substituting \(3\) for \(x\) in their derivative. There were also a few candidates who incorrectly tried to work </span><span style="font-family: 'times new roman', times; font-size: medium;">with \(f(x)\) , rather than \(f'(x)\) , in part (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f(x) = \cos x\), for \(0\) \(\le \) \(x\) \( \le \) \(2\pi \). The following diagram shows the graph of \(f\).</p>
<p class="p1">There are <em>\(x\)</em>-intercepts at \(x = \frac{\pi }{2},{\text{ }}\frac{{3\pi }}{2}\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_07.44.53.png" alt></p>
<p class="p1">The shaded region \(R\) is enclosed by the graph of \(f\), the line \(x = b\), where \(b > \frac{{3\pi }}{2}\), and the \(x\)-axis. The area of \(R\) <span class="s1">is \(\left( {1 - \frac{{\sqrt 3 }}{2}} \right)\)</span>. Find the value of \(b\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to set up integral (accept missing or incorrect limits and missing \({\text{d}}x\)) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_{\frac{{3\pi }}{2}}^b {\cos x{\text{d}}x,{\text{ }}\int_a^b {\cos x{\text{d}}x,{\text{ }}\int_{\frac{{3\pi }}{2}}^b {f{\text{d}}x,{\text{ }}\int {\cos x} } } } \)</p>
<p>correct integration (accept missing or incorrect limits) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;[\sin x]_{\frac{{3\pi }}{2}}^b,{\text{ }}\sin x\)</p>
<p>substituting correct limits into <strong>their </strong>integrated function and subtracting (in any order) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\sin b - \sin \left( {\frac{{3\pi }}{2}} \right),{\text{ }}\sin \left( {\frac{{3\pi }}{2}} \right) - \sin b\)</p>
<p>\(\sin \left( {\frac{{3\pi }}{2}} \right) = - 1\;\;\;\)(seen anywhere) <strong><em>(A1)</em></strong></p>
<p>setting <strong>their </strong>result from an integrated function equal to \(\left( {1 - \frac{{\sqrt 3 }}{2}} \right)\) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;\sin b = - \frac{{\sqrt 3 }}{2}\)</p>
<p>evaluating \({\sin ^{ - 1}}\left( {\frac{{\sqrt 3 }}{2}} \right) = \frac{\pi }{3}\) or \({\sin ^{ - 1}}\left( { - \frac{{\sqrt 3 }}{2}} \right) = - \frac{\pi }{3}\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;b = \frac{\pi }{3},{\text{ }} - 60^\circ \)</p>
<p>identifying correct value <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;2\pi - \frac{\pi }{3},{\text{ }}360 - 60\)</p>
<p>\(b = \frac{{5\pi }}{3}\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Most candidates recognised that a definite integral was required and many were able to set up a correct equation. Incorrect integration leading to \( - \sin x\) was quite common and poor notation was frequently seen. Some candidates appeared to guess their value from the graph, showing little supporting work.</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = p{x^3} + p{x^2} + qx\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f'(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(f'(x) \geqslant 0\), show that \({p^2} \leqslant 3pq\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = 3p{x^2} + 2px + q\) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px '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: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Note:</em></strong> Award <strong><em>A1</em></strong> if only 1 error.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px '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: 21.0px 'Times New Roman'; color: #3f3f3f;"><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 10.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of discriminant (must be seen explicitly, not in quadratic formula) <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; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({b^2} - 4ac\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into discriminant (may be seen in inequality) <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; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({(2p)^2} - 4 \times 3p \times q,{\text{ }}4{p^2} - 12pq\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) \geqslant 0\) then \(f'\) has two equal roots or no roots <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing discriminant less or equal than zero <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\Delta \leqslant 0,{\text{ }}4{p^2} - 12pq \leqslant 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;">correct working that clearly leads 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; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({p^2} - 3pq \leqslant 0,{\text{ }}4{p^2} \leqslant 12pq\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f; background-color: #f7f7f7;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p^2} \leqslant 3pq\) <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; background-color: #f7f7f7;"><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>
<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;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3 + \frac{{20}}{{{x^2} - 4}}\) , for \(x \ne \pm 2\) . The graph of <em>f</em> is given below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/fajita.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>y</em>-intercept is at the point A.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the coordinates of A.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(f'(x) = 0\) at A.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The second derivative \(f''(x) = \frac{{40(3{x^2} + 4)}}{{{{({x^2} - 4)}^3}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> . Use this to</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) justify that the graph of <em>f</em> has a local maximum at A;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) explain why the graph of <em>f</em> does <strong>not</strong> have a point of inflexion.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Describe the behaviour of the graph of \(f\) for large \(|x|\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<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) coordinates of A are \((0{\text{, }} - 2)\) <em><strong>A1A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) derivative of \({x^2} - 4 = 2x\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p><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. quotient rule, chain rule</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(f'(x)\) <em><strong>A2</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">e.g. \(f'(x) = 20 \times ( - 1) \times {({x^2} - 4)^{ - 2}} \times (2x)\) , \(\frac{{({x^2} - 4)(0) - (20)(2x)}}{{{{({x^2} - 4)}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(x = 0\) into \(f'(x)\) (do not accept solving \(f'(x) = 0\) ) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">at A \(f'(x) = 0\) <em><strong>AG N0</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 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) reference to \(f'(x) = 0\) (seen anywhere) <em><strong> (R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">reference to \(f''(0)\) is negative (seen anywhere) <em><strong>R1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(x = 0\) into \(f''(x)\) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(f''(0) = \frac{{40 \times 4}}{{{{( - 4)}^3}}}\) \(\left( { = - \frac{5}{2}} \right)\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">then the graph must have a local maximum <em><strong>AG</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) reference to \(f''(x) = 0\) at point of inflexion <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that the second derivative is never 0 <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(40(3{x^2} + 4) \ne 0\) , \(3{x^2} + 4 \ne 0\) , \({x^2} \ne - \frac{4}{3}\) , the numerator </span><span style="font-family: times new roman,times; font-size: medium;">is always positive </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Do not accept the use of the first derivative in part (b). </span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[6 marks]</em> </span></strong></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 (informal) statement, including reference to approaching \(y = 3\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. getting closer to the line \(y = 3\) , horizontal asymptote at \(y = 3\)</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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>correct</strong> inequalities, \(y \le - 2\) , \(y > 3\) , <em><strong>FT</strong></em> from (a)(i) and (c) <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">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;">Almost all candidates earned the first two marks in part (a) (i), although fewer were able to apply the quotient rule correctly. </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 were able to state how the second derivative can be used to identify maximum and inflection points, but fewer were actually able to demonstrate this with the given function. For example, in (b)(ii) candidates often simply said "the second derivative cannot equal 0" but did not justify or explain why this was true. </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;">Not too many candidates could do part (c) correctly.</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 (d) even those who knew what the range was had difficulty expressing the inequalities correctly. </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 \(f(x) = \frac{x}{{ - 2{x^2} + 5x - 2}}\) for \( - 2 \le x \le 4\) , \(x \ne \frac{1}{2}\) , \(x \ne 2\) . The graph of \(f\) is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P1TZ2Q10.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of \(f\) has a local minimum at A(\(1\), \(1\)) and a local maximum at B.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the quotient rule to show that \(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence find the coordinates of B.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that the line \(y = k\) does not meet the graph of <em>f</em> , find the possible values </span><span style="font-family: times new roman,times; font-size: medium;">of <em>k</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<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 derivatives <strong>applied</strong> in quotient rule <em><strong>(A1)A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(1\), \( - 4x + 5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 1, <em><strong>A1</strong></em> for \( - 4x\) and <em><strong>A1</strong></em> for \(5\), <strong>only</strong> if it is clear candidates are using the quotient rule. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into quotient rule <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> \(\frac{{1 \times ( - 2{x^2} + 5x - 2) - x( - 4x + 5)}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) , \(\frac{{ - 2{x^2} + 5x - 2 - x( - 4x + 5)}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\)</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;"><em>e.g.</em> \(\frac{{ - 2{x^2} + 5x - 2 - ( - 4{x^2} + 5x)}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">expression clearly leading to the answer <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>e.g.</em> \(\frac{{ - 2{x^2} + 5x - 2 + 4{x^2} - 5x}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{2{x^2} - 2}}{{{{( - 2{x^2} + 5x - 2)}^2}}}\) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempting to solve \(f'(x) = 0\) <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2{x^2} - 2 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct working <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({x^2} = 1,\frac{{ \pm \sqrt {16} }}{4}{\text{, }}2(x - 1)(x + 1)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct solution to quadratic <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = \pm 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct <em>x</em>-coordinate \(x = - 1\) (may be seen in coordinate form \(\left( { - 1,\frac{1}{9}} \right)\) ) <em><strong>A1 N2</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute \( - 1\) into <em>f</em> (do not accept any other value) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f( - 1) = \frac{{ - 1}}{{ - 2 \times {{( - 1)}^2} + 5 \times ( - 1) - 2}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 1}}{{ - 2 - 5 - 2}}\) <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct <em>y</em>-coordinate \(y = \frac{1}{9}\) (may be seen in coordinate form \(\left( { - 1,\frac{1}{9}} \right)\) ) <strong><em>A1 N2</em></strong> </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;">recognizing values between max and min <em><strong>(R1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{1}{9} < k < 1\) <em><strong>A2 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>
<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 most candidates answered part (a) correctly, there were some who did not show quite enough work for a "show that" question. A very small number of candidates did not follow the instruction to use the quotient rule. </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 that they needed to solve the equation \(f'(x) = 0\) , and many were successful in answering this question correctly. However, some candidates failed to find both values of <em>x</em>, or made other algebraic errors in their solutions. One common error was to find only one solution for \({x^2} = 1\) ; another was to work with the denominator equal to zero, rather than the numerator.</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), a significant number of candidates seemed to think that the line \(y = k\) was a vertical line, and attempted to find the vertical asymptotes. Others tried looking for a horizontal asymptote. Fortunately, there were still a good number of intuitive candidates who recognized the link with the graph and with part (b), and realized that the horizontal line must pass through the space between the given local minimum and the local maximum they had found in part (b). </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f'(x) = 6{x^2} - 5\). Given that \(f(2) = - 3\), find \(f(x)\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>evidence of antidifferentiation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f = \int {f'} \)</p>
<p>correct integration (accept absence of \(C\)) <strong><em>(A1)(A1)</em></strong></p>
<p>\(f(x) = \frac{{6{x^3}}}{3} - 5x + C,{\text{ }}2{x^3} - 5x\)</p>
<p>attempt to substitute \((2,{\text{ }} - 3)\) into <strong>their </strong>integrated expression (must have \(C\)) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;2{(2)^3} - 5(2) + C = - 3,{\text{ }}16 - 10 + C = - 3\)</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M0 </em></strong>if substituted into original or differentiated function.</p>
<p> </p>
<p>correct working to find \(C\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;16 - 10 + C = - 3,{\text{ }}6 + C = - 3,{\text{ }}C = - 9\)</p>
<p>\(f(x) = 2{x^3} - 5x - 9\) <strong><em>A1 N4</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="question">
<p>Consider \(f(x) = \log k(6x - 3{x^2})\), for \(0 < x < 2\), where \(k > 0\).</p>
<p>The equation \(f(x) = 2\) has exactly one solution. Find the value of \(k\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 – using discriminant</strong></p>
<p>correct equation without logs <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(6x - 3{x^2} = {k^2}\)</p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\( - 3{x^2} + 6x - {k^2} = 0,{\text{ }}3{x^2} - 6x + {k^2} = 0\)</p>
<p>recognizing discriminant must be zero (seen anywhere) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\Delta = 0\)</p>
<p>correct discriminant <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({6^2} - 4( - 3)( - {k^2}),{\text{ }}36 - 12{k^2} = 0\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(12{k^2} = 36,{\text{ }}{k^2} = 3\)</p>
<p>\(k = \sqrt 3 \) <strong><em>A2 N2</em></strong></p>
<p><strong>METHOD 2 – completing the square</strong></p>
<p>correct equation without logs <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(6x - 3{x^2} = {k^2}\)</p>
<p>valid approach to complete the square <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3({x^2} - 2x + 1) = - {k^2} + 3,{\text{ }}{x^2} - 2x + 1 - 1 + \frac{{{k^2}}}{3} = 0\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3{(x - 1)^2} = - {k^2} + 3,{\text{ }}{(x - 1)^2} - 1 + \frac{{{k^2}}}{3} = 0\)</p>
<p>recognizing conditions for one solution <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({(x - 1)^2} = 0,{\text{ }} - 1 + \frac{{{k^2}}}{3} = 0\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^2}}}{3} = 1,{\text{ }}{k^2} = 3\)</p>
<p>\(k = \sqrt 3 \) <strong> <em>A2 N2</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f'(x) = 3{x^2} + 2\) . Given that \(f(2) = 5\) , find \(f(x)\) .</span></p>
</div>
<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 anti-differentiation <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {f'(x)} \) , \(\int {(3{x^2} + 2){\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = {x^3} + 2x + c\) (seen anywhere, including the answer) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute (2, 5) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(2) = {(2)^3} + 2(2)\) , \(5 = 8 + 4 + c\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding the value of <em>c</em> <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(5 = 12 + c\) , \(c = - 7\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = {x^3} + 2x - 7\) <em><strong>A1 N5</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, which required candidates to integrate a simple polynomial and then substitute an initial condition to solve for "<em>c</em>", was very well done. Nearly all candidates who attempted this question were able to earn full marks. The very few mistakes that were seen involved arithmetic errors when solving for "<em>c</em>", or failing to write the final answer as the equation of the function. </span></p>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \int {\frac{{12}}{{2x - 5}}} {\rm{d}}x\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(x > \frac{5}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> . The graph of \(f\) passes through (\(4\), \(0\)) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f(x)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to integrate which involves \(\ln \) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\ln (2x - 5)\) , \(12\ln 2x - 5\) , \(\ln 2x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression (accept absence of \(C\))</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(12\ln (2x - 5)\frac{1}{2} + C\) , \(6\ln (2x - 5)\) <strong><em>A2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute (4,0) into <strong>their</strong> integrated <em><strong>f </strong></em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(0 = 6\ln (2 \times 4 - 5)\) , \(0 = 6\ln (8 - 5) + C\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(C = - 6\ln 3\) </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;">\(f(x) = 6\ln (2x - 5) - 6\ln 3\) \(\left( { = 6\ln \left( {\frac{{2x - 5}}{3}} \right)} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">(accept \(6\ln (2x - 5) - \ln {3^6}\) ) <em><strong>A1 N5</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Exception to the <em><strong>FT</strong></em> rule. Allow full <em><strong>FT</strong></em> on incorrect integration which must involve \(\ln\). </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;">While some candidates correctly integrated the function, many missed the division by \(2\) and answered \(12\ln \left( {2x - 5} \right)\) . Other common incorrect responses included \(\frac{{12x}}{{{x^2} - 5x}}\) and \( - 122{\left( {x - 5} \right)^{ - 2}}\) . Finding the constant of integration also proved elusive for many. Some either did not remember the \(+C\) or did not try to find its value, while others misunderstood the boundary condition and attempted to calculate the definite integral from \(0\) to \(4\).</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f'(x) = {\sin ^3}(2x)\cos (2x)\). Find \(f(x)\), given that \(f\left( {\frac{\pi }{4}} \right) = 1\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">evidence of integration <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\int {f'(x){\text{d}}x} \)</p>
<p class="p1">correct integration (accept missing \(C\)) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A2)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{2} \times \frac{{{{\sin }^4}(2x)}}{4},{\text{ }}\frac{1}{8}{\sin ^4}(2x) + C\)</p>
<p class="p1">substituting initial condition into their integrated expression (must have \( + C\)) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(1 = \frac{1}{8}{\sin ^4}\left( {\frac{\pi }{2}} \right) + C\)</p>
<p class="p3"> </p>
<p class="p4"><span class="s2"><strong>Note: </strong>Award </span><span class="s1"><strong><em>M0 </em></strong></span>if they substitute into the original or differentiated function.</p>
<p class="p5"> </p>
<p class="p4">recognizing \(\sin \left( {\frac{\pi }{2}} \right) = 1\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(1 = \frac{1}{8}{(1)^4} + C\)</p>
<p class="p2"><span class="Apple-converted-space">\(C = \frac{7}{8}\) </span><strong><em>(A1)</em></strong></p>
<p class="p2"><span class="s3">\(f(x) = \frac{1}{8}{\sin ^4}(2x) + \frac{7}{8}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N5</em></strong></p>
<p class="p2"><strong><em>[7 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>A closed cylindrical can with radius r centimetres and height h centimetres has a volume of 20\(\pi \) cm<sup>3</sup>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The material for the base and top of the can costs 10 cents per cm<sup>2</sup> and the material for the curved side costs 8 cents per cm<sup>2</sup>. The total cost of the material, in cents, is <em>C</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>h</em> in terms of <em>r.</em></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(C = 20\pi {r^2} + \frac{{320\pi }}{r}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that there is a minimum value for <em>C</em>, find this minimum value in terms of \(\pi \).</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct equation for volume   <strong><em>(A1)</em></strong><br><em>eg</em>  \(\pi {r^2}h = 20\pi \)</p>
<p>\(h = \frac{{20}}{{{r^2}}}\)Â Â Â <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p>Â </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find formula for cost of parts   <em><strong> (M1)</strong></em><br><em>eg </em> 10 × two circles, 8 × curved side</p>
<p>correct expression for cost of two circles in terms of <em>r</em> (seen anywhere)   <em><strong>A1</strong></em><br><em>eg  </em>\(2\pi {r^2} \times 10\)</p>
<p>correct expression for cost of curved side (seen anywhere)    <em><strong>(A1)</strong></em><br><em>eg  </em>\(2\pi r \times h \times 8\)</p>
<p>correct expression for cost of curved side in terms of <em>r </em>   <em><strong>A1</strong></em><br>eg  \(8 \times 2\pi r \times \frac{{20}}{{{r^2}}},\,\,\frac{{320\pi }}{{{r^2}}}\)</p>
<p>\(C = 20\pi {r^2} + \frac{{320\pi }}{r}\)Â Â Â Â <em><strong>AG N0</strong></em></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>recognize \(C' = 0\) at minimum    <em><strong>(R1)</strong></em><br>eg  \(C' = 0,\,\,\frac{{{\text{d}}C}}{{{\text{d}}r}} = 0\)</p>
<p>correct differentiation (may be seen in equation)</p>
<p>\(C' = 40\pi r - \frac{{320\pi }}{{{r^2}}}\) Â Â Â Â <em><strong>A1A1</strong></em></p>
<p>correct equation   <em><strong>A1</strong></em><br>eg  \(40\pi r - \frac{{320\pi }}{{{r^2}}} = 0,\,\,40\pi r\frac{{320\pi }}{{{r^2}}}\)</p>
<p>correct working   <em><strong>(A1)</strong></em><br>eg  \(40{r^3} = 320,\,\,{r^3} = 8\)</p>
<p><em>r</em> = 2 (m)Â Â Â <em><strong>A1</strong></em></p>
<p>attempt to substitute <strong>their</strong> value of <em>r</em> into <em>C</em><br>eg  \(20\pi \times 4 + 320 \times \frac{\pi }{2}\)   <em><strong>(M1)</strong></em></p>
<p>correct working<br>eg  \(80\pi + 160\pi \)    <em><strong>  (A1)</strong></em></p>
<p>\(240\pi \)Â (cents)Â Â Â <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> Do not accept 753.6, 753.98 or 754, even if 240\(\pi \) is seen.</p>
<p><em><strong>[9 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="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\int {\frac{{{{\rm{e}}^x}}}{{1 + {{\rm{e}}^x}}}} {\rm{d}}x\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(\int {\sin 3x\cos 3x{\rm{d}}x} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<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 substitution or inspection <em><strong> M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(u = 1 + {{\rm{e}}^x}\) so \(\frac{{{\rm{d}}u}}{{{\rm{d}}x}} = {{\rm{e}}^x}\)</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. \(\int {\frac{{{\rm{d}}u}}{u}} = \ln u\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\ln (1 + {{\rm{e}}^x}) + C\) <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">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;">attempt to use substitution or inspection <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. let \(u = \sin 3x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\rm{d}}u}}{{{\rm{d}}x}} = 3\cos 3x\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{1}{3}\int {u{\rm{d}}u = } \frac{1}{3} \times \frac{{{u^2}}}{2} + C\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\int {\sin 3x\cos 3x{\rm{d}}x = } \frac{{{{\sin }^2}3x}}{6} + C\) </span><span style="font-family: times new roman,times; font-size: medium;"> <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;">attempt to use substitution or inspection <em><strong> M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. let \(u = \cos 3x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\rm{d}}u}}{{{\rm{d}}x}} = - 3\sin 3x\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( - \frac{1}{3}\int {u{\rm{d}}u = } - \frac{1}{3} \times \frac{{{u^2}}}{2} + C\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\int {\sin 3x\cos 3x{\rm{d}}x = } \frac{{{{\cos }^2}3x}}{6} + C\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1 N2</span></strong></em></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;">recognizing double angle <em><strong>M1</strong></em></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. \(\frac{1}{2}\sin 6x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\int {\sin 6x{\rm{d}}x = } \frac{{ - \cos 6x}}{6} + C\) </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;">\(\int {\frac{1}{2}\sin 6x{\rm{d}}x = - \frac{{\cos 6x}}{{12}}} + C\) </span><span style="font-family: times new roman,times; font-size: medium;"> <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;">
[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="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_4^{10} {(x - 4){\rm{d}}x} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Part of the graph of \(f(x) = \sqrt {{x^{}} - 4} \) , for \(x \ge 4\) , is shown below. The shaded region </span><em style="font-family: 'times new roman', times; font-size: medium;">R</em><span style="font-family: 'times new roman', times; font-size: medium;"> is enclosed by the graph of \(f\) , the line \(x = 10\) , and the </span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;">-axis.</span></p>
<p><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">The region <em>R</em> is rotated \({360^ \circ }\) about the <em>x</em>-axis. Find the volume of the solid </span><span style="font-family: 'times new roman', times; font-size: medium;">formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<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 integration <em><strong> A1A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. \(\frac{{{x^2}}}{2} - 4x\), \(\left[ {\frac{{{x^2}}}{2} - 4x} \right]_4^{10}\), \(\frac{{{{(x - 4)}^2}}}{2}\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> In the first 2 examples, award <em><strong>A1</strong></em> for each correct term.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">In the third example, award <em><strong>A1</strong></em> for \(\frac{1}{2}\) and <strong><em>A1</em></strong> for \({(x - 4)^2}\).</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">substituting limits into <strong>their</strong> integrated function and subtracting (in any order) <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. \(\left( {\frac{{{{10}^2}}}{2} - 4(10)} \right) - \left( {\frac{{{4^2}}}{2} - 4(4)} \right),10 - ( - 8),\frac{1}{2}({6^2} - 0)\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_4^{10} {(x - 4){\rm{d}}x = 18} \) <em><strong>A1 N2</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;">attempt to substitute either limits or the function into volume formula <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\pi \int_4^{10} {{f^2}} {\rm{d}}x{\text{, }}{\int_a^b {(\sqrt {x - 4} )} ^2}{\text{, }}\pi \int_4^{10} {\sqrt {x - 4} } \)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not penalise for missing \(\pi \) or d</span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;">.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution (accept absence of d</span><em style="font-family: 'times new roman', times; font-size: medium;">x</em><span style="font-family: 'times new roman', times; font-size: medium;"> and \(\pi \)) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(A1)</em></strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">e.g. </span><span style="font-family: 'times new roman', times; font-size: medium;">\(\pi {\int_4^{10} {(\sqrt {x - 4} )} ^2}{\text{, }}\pi \int_4^{10} {(x - 4){\rm{d}}x} {\text{, }}\int_4^{10} {(x - 4){\rm{d}}x} \)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">volume = \(18\pi \) <em><strong>A1 N2</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">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 candidates answered both parts of this question correctly. In part (b), a large number of </span><span style="font-family: 'times new roman', times; font-size: medium;">successful candidates did not seem to notice the link between parts (a) and (b), and duplicated the </span><span style="font-family: 'times new roman', times; font-size: medium;">work they had already done in part (a). Also in part (b), a good number of candidates squared </span><span style="font-family: 'times new roman', times; font-size: medium;">\((x - 4)\) in their integral, rather than squaring \(\sqrt {x - 4} \) , which of course prevented them from noting the link </span><span style="font-family: 'times new roman', times; font-size: medium;">between the two parts and obtaining the correct answer.</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 both parts of this question correctly. In part (b), a large number of </span><span style="font-family: 'times new roman', times; font-size: medium;">successful candidates did not seem to notice the link between parts (a) and (b), and duplicated the </span><span style="font-family: 'times new roman', times; font-size: medium;">work they had already done in part (a). Also in part (b), a good number of candidates squared \((x - 4)\) in their integral, rather than squaring \(\sqrt {x - 4} \) , which of course prevented them from noting the link </span><span style="font-family: 'times new roman', times; font-size: medium;">between the two parts and obtaining the correct answer.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A particle moves along a straight line so that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\)at time <em>t</em> seconds is given by \(v = 6{{\rm{e}}^{3t}} + 4\) . When \(t = 0\) , the displacement, <em>s</em>, of the particle is 7 metres. Find an expression for <em>s</em> in terms of <em>t</em>.</span></p>
</div>
<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 anti-differentiation <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s = \int {(6{{\rm{e}}^{3x}} + 4)} {\rm{d}}x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(s = 2{{\rm{e}}^{3t}} + 4t + C\) <strong><em>A2A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting \(t = 0\) , <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(7 = 2 + C\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(C = 5\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(s = 2{{\rm{e}}^{3t}} + 4t + 5\) <em><strong>A1 N3</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There were a number of completely correct solutions to this question. However, there were </span><span style="font-family: times new roman,times; font-size: medium;">many who did not know the relationship between velocity and position. Many students </span><span style="font-family: times new roman,times; font-size: medium;">differentiated rather than integrated and those who did integrate often had difficulty with the </span><span style="font-family: times new roman,times; font-size: medium;">term involving e. Many who integrated correctly neglected the <em>C</em> or made \(C = 7\) .</span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the function <em>f</em> with second derivative \(f''(x) = 3x - 1\) . The graph of <em>f</em> has a </span><span style="font-family: times new roman,times; font-size: medium;">minimum point at A(2, 4) and a maximum point at \({\rm{B}}\left( { - \frac{4}{3},\frac{{358}}{{27}}} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Use the second derivative to justify that B is a maximum.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(f'(x) = \frac{3}{2}{x^2} - x + p\)</span><span style="font-family: times new roman,times; font-size: medium;"> , show that \(p = - 4\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f(x)\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<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 into the second derivative <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3 \times \left( { - \frac{4}{3}} \right) - 1\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f''\left( { - \frac{4}{3}} \right) = - 5\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since the second derivative is negative, B is a maximum <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">setting \(f'(x)\) equal to zero <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(x = 2\) (or \(x = - \frac{4}{3}\) )</span><span style="font-family: times new roman,times; font-size: medium;"> <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;">correct substitution <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{3}{2}{(2)^2} - 2 + p\) , \(\frac{3}{2}{\left( { - \frac{4}{3}} \right)^2} - \left( { - \frac{4}{3}} \right) + p\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct simplification</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6 - 2 + p = 0\) , \(\frac{8}{3} + \frac{4}{3} + p = 0\) , \(4 + p = 0\) <em><strong> A1</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(p = - 4\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG N0</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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of integration <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f(x) = \frac{1}{2}{x^3} - \frac{1}{2}{x^2} - 4x + c\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting (2, 4) or \(\left( { - \frac{4}{3},\frac{{358}}{{27}}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">into <strong>their</strong> expression <em><strong>(M1)</strong></em></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. \(\frac{1}{2} \times {2^3} - \frac{1}{2} \times {2^2} - 4 \times 2 + c = 4\) , \(\frac{1}{2} \times 8 - \frac{1}{2} \times 4 - 4 \times 2 + c = 4\) , \(4 - 2 - 8 + c = 4\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f(x) = \frac{1}{2}{x^3} - \frac{1}{2}{x^2} - 4x + 10\) </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;"> [7 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 were successful with this question. In part (a), some candidates found </span><span style="font-family: times new roman,times; font-size: medium;">\(f''\left( { - \frac{4}{3}} \right)\) and were unclear how to conclude, but most demonstrated a good understanding </span><span style="font-family: times new roman,times; font-size: medium;">of the second derivative test.</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;">A large percentage of candidates were successful in showing that \(p = - 4\) but there were still </span><span style="font-family: times new roman,times; font-size: medium;">some who worked backwards from the answer. Others did not use the given information and </span><span style="font-family: times new roman,times; font-size: medium;">worked from the second derivative, integrated, and then realized that <em>p</em> was the constant of </span><span style="font-family: times new roman,times; font-size: medium;">integration. Candidates who evaluated the derivative at \(x = 2\) but set the result equal to 4 </span><span style="font-family: times new roman,times; font-size: medium;">clearly did not understand the concept being assessed. Few candidates used the point B with </span><span style="font-family: times new roman,times; font-size: medium;">fractional coordinates.</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;">Candidates often did well on the first part of (c), knowing to integrate and successfully </span><span style="font-family: times new roman,times; font-size: medium;">finding some or all terms. Some had trouble with the fractions or made careless errors with </span><span style="font-family: times new roman,times; font-size: medium;">the signs; others did not use the value of \(p = - 4\) and so could not find the third term when </span><span style="font-family: times new roman,times; font-size: medium;">integrating. It was very common for candidates to either forget the constant of integration or </span><span style="font-family: times new roman,times; font-size: medium;">to leave it in without finding its value.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f'(x) = \frac{{6 - 2x}}{{6x - {x^2}}}\), for \(0 < x < 6\).</p>
<p class="p1"><span class="s1">The graph of \(f\) </span>has a maximum point at P<span class="s1">.</span></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">The \(y\)</span>-coordinate of P is \(\ln 27\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the \(x\)-coordinate of <span class="s1">P</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f(x)\), expressing your answer as a single logarithm.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">The graph of \(f\) </span>is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\). The image of P under this transformation has coordinates \((a,{\text{ }}b)\).</p>
<p class="p1">Find the value of \(a\) and of \(b\), where \(a,{\text{ }}b \in \mathbb{N}\).</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing \(f'(x) = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(6 - 2x = 0\)</p>
<p class="p1"><span class="Apple-converted-space">\(x = 3\) </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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of integration <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {f',{\text{ }}\int {\frac{{6 - 2x}}{{6x - {x^2}}}{\text{d}}x} } \)</p>
<p class="p1">using substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {\frac{1}{u}{\text{d}}u} \) where \(u = 6x - {x^2}\)</p>
<p class="p1">correct integral <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln (u) + c,{\text{ }}\ln (6x - {x^2})\)</p>
<p class="p1">substituting \((3,{\text{ }}\ln 27)\) into <strong>their </strong>integrated expression (must have \(c\)) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln (6 \times 3 - {3^2}) + c = \ln 27,{\text{ }}\ln (18 - 9) + \ln k = \ln 27\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\,\,\,\,\,\)\(c = \ln 27 - \ln 9\)</p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p2"><span class="Apple-converted-space">\(c = \ln 3\) </span><span class="s2"><strong><em>(A1)</em></strong></span></p>
<p class="p1">attempt to substitute <strong>their </strong>value of \(c\) into \(f(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = \ln (6x - {x^2}) + \ln 3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">attempt to substitute <strong>their </strong>value of \(c\) into \(f(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = \ln (6x - {x^2}) + \ln 27 - \ln 9\)</p>
<p class="p1">correct use of a log law <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = \ln (6x - {x^2}) + \ln \left( {\frac{{27}}{9}} \right),{\text{ }}f(x) = \ln \left( {27(6x - {x^2})} \right) - \ln 9\)</p>
<p class="p1"><span class="Apple-converted-space">\(f(x) = \ln \left( {3(6x - {x^2})} \right)\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(a = 3\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\ln 27}}{{\ln 3}}\)</p>
<p class="p1">correct use of log law <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{3\ln 3}}{{\ln 3}},{\text{ }}{\log _3}27\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = 3\) </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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part a) was well answered.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part b) most candidates realised that integration was required but fewer recognised the need to use integration by substitution. Quite a number of candidates who integrated correctly omitted finding the constant of integration.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part c) many candidates showed good understanding of transformations and could apply them correctly, however, correct use of the laws of logarithms was challenging for many. In particular, a common error was \(\frac{{\ln 27}}{{\ln 3}} = \ln 9\).</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: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = {x^2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The following diagram shows part of the graph of \(f\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/maths_3b.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The shaded region \(R\)<em> </em>is enclosed by the graph of \(f\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">Find the volume of the solid formed when </span><span style="font-family: 'times new roman', times; font-size: medium;">\(R\)</span><em> </em><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">is revolved \({360^ \circ }\) about the </span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><span style="background-color: #ffffff;">\(x\)</span>-axis.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<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'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">substituting for \({\left( {f(x)} \right)^2}\) (may be seen in integral) <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> \({\left( {{x^2}} \right)^2}{\text{, }}{x^4}\)</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 integration, \(\int {{x^4}{\text{d}}x = \frac{1}{5}{x^5}} \) <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;">substituting limits into <strong>their integrated </strong>function and subtracting (in any order) <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> \(\frac{{{2^5}}}{5} - \frac{1}{5}{\text{, }}\frac{1}{5}(1 - 4)\)</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;">\(\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x} = \frac{{31}}{5}( = 6.2) \) <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'; 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">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'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute limits or 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: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x{\text{, }}\pi \int {{x^4}{\text{d}}x} } \)</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;">\(\frac{{31}}{5}\pi {\text{ }}( = 6.2\pi )\) <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'; color: #3f3f3f;"><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>
<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="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(\int_0^5 {\frac{2}{{2x + 5}}} {\rm{d}}x = \ln k\)</span><span style="font-family: times new roman,times; font-size: medium;"> , find the value of <em>k</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">correct integration, \(2 \times \frac{1}{2}\ln (2x + 5)\) <em><strong>A1A1</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 \(2 \times \frac{1}{2}( = 1)\) and <em><strong>A1</strong></em> for \(\ln (2x + 5)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting limits into integrated function and subtracting <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln (2 \times 5 + 5) - \ln (2 \times 0 + 5)\)</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. \(\ln 15 - \ln 5\)</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. \(\ln \frac{{15}}{5},\ln 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 3\) <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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Knowing that the answer to this integration led to a natural logarithm function helped many candidates make progress on this more challenging question, although some candidates simply substituted the limits straight away without integrating. Although some candidates incorrectly simplified \(\ln 15 - \ln 5\) as \(\ln 10\) or \(\frac{{\ln 15}}{{\ln 5}} = \ln 3\) , a pleasing number applied the logarithm property correctly. Some candidates had difficulty with missing brackets which typically led to \(\ln 0\) in their answer. </span></p>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \frac{1}{{\sqrt {2x - 1} }}\), for \(x > \frac{1}{2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int {{{\left( {f\left( x \right)} \right)}^2}{\text{d}}x} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Part of the graph of <em>f</em> is shown in the following diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<p>The shaded region <em>R</em> is enclosed by the graph of <em>f</em>, the <em>x</em>-axis, and the lines <em>x</em> = 1 and <em>x</em> = 9 . Find the volume of the solid formed when <em>R</em> is revolved 360° about the <em>x</em>-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct working   <em><strong>(A1)</strong></em></p>
<p>eg  \(\int {\frac{1}{{2x - 1}}{\text{d}}x,\,\,\int {{{\left( {2x - 1} \right)}^{ - 1}},\,\,\frac{1}{{2x - 1}},\,\,\int {{{\left( {\frac{1}{{\sqrt u }}} \right)}^2}\frac{{{\text{d}}u}}{2}} } } \)</p>
<p>\({\int {\left( {f\left( x \right)} \right)} ^2}{\text{d}}x = \frac{1}{2}{\text{ln}}\left( {2x - 1} \right) + c\)Â Â Â <em><strong>A2 N3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for \(\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)\).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving <em>f </em><sup>2</sup> (accept absence of \(\pi \) / d<em>x</em>)   <strong><em>(M1)</em></strong></p>
<p><em>eg  </em>\(\int_1^9 {{y^2}{\text{d}}x,\,\,} \pi {\int {\left( {\frac{1}{{\sqrt {2x - 1} }}} \right)} ^2}{\text{d}}x,\,\,\left[ {\frac{1}{2}{\text{ln}}\left( {2x - 1} \right)} \right]_1^9\)</p>
<p>substituting limits into <strong>their</strong> integral and subtracting (in any order)Â Â Â <strong><em>(M1)</em></strong></p>
<p><em>eg</em>Â Â \(\frac{\pi }{2}\left( {{\text{ln}}\left( {17} \right) - {\text{ln}}\left( 1 \right)} \right),\,\,\pi \left( {0 - \frac{1}{2}{\text{ln}}\left( {2 \times 9 - 1} \right)} \right)\)</p>
<p>correct working involving calculating a log value or using log law   <em><strong>(A1)</strong></em></p>
<p><em>eg</em>Â Â \({\text{ln}}\left( 1 \right) = 0,\,\,{\text{ln}}\left( {\frac{{17}}{1}} \right)\)</p>
<p>\(\frac{\pi }{2}{\text{ln}}17\,\,\,\,\left( {{\text{accept }}\pi {\text{ln}}\sqrt {17} } \right)\)Â Â Â <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> Full <em><strong>FT</strong></em> may be awarded as normal, from their incorrect answer in part (a), however, do not award the final two <em><strong>A</strong></em> marks unless they involve logarithms.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{1}{2}{x^3} - {x^2} - 3x\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sheldon.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum point at A and a minimum point at B(3, − 9) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the coordinates of A.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the coordinates of</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) the image of B after reflection in the <em>y</em>-axis;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the image of B after translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>{ - 2}\\<br>5<br>\end{array}} \right)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the image of B after reflection in the <em>x</em>-axis followed by </span><span style="font-family: times new roman,times; font-size: medium;">a horizontal stretch with scale factor \(\frac{1}{2}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<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;">\(f(x) = {x^2} - 2x - 3\) <em><strong>A1A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of solving \(f'(x) = 0\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({x^2} - 2x - 3 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct working <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \((x + 1)(x - 3)\) , \(\frac{{2 \pm \sqrt {16} }}{2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = - 1\) (ignore \(x = 3\) ) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting <strong>their</strong> negative <em>x</em>-value into \(f(x)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{3}{( - 1)^3} - {( - 1)^2} - 3( - 1)\) , \( - \frac{1}{3} - 1 + 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = \frac{5}{3}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">coordinates are \(\left( { - 1,\frac{5}{3}} \right)\) <em><strong>N3</strong> </em></span></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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(( - 3{\text{, }} - 9)\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \((1{\text{, }} - 4)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) reflection gives \((3{\text{, }}9)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">stretch gives \(\left( {\frac{3}{2}{\text{, }}9} \right)\) <em><strong>A1A1 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">b(i), (ii) and (iii).</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 majority of candidates answered part (a) completely.</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;">Candidates were generally successful in </span><span style="font-family: times new roman,times; font-size: medium;">finding images after single transformations in part (b). Common incorrect answers for (biii) </span><span style="font-family: times new roman,times; font-size: medium;">included \(\left( {\frac{3}{2},\frac{9}{2}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">, (6, 9) and (6, 18) , demonstrating difficulty with images from horizontal </span><span style="font-family: times new roman,times; font-size: medium;">stretches.</span></p>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = \sqrt x \) between \(x = 0\) and \(x = a\) is rotated \(360^\circ \) about the <em>x</em>-axis. </span><span style="font-family: times new roman,times; font-size: medium;">The volume of the solid formed is \(32\pi \) . Find the value of <em>a</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into formula \(V = \int {\pi {y^2}{\rm{d}}x} \) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">integral expression <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi \int_0^a {(\sqrt x } {)^2}{\rm{d}}x\) , \(\pi \int x \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct integration <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int x{{\rm{d}}x = \frac{1}{2}{x^2}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution \(V = \pi \left[ {\frac{1}{2}{a^2}} \right]\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equating <strong>their</strong> expression to \(32\pi \) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi \left[ {\frac{1}{2}{a^2}} \right] = 32\pi \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({a^2} = 64\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 8\) <em><strong>A2 N2</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[7 marks]</em></span></strong></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;">Despite the “reverse” nature of this question, many candidates performed well with the integration. Some forgot to square the function, while others did not discard the negative value of <em>a</em>. Some attempted to equate \(32\pi \) to the formula for volume of a sphere, which suggests this topic was not fully covered in some centres. </span></p>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = 6{x^2} - 3x\). The graph of \(f\) is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\int {\left( {6{x^2} - 3x} \right){\text{d}}x} \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of \(f\), the <em>x</em>-axis and the lines <em>x</em> = 1 and <em>x</em> = 2 .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(2{x^3} - \frac{{3{x^2}}}{2} + c\,\,\,\left( {{\text{accept}}\,\,\frac{{6{x^3}}}{3} - \frac{{3{x^2}}}{2} + c} \right)\)Â Â Â <em><strong>A1A1 N2</strong></em></p>
<p><strong>Notes:</strong> Award <em><strong>A1A0</strong></em> for both correct terms if +<em>c</em> is omitted.<br>Award <em><strong>A1A0</strong></em> for one correct term <em>eg</em> \(2{x^3} + c\).<br>Award <em><strong>A1A0</strong></em> if both terms are correct, but candidate attempts further working to solve for <em>c</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>substitution of limits or function <em><strong>(A1)</strong></em></p>
<p><em>eg</em>Â Â \(\int_1^2 {f\left( x \right)} \,{\text{d}}x,\,\,\left[ {2{x^3} - \frac{{3{x^2}}}{2}} \right]_1^2\)</p>
<p>substituting limits into their integrated function and subtracting   <em><strong>(M1)</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{6 \times {2^3}}}{3} - \frac{{3 \times {2^2}}}{2} - \left( {\frac{{6 \times {1^3}}}{3} + \frac{{3 \times {1^2}}}{2}} \right)\)</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if substituted into original function.</p>
<p>correct working   <em><strong>(A1)</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{6 \times 8}}{3} - \frac{{3 \times 4}}{2} - \frac{{6 \times 1}}{3} + \frac{{3 \times 1}}{2},\,\,\left( {16 - 6} \right) - \left( {2 - \frac{3}{2}} \right)\)</p>
<p>\(\frac{{19}}{2}\)Â Â Â <em><strong>A1 N3</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>The following table shows the probability distribution of a discrete random variable \(A\), in terms of an angle \(\theta \).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-11_om_09.10.36.png" alt="M17/5/MATME/SP1/ENG/TZ1/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\cos \theta = \frac{3}{4}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\tan \theta > 0\), find \(\tan \theta \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(y = \frac{1}{{\cos x}}\), for \(0 < x < \frac{\pi }{2}\). The graph of \(y\)between \(x = \theta \) and \(x = \frac{\pi }{4}\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of summing to 1 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\sum {p = 1} \)</p>
<p>correct equation <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta + 2\cos 2\theta = 1\)</p>
<p>correct equation in \(\cos \theta \) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta + 2(2{\cos ^2}\theta - 1) = 1,{\text{ }}4{\cos ^2}\theta + \cos \theta - 3 = 0\)</p>
<p>evidence of valid approach to solve quadratic <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)factorizing equation set equal to \(0,{\text{ }}\frac{{ - 1 \pm \sqrt {1 - 4 \times 4 \times ( - 3)} }}{8}\)</p>
<p>correct working, clearly leading to required answer <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\((4\cos \theta - 3)(\cos \theta + 1),{\text{ }}\frac{{ - 1 \pm 7}}{8}\)</p>
<p>correct reason for rejecting \(\cos \theta \ne - 1\) <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\cos \theta \) is a probability (value must lie between 0 and 1), \(\cos \theta > 0\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>R0 </em></strong>for \(\cos \theta \ne - 1\) without a reason.</p>
<p> </p>
<p>\(\cos \theta = \frac{3}{4}\) <em><strong>AG N0</strong></em></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>\(\,\,\,\,\,\)sketch of right triangle with sides 3 and 4, \({\sin ^2}x + {\cos ^2}x = 1\)</p>
<p>correct working </p>
<p><strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)missing side \( = \sqrt 7 ,{\text{ }}\frac{{\frac{{\sqrt 7 }}{4}}}{{\frac{3}{4}}}\)</p>
<p>\(\tan \theta = \frac{{\sqrt 7 }}{3}\) <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.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula involving \({f^2}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\pi \int_\theta ^{\frac{\pi }{4}} {{f^2},{\text{ }}\int {{{\left( {\frac{1}{{\cos x}}} \right)}^2}} } \)</p>
<p>correct substitution of both limits and function <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\pi \int_\theta ^{\frac{\pi }{4}} {{{\left( {\frac{1}{{\cos x}}} \right)}^2}{\text{d}}x} \)</p>
<p>correct integration <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\tan x\)</p>
<p>substituting <strong>their </strong>limits into <strong>their </strong>integrated function and subtracting <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\tan \frac{\pi }{4} - \tan \theta \)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M0 </em></strong>if they substitute into original or differentiated function.</p>
<p> </p>
<p>\(\tan \frac{\pi }{4} = 1\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(1 - \tan \theta \)</p>
<p>\(V = \pi - \frac{{\pi \sqrt 7 }}{3}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p> </p>
<p><strong><em>[6 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 part of the graph of the function \(f(x) = 2{x^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/curve.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The line <em>T</em> is the tangent to the graph of <em>f</em> at \(x = 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that the equation of <em>T</em> is \(y = 4x - 2\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</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>-intercept of <em>T</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 shaded region <em>R</em> is enclosed by the graph of <em>f</em> , the line <em>T</em> , and the <em>x</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for the area of <em>R</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the area of <em>R</em> .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">c(i) and (ii).</div>
</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;">\(f(1) = 2\) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 4x\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding the gradient of <em>f</em> at \(x = 1\) <strong><em> M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. substituting \(x = 1\) into \(f'(x)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding gradient of <em>f </em>at \(x = 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(1) = 4\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding equation of the line <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y - 2 = 4(x - 1)\) , \(2 = 4(1) + b\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 4x - 2\) <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>
<p align="LEFT"> </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;">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. \(4x - 2 = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = \frac{1}{2}\) <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;">(i) bottom limit \(x = 0\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction of integrals/areas <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {f(x) - {\text{area of triangle}}} \) , \(\int {f - \int l } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong> A2 N4</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^1 {2{x^2}{\rm{d}}x - } \int_{0.5}^1 {(4x - 2){\rm{d}}x} \) , \(\int_0^1 {f(x){\rm{d}}x - \frac{1}{2}} \) , \(\int_0^{0.5} {2{x^2}{\rm{d}}x} + \int_{0.5}^1 {(f(x) - (4x - 2)){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>METHOD 1 (using only integrals)</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct integration <em><strong>(A1)(A1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(\int {2{x^2}{\rm{d}}x} = \frac{{2{x^3}}}{3}\) , \(\int {(4x - 2){\rm{d}}x = } 2{x^2} - 2x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substitution of limits <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{1}{{12}} + \frac{2}{3} - 2 + 2 - \left( {\frac{1}{{12}} - \frac{1}{2} + 1} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area = \(\frac{1}{6}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N4</span></strong></em></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2 (using integral and triangle)</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area of triangle = \(\frac{1}{2}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct integration <strong><em>(A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \(\int {2{x^2}{\rm{d}}x = } \frac{{2{x^3}}}{3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substitution of limits <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{2}{3}{(1)^3} - \frac{2}{3}{(0)^3}\) , \(\frac{2}{3} - 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct simplification <strong><em> (A1)</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{2}{3} - \frac{1}{2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area = \(\frac{1}{6}\) </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;"> [9 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><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates seemed to know what was meant by the tangent to the graph in part (a), but there were many who did not fully show their work, which is of course necessary on a "show that" question. While many candidates knew they needed to find the derivative of <em>f</em> , some failed to substitute the given value of <em>x</em> in order to find the gradient of the tangent. </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), finding the <em>x</em>-intercept, was answered correctly by nearly every candidate. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
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<![endif]--><span style="font-size: 11.0pt; font-family: 'Calibri','sans-serif'; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-ansi-language: EN-GB; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">–</span>0.5 only required the use of function <em>f</em> . Many of these candidates were able to earn </span><span style="font-family: times new roman,times; font-size: medium;">follow-through marks in the second part of (c) for their correct integration. There were a few </span><span style="font-family: times new roman,times; font-size: medium;">candidates who successfully found the area under the line as the area of a triangle.</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 \(g(x) = \frac{{\ln x}}{{{x^2}}}\) , 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;">Use the quotient rule to show that \(g'(x) = \frac{{1 - 2\ln x}}{{{x^3}}}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</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>g</em> has a maximum point at A. Find the <em>x</em>-coordinate of A.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</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;">\(\frac{{\rm{d}}}{{{\rm{d}}x}}\ln x = \frac{1}{x}\) , \(\frac{{\rm{d}}}{{{\rm{d}}x}}{x^2} = 2x\) (seen anywhere) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into the quotient rule (do <strong>not</strong> accept product rule) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{x^2}\left( {\frac{1}{x}} \right) - 2x\ln x}}{{{x^4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct manipulation that clearly leads to result <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{x - 2x\ln x}}{{{x^4}}}\) , \(\frac{{x(1 - 2\ln x)}}{{{x^4}}}\) , \(\frac{x}{{{x^4}}}\) , \(\frac{{2x\ln x}}{{{x^4}}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{{1 - 2\ln x}}{{{x^3}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">AG N0</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;">evidence of setting the derivative 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. \(g'(x) = 0\) , \(1 - 2\ln x = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\ln x = \frac{1}{2}\) </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;">\(x = {{\rm{e}}^{\frac{1}{2}}}\) <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">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 candidates clearly knew their quotient rule, although a common error was to simplify \(2x\ln x\) as \(2\ln {x^2}\) and then "cancel" the exponents. </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 (b), those who knew to set the derivative to zero typically went on find the correct <em>x</em>-coordinate, which must be in terms of e, as this is the calculator-free paper. Occasionally, students would take \(\frac{{1 - 2\ln x}}{{{x^3}}} = 0\) and attempt to solve from \(1 - 2\ln x = {x^3}\) . </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider <em>f</em>(<em>x</em>), <em>g</em>(<em>x</em>) and <em>h</em>(<em>x</em>), for x∈\(\mathbb{R}\) where <em>h</em>(<em>x</em>) = \(\left( {f \circ g} \right)\)(<em>x</em>).</p>
<p>Given that <em>g</em>(3) = 7 , <em>g′</em> (3) = 4 and <em>f ′ </em>(7) = −5 , find the gradient of the normal to the curve of <em>h</em> at <em>x</em> = 3.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>recognizing the need to find <em>h′</em>  <em><strong> (M1)</strong></em></p>
<p>recognizing the need to find <em>h′ </em>(3) (seen anywhere)  <em><strong> (M1)</strong></em></p>
<p>evidence of choosing chain rule   <em><strong> (M1)</strong></em></p>
<p><em>eg  </em>\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{\text{d}}y}}{{{\text{d}}u}} \times \frac{{{\text{d}}u}}{{{\text{d}}x}},\,\,f'\left( {g\left( 3 \right)} \right) \times g'\left( 3 \right),\,\,f'\left( g \right) \times g'\)</p>
<p>correct working    <em><strong>(A1)</strong></em></p>
<p><em>eg  </em>\(f'\left( 7 \right) \times 4,\,\, - 5 \times 4\)</p>
<p>\(h'\left( 3 \right) =Â - 20\)Â Â Â Â <strong><em>(A1)</em></strong></p>
<p>evidence of taking <strong>their</strong> negative reciprocal for normal    <em><strong>(M1)</strong></em></p>
<p><em>eg  </em>\( - \frac{1}{{h'\left( 3 \right)}},\,\,{m_1}{m_2} = - 1\)</p>
<p>gradient of normal is \(\frac{1}{{20}}\)    <em><strong>A1 N4</strong></em></p>
<p><em><strong>[7 marks]</strong></em></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;">The graph of \(f(x) = \sqrt {16 - 4{x^2}} \) , for \( - 2 \le x \le 2\) , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/crying.png" alt></span></p>
</div>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region enclosed by the curve of <em>f</em> and the <em>x</em>-axis is rotated \(360^\circ \) about the <em>x</em>-axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the volume of the solid formed.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up integral expression <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi {\int {\sqrt {16 - 4{x^2}} } ^2}{\rm{d}}x\) , \(2\pi \int_0^2 {(16 - 4{x^2})} \) , \({\int {\sqrt {16 - 4{x^2}} } ^2}{\rm{d}}x\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\int {16} {\rm{d}}x = 16x\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(\int {4{x^2}{\rm{d}}x = } \frac{{4{x^3}}}{3}\) (seen anywhere) <em><strong>A1A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting limits into the integrand <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {32 - \frac{{32}}{3}} \right) - \left( { - 32 + \frac{{32}}{3}} \right)\) , \(64 - \frac{{64}}{3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">volume \(= \frac{{128\pi }}{3}\) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates correctly integrated using \(f(x)\) , although some neglected to square the </span><span style="font-family: times new roman,times; font-size: medium;">function and mired themselves in awkward integration attempts. Upon substituting the limits, </span><span style="font-family: times new roman,times; font-size: medium;">many were unable to carry the calculation to completion. Occasionally the \(\pi \) was neglected in </span><span style="font-family: times new roman,times; font-size: medium;">a final answer. Weaker candidates considered the solid formed to be a sphere and did not use </span><span style="font-family: times new roman,times; font-size: medium;">integration.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = {{\text{e}}^{2x}}\). The line \(L\) is the tangent to the curve of \(f\) at \((1,{\text{ }}{{\text{e}}^2})\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the equation of \(L\) in the form \(y = ax + b\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<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;">recognising need to differentiate (seen anywhere) <strong><em>R1</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> \(f',{\text{ }}2{{\text{e}}^{2x}}\)</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;">attempt to find the gradient when \(x = 1\) <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> \(f'(1)\)</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;">\(f'(1) = 2{{\text{e}}^2}\) <strong><em>(A1)</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;">attempt to substitute coordinates (in any order) into equation of a straight line <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> \(y - {{\text{e}}^2} = 2{{\text{e}}^2}(x - 1),{\text{ }}{{\text{e}}^2} = 2{{\text{e}}^2}(1) + b\)</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;">correct working <strong><em>(A1) </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> \(y - {{\text{e}}^2} = 2{{\text{e}}^2}x - 2{{\text{e}}^2},{\text{ }}b = - {{\text{e}}^2}\)</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;">\(y = 2{{\text{e}}^2}x - {{\text{e}}^2}\) <strong><em>A1 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>[6 marks] </em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}\cos x\) . Find the gradient of the normal to the curve of <em>f</em> at \(x = \pi \) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<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;">\(f'(x) = {{\rm{e}}^x} \times ( - \sin x) + \cos x \times {{\rm{e}}^x}\) \(( = {{\rm{e}}^x}\cos x - {{\rm{e}}^x}\sin x)\) <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(\pi \) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(\pi ) = {{\rm{e}}^\pi }\cos \pi - {{\rm{e}}^\pi }\sin \pi \) , \({{\rm{e}}^\pi }( - 1 - 0)\) , \( - {{\rm{e}}^\pi }\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">taking negative reciprocal <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - \frac{1}{{f'(\pi )}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">gradient is \(\frac{1}{{{{\rm{e}}^\pi }}}\) <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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates familiar with the product rule easily found the correct derivative function. Many substituted \(\pi \) to find the tangent gradient, but surprisingly few candidates correctly considered that the gradient of the normal is the negative reciprocal of this answer.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{{6x}}{{\cos x}}\) . Find \(h'(0)\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1 (quotient)</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of numerator is 6 <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of denominator is \( - \sin x\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into quotient rule <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. \(\frac{{(\cos x)(6) - (6x)( - \sin x)}}{{{{(\cos x)}^2}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting \(x = 0\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{(\cos 0)(6) - (6 \times 0)( - \sin 0)}}{{{{(\cos 0)}^2}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h'(0) = 6\) <strong><em>A1 N2</em></strong></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 (product)</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h(x) = 6x \times {(\cos x)^{ - 1}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of 6<em>x</em> is 6 <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of \({(\cos x)^{ - 1}}\) is \(( - {(\cos x)^{ - 2}}( - \sin x))\) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute into product rule <strong>(M1)</strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \((6x)( - {(\cos x)^{ - 2}}( - \sin x)) + (6){(\cos x)^{ - 1}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting \(x = 0\) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \((6 \times 0)( - {(\cos 0)^{ - 2}}( - \sin 0)) + (6){(\cos 0)^{ - 1}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(h'(0) = 6\) <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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates were successful in using the quotient rule, and were able to earn most of the marks for this question. However, there were a large number of candidates who substituted correctly into the quotient rule, but then went on to make mistakes in simplifying this expression. These algebraic errors kept the candidates from earning the final mark for the correct answer. A few candidates tried to use the product rule to find the derivative, but they were generally not as successful as those who used the quotient rule. It was pleasing to note that most candidates did know the correct values for the sine and cosine of zero. </span></p>
</div>
<br><hr><br><div class="specification">
<p>A function <em>f </em>(<em>x</em>) has derivative <em>f ′</em>(<em>x</em>) = 3<em>x</em><sup>2</sup> + 18<em>x</em>. The graph of <em>f</em> has an <em>x</em>-intercept at <em>x</em> = −1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f </em>(<em>x</em>).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <em>f</em> has a point of inflexion at <em>x</em> = <em>p</em>. Find <em>p</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <em>x</em> for which the graph of <em>f</em> is concave-down.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of integration    <em><strong>(M1)</strong></em></p>
<p><em>eg  </em>\(\int {f'\left( x \right)} \)</p>
<p>correct integration (accept absence of <em>C</em>)Â Â Â Â <em><strong>(A1)(A1)</strong></em></p>
<p><em>eg </em> \({x^3} + \frac{{18}}{2}{x^2} + C,\,\,{x^3} + 9{x^2}\)</p>
<p>attempt to substitute <em>x</em> = −1 into <strong>their</strong> <em>f </em>= 0 (must have <em>C</em>)   <em><strong>M1</strong></em></p>
<p><em>eg  </em>\({\left( { - 1} \right)^3} + 9{\left( { - 1} \right)^2} + C = 0,\,\, - 1 + 9 + C = 0\)</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>if they substitute into original or differentiated function.</p>
<p>correct working    <em><strong>(A1)</strong></em></p>
<p><em>eg  </em>\(8 + C = 0,\,\,\,C = - 8\)</p>
<p>\(f\left( x \right) = {x^3} + 9{x^2} - 8\)Â Â Â Â <em><strong>A1 N5</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong> (using 2<sup>nd</sup> derivative)</p>
<p>recognizing that <em>f"</em> = 0 (seen anywhere)Â Â Â <em><strong>M1</strong></em></p>
<p>correct expression for <em>f"   <strong>(A1)</strong></em></p>
<p><em>eg </em> 6<em>x</em> + 18<em>, </em>6<em>p </em>+ 18</p>
<p>correct working   <em><strong> (A1)</strong></em></p>
<p>6<em>p </em>+ 18 = 0</p>
<p><em>p</em> = −3    <em><strong>A1 N3</strong></em></p>
<p>Â </p>
<p><strong>METHOD 1</strong> (using 1<sup>st</sup> derivative)</p>
<p>recognizing the vertex of <em>f′</em> is needed    <em><strong>(M2)</strong></em></p>
<p><em>eg</em>  \( - \frac{b}{{2a}}\) (must be clear this is for <em>f′</em>)</p>
<p>correct substitution    <em><strong>(A1)</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{ - 18}}{{2 \times 3}}\)</p>
<p><em>p</em> = −3    <em><strong>A1 N3</strong></em></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to use <em>f"</em> (<em>x</em>) to determine concavity   <strong><em>(M1)</em></strong></p>
<p><em>eg </em>  <em>f"</em> (<em>x</em>) < 0, <em>f"</em> (−2), <em>f"</em> (−4), 6<em>x</em> + 18 ≤ 0 <img src="data:image/png;base64,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"></p>
<p>correct working    <em><strong>(A1)</strong></em></p>
<p><em>eg  </em>6<em>x</em> + 18 < 0, <em>f"</em> (−2) = 6, <em>f"</em> (−4) = −6 <img src="data:image/png;base64,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"></p>
<p><em>f</em> concave down for <em>x</em> < −3 (do not accept <em>x </em>≤ −3)    <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><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \sin x + \frac{1}{2}{x^2} - 2x\) , for \(0 \le x \le \pi \) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g\) be a quadratic function such that \(g(0) = 5\) . The line \(x = 2\) is the axis of </span><span style="font-family: times new roman,times; font-size: medium;">symmetry of the graph of \(g\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The function \(g\) can be expressed in the form \(g(x) = a{(x - h)^2} + 3\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(g(4)\) .</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) Write down the value of \(h\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the value of \(a\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</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 \(x\) for which the tangent to the graph of \(f\) is parallel to the </span><span style="font-family: times new roman,times; font-size: medium;">tangent to the graph of \(g\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</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;">\(f'(x) = \cos x + x - 2\) <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 each term. </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 \(g(0) = 5\) gives the point (\(0\), \(5\)) <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognize symmetry <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> vertex, sketch </span></p>
<p><br><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g(4) = 5\) <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">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(h = 2\) <em><strong> A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) substituting into \(g(x) = a{(x - 2)^2} + 3\) (not the vertex) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(5 = a{(0 - 2)^2} + 3\) , \(5 = a{(4 - 2)^2} + 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working towards solution <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(5 = 4a + 3\) , \(4a = 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = \frac{1}{2}\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </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">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(g(x) = \frac{1}{2}{(x - 2)^2} + 3 = \frac{1}{2}{x^2} - 2x + 5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivative of \(g\) <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(2 \times \frac{1}{2}(x - 2)\) , \(x - 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of equating both derivatives <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f' = g'\)</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> \(\cos x + x - 2 = x - 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">working towards a solution <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\cos x = 0\) , combining like terms </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = \frac{\pi }{2}\) <em><strong>A1 N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Do not award final <em><strong>A1</strong></em> if additional values are given. </span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></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 style="font-size: 13.28px; font-family: sans-serif; left: 506.013px; top: 127.867px; transform: scale(1.03286, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="91.92416273955348"><span style="font-family: times new roman,times; font-size: medium;">In part (a), most candidates were able to correctly find the derivative of the 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;">In part (b), many candidates did not understand the significance of the axis of symmetry and the known point (\(0\), \(5\)), and so were unable to find \(g(4)\) using symmetry. A few used more complicated manipulations of the function, but many algebraic errors were seen.</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), a large number of candidates were able to simply write down the correct value of \(h\), as intended by the command term in this question. A few candidates wrote down the incorrect negative value. Most candidates attempted to substitute the \(x\) and \(y\) values of the known point correctly into the function, but again many arithmetic and algebraic errors kept them from finding the correct value for \(a\).</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) required the candidates to find the derivative of \(g\), and to equate that to their answer from part (a). Although many candidates were able to simplify their equation to \(\cos x = 0\), many did not know how to solve for \(x\) at this point. Candidates who had made errors in parts (a) and/or (c) were still able to earn follow-through marks in part (d).</span></p>
<div class="question_part_label">d.</div>
</div>
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