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</div><h2>HL Paper 2</h2><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The population of mosquitoes in a specific area around a lake is controlled by pesticide. The rate of decrease of the number of mosquitoes is proportional to the number of mosquitoes at any time <em>t</em>. Given that the population decreases from \({\text{500}}\,{\text{000}}\) to \({\text{400}}\,{\text{000}}\) in a five year period, find the time it takes in years for the population of mosquitoes to decrease by half.</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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let the number of mosquitoes be <em>y</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}y}}{{{\text{d}}t}} = - ky\) <strong><em>M1</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;">\(\int {\frac{1}{y}{\text{d}}y = \int { - k{\text{d}}t} } \) <strong><em>M1</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;">\(\ln y = - kt + c\) <strong><em>A1</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;">\(y = {{\text{e}}^{ - kt + c}}\)</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;">\(y = A{{\text{e}}^{ - kt}}\)</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;">when \(t = 0,{\text{ }}y = 500\,000 \Rightarrow A = 500\,000\) <strong><em>A1</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;">\(y = {\text{500}}\,{\text{000}}{{\text{e}}^{ - kt}}\)</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;">when \(t = 5,{\text{ }}y = {\text{400}}\,{\text{000}}\)</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;">\({\text{400}}\,000 = 500\,{\text{000}}{{\text{e}}^{ - 5k}}\) <strong><em>M1</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;">\(\frac{4}{5} = {{\text{e}}^{ - 5k}}\)</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;">\( - 5k = \ln \frac{4}{5}\)</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;">\(k = - \frac{1}{5}\ln \frac{4}{5}\,\,\,\,\,{\text{( = 0.0446)}}\) <strong><em>A1</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;">\(250\,000 = 500\,{\text{000}}{{\text{e}}^{ - kt}}\) <strong><em>M1</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;">\(\frac{1}{2} = {{\text{e}}^{ - kt}}\)</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;">\(\ln \frac{1}{2} = - kt\)</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;">\(t = \frac{5}{{\ln \frac{4}{5}}}\ln \frac{1}{2} = 15.5{\text{ years}}\) <strong><em>A1</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>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Some candidates assumed that the decrease in population size was exponential / geometric and were therefore unable to gain the first 4 marks. Apart from this, reasonably good attempts were made by many candidates.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The acceleration in ms<sup>−2</sup> of a particle moving in a straight line at time \(t\) seconds, </span><span style="font-family: times new roman,times; font-size: medium;">\(t \geqslant 0\) , is given by the formula \(a = - \frac{1}{2}v\)</span><span style="font-family: times new roman,times; font-size: medium;">. When \(t = 0\) , the velocity is \(40\) ms<sup>−1</sup> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find an expression for \(v\) 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;">\(\frac{{{\text{d}}v}}{{{\text{d}}t}} = - \frac{1}{2}v\) </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;">\(\int {\frac{{{\text{d}}v}}{v}} = \int { - \frac{1}{2}} {\text{d}}t\) </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;">\(\ln v = - \frac{1}{2}t + c\) </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;">\(v = {{\text{e}}^{ - \frac{1}{2}t + c}}\) \(\left( { = A{{\text{e}}^{ - \frac{1}{2}t}}} \right)\) </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;">\(t = 0\), \(v = 40\), so \(A = 40\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({v = 40{{\text{e}}^{ - \frac{1}{2}t}}}\) </span><span style="font-family: times new roman,times; font-size: medium;">(or equivalent) <em><strong>A1</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 was a poorly answered question which linked the topic of kinematics with that of first order differential equations. Many candidates seemed unaware that the acceleration is the time derivative of the velocity. This was often followed by a failure to recognize a separable differential equation and/or integration with respect to the wrong variable.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">An open glass is created by rotating the curve \(y = {x^2}\) , defined in the domain \(x \in [0,10]\), \(2\pi \) radians about the <em>y</em>-axis. Units on the coordinate axes are defined to be in centimetres.</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;">When the glass contains water to a height \(h\) cm, find the volume \(V\) of water in terms of \(h\) .</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;">If the water in the glass evaporates at the rate of 3 cm<sup>3</sup> per hour for each cm<sup>2</sup> of exposed surface area of the water, show that,</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\sqrt {2\pi V} \)</span><span style="font-family: times new roman,times; font-size: medium;"> , where \(t\) is measured in hours.</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;">If the glass is filled completely, how long will it take for all the water to evaporate?</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;">volume \( = \pi \int_0^h {{x^2}{\text{d}}y} \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\(\pi \int_0^h {y{\text{d}}y} \) </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;">\(\pi \left[ {\frac{{{y^2}}}{2}} \right]_0^h = \frac{{\pi {h^2}}}{2}\) <em><strong>A1</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;">\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3 \times \)</span><span style="font-family: times new roman,times; font-size: medium;"> surface area </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;">surface area \( = \pi {x^2}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \pi h\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = \frac{{\pi {h^2}}}{2} \Rightarrow h\sqrt {\frac{{2V}}{\pi }} \) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\pi \sqrt {\frac{{2V}}{\pi }} \) </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;">\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\sqrt {2\pi V} \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> AG</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Assuming that </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\text{d}}h}}{{{\text{d}}t}} = - 3\)</span><span style="font-family: times new roman,times; font-size: medium;"> without justification gains no marks.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({V_0} = 5000\pi \) (\( = 15700\) cm<sup>3</sup>) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\text{d}}V}}{{{\text{d}}t}} = - 3\sqrt {2\pi V} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempting to separate variables <em><strong>M1</strong></em></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;">\(\int {\frac{{{\text{d}}V}}{{\sqrt V }}} = - 3\sqrt {2\pi } \int {{\text{d}}t} \) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2\sqrt V = - 3\sqrt {2\pi t} + c\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(c = 2\sqrt {5000\pi } \) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = 0\) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow t = \frac{2}{3}\sqrt {\frac{{5000\pi }}{{2\pi }}} = 33\frac{1}{3}\) hours </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </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;">\(\int_{5000\pi }^0 {\frac{{{\text{d}}V}}{{\sqrt V }}} = - 3\sqrt {2\pi } \int_0^T {{\text{d}}t} \) <em><strong>M1A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt to use definite integrals, <em><strong>A1</strong></em> for correct limits </span><span style="font-family: times new roman,times; font-size: medium;">and <em><strong>A1</strong></em> for correct integrands.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left[ {2\sqrt V } \right]_{5000\pi }^0 = 3\sqrt {2\pi } T\) </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;">\(T = \frac{2}{3}\sqrt {\frac{{5000\pi }}{{2\pi }}} = 33\frac{1}{3}\) hours</span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</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;">[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 found to be challenging by many candidates and there were very few completely correct solutions. Many candidates did not seem able to find the volume of revolution when taken about the <em>y</em>-axis in (a). Candidates did not always recognize that part (b) did not involve related rates. Those candidates who attempted the question made some progress by separating the variables and integrating in (c) but very few were able to identify successfully the values necessary to find 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;">This question was found to be challenging by many candidates and there were very few completely correct solutions. Many candidates did not seem able to find the volume of revolution when taken about the <em>y</em>-axis in (a). Candidates did not always recognize that part (b) did not involve related rates. Those candidates who attempted the question made some progress by separating the variables and integrating in (c) but very few were able to identify successfully the values necessary to find the correct 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;">This question was found to be challenging by many candidates and there were very few completely correct solutions. Many candidates did not seem able to find the volume of revolution when taken about the <em>y</em>-axis in (a). Candidates did not always recognize that part (b) did not involve related rates. Those candidates who attempted the question made some progress by separating the variables and integrating in (c) but very few were able to identify successfully the values necessary to find the correct answer.</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the differential equation \(y\frac{{{\text{d}}y}}{{{\text{d}}x}} = \cos 2x\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that the function \(y = \cos x + \sin x\) satisfies the differential equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the general solution of the differential equation. Express your solution in the form \(y = f(x)\), involving a constant of integration.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) For which value of the constant of integration does your solution coincide with the function given in part (i)?</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A different solution of the differential equation, satisfying <em>y</em> = 2 when \(x = \frac{\pi }{4}\), defines a curve <em>C</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the equation of <em>C</em> in the form \(y = g(x)\) , and state the range of the function <em>g</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A region <em>R</em> in the <em>xy</em> plane is bounded by <em>C</em>, the <em>x</em>-axis and the vertical lines <em>x</em> = 0 and \(x = \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the area of <em>R</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the volume generated when that part of <em>R</em> above the line <em>y</em> = 1 is rotated about the <em>x</em>-axis through \(2\pi \) radians.</span></p>
<div class="marks">[12]</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: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \sin x + \cos x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y\frac{{{\text{d}}y}}{{{\text{d}}x}} = (\cos x + \sin x)( - \sin x + \cos x)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\cos ^2}x - {\sin ^2}x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \cos 2x\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({y^2} = {(\sin x + \cos x)^2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2y\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2(\cos x + \sin x)(\cos x - \sin x)\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y\frac{{{\text{d}}y}}{{{\text{d}}x}} = {\cos ^2}x - {\sin ^2}x\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \cos 2x\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) attempting to separate variables \(\int {y{\text{ d}}y = \int {\cos 2x{\text{ d}}x} } \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{2}{y^2} = \frac{1}{2}\sin 2x + C\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for a correct LHS and </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for a correct RHS.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = \pm {(\sin 2x + A)^{\frac{1}{2}}}\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(\sin 2x + A \equiv {(\cos x + \sin x)^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({(\cos x + \sin x)^2} = {\cos ^2}x + 2\sin x\cos x + {\sin ^2}x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">use of \(\sin 2x \equiv 2\sin x\cos x\). <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>A</em> = 1 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) substituting \(x = \frac{\pi }{4}\) and <em>y</em> = 2 into \(y = {(\sin 2x + A)^{\frac{1}{2}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(g(x) = {(\sin 2x + 3)^{\frac{1}{2}}}\). <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">range <em>g</em> is \(\left[ {\sqrt 2 ,{\text{ }}2} \right]\) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept [1.41, 2]. Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for each correct endpoint and </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for the correct closed interval.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\int_0^{\frac{\pi }{2}} {{{(\sin 2x + 3)}^{\frac{1}{2}}}{\text{d}}x} \) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)(A1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 2.99 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(\pi \int_0^{\frac{\pi }{2}} {(\sin 2x + 3){\text{d}}x - \pi (1)\left( {\frac{\pi }{2}} \right)} \) (or equivalent) <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)(A1)(A1)</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \(\pi \int_0^{\frac{\pi }{2}} {(\sin 2x + 2){\text{d}}x} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 17.946 - 4.935{\text{ }}( = \frac{\pi }{2}(3\pi + 2) - \pi \left( {\frac{\pi }{2}} \right))\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \(\pi (\pi + 1)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[12 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was not well done and was often difficult to mark. In part (a) (i), a large number of candidates did not know how to verify a solution, \(y(x)\), to the given differential equation. Instead, many candidates attempted to solve the differential equation. In part (a) (ii), a large number of candidates began solving the differential equation by correctly separating the variables but then either neglected to add a constant of integration or added one as an afterthought. Many simple algebraic and basic integral calculus errors were seen. In part (a) (iii), many candidates did not realize that the solution given in part (a) (i) and the general solution found in part (a) (ii) were to be equated. Those that did know to equate these two solutions, were able to square both solution forms and correctly use the trigonometric identity \(\sin 2x = 2\sin x\cos x\). Many of these candidates however started with incorrect solution(s).</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: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (b), a large number of candidates knew how to find a required area and a required volume of solid of revolution using integral calculus. Many candidates, however, used incorrect expressions obtained in part (a). In part (b) (ii), a number of candidates either neglected to state ‘π’ or attempted to calculate the volume of a solid of revolution of ‘radius’ \(f(x) - g(x)\).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The acceleration of a car is \(\frac{1}{{40}}(60 - v){\text{ m}}{{\text{s}}^{ - 2}}\), when its velocity is \(v{\text{ m}}{{\text{s}}^{ - 2}}\). Given the car starts from rest, find the velocity of the car after 30 seconds.</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: 24.0px Helvetica;"><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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{1}{{40}}(60 - v)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to separate variables \(\int {\frac{{{\text{d}}v}}{{60 - v}} = \int {\frac{{{\text{d}}t}}{{40}}} } \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - \ln (60 - v) = \frac{t}{{40}} + c\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(c = - \ln 60\) (or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve for <em>v</em> when <em>t</em> = 30 <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 60 - 60{e^{ - \frac{3}{4}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 31.7{\text{ (m}}{{\text{s}}^{ - 1}})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{1}{{40}}(60 - v)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}t}}{{{\text{d}}v}} = \frac{{40}}{{60 - v}}\) (or equivalent) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^{{v_f}} {\frac{{40}}{{60 - v}}{\text{d}}v = 30} \) where \({v_f}\) is the velocity of the car after 30 seconds. <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve \(\int_0^{{v_f}} {\frac{{40}}{{60 - v}}{\text{d}}v = 30} \) for \({v_f}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = 31.7{\text{ (m}}{{\text{s}}^{ - 1}})\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates experienced difficulties with this question. A large number of candidates did not attempt to separate the variables and instead either attempted to integrate with respect to <em>v </em>or employed constant acceleration formulae. Candidates that did separate the variables and attempted to integrate both sides either made a sign error, omitted the constant of integration or found an incorrect value for this constant. Almost all candidates were not aware that this question could be solved readily on a GDC.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Solve the differential equation \(\frac{{{{\cos }^2}x}}{{{{\text{e}}^y}}} - {{\text{e}}^{{{\text{e}}^y}}}\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> , given that \(y = 0\) when \(x = \pi\).<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the value of y when \(x = \frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;">.</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;">(a) rearrange \(\frac{{{{\cos }^2}x}}{{{{\text{e}}^y}}} - {{\text{e}}^{{{\text{e}}^y}}}\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> to obtain \({\cos ^2}x{\text{d}}x = {{\text{e}}^y}{{\text{e}}^{{{\text{e}}^y}}}{\text{d}}y\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">as \(\int {{{\cos }^2}x{\text{d}}x} = \int {\frac{{1 + \cos \left( {2x} \right)}}{2}{\text{d}}x = \frac{1}{2}x + \frac{1}{4}\sin } \left( {2x} \right) + {C_1}\) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">and \(\int {{{\text{e}}^y}{{\text{e}}^{{{\text{e}}^y}}}{\text{d}}y} = {{\text{e}}^{{{\text{e}}^y}}} + {C_2}\) </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,times; font-size: medium;"><strong>Note:</strong> The above two integrations are independent and should not </span><span style="font-family: times new roman,times; font-size: medium;">be penalized for missing.</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;">a general solution of \(\frac{{{{\cos }^2}x}}{{{{\text{e}}^y}}} - {{\text{e}}^{{{\text{e}}^y}}}\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> is \(\frac{1}{2}x + \frac{1}{4}\sin \left( {2x} \right) - {{\text{e}}^{{{\text{e}}^y}}} = C\) </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;">given that \(y = 0\) when </span><span style="font-family: times new roman,times; font-size: medium;">\(x = \pi\), \(C = \frac{\pi }{2} + \frac{1}{4}\sin \left( {2\pi } \right) - {{\text{e}}^{{{\text{e}}^0}}} = \frac{\pi }{2} - {\text{e}}\),</span><span style="font-family: times new roman,times; font-size: medium;"> (or \( - 1.15\)) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so, the required solution is defined by the equation</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{1}{2}x + \frac{1}{4}\sin \left( {2x} \right) - {{\text{e}}^{{{\text{e}}^y}}} = \frac{\pi }{2} - {\text{e}}\) or \(y = \ln \left( {\ln \left( {\frac{1}{2}x + \frac{1}{4}\sin \left( {2x} \right){\text{ + }}{{\text{e}}^{{{\text{e}}^y}}} - \frac{\pi }{2}} \right)} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N0</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(or equivalent)</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;">(b) for \(x = \frac{\pi }{2}\)</span><span style="font-family: times new roman,times; font-size: medium;">, \(y = \ln \left( {\ln \left( {{\text{e}} - \frac{\pi }{4}} \right)} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">(or \( - 0.417\) ) <em><strong>A1</strong></em></span></p>
<p><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;">[8 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 was a more difficult question and it was apparent that students did find it so. For those that managed to rearrange the equation to separate the variables, few could manage to successfully integrate both sides. The unfamiliarity of \({{\text{e}}^{{{\text{e}}^y}}}\) seemed to disturb some students.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove by mathematical induction that, for \(n \in {\mathbb{Z}^ + }\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}.\]</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using integration by parts, show that \(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}} (2\sin x - \cos x) + C\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Solve the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = \sqrt {1 - {y^2}} {{\text{e}}^{2x}}\sin x\), given that <em>y</em> = 0 when <em>x</em> = 0,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">writing your answer in the form \(y = f(x)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Sketch the graph of \(y = f(x)\) , found in part (b), for \(0 \leqslant x \leqslant 1.5\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the coordinates of the point P, the first positive intercept on the <em>x</em>-axis, and mark it on your sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The region bounded by the graph of \(y = f(x)\) and the <em>x</em>-axis, between the origin and P, is rotated 360° about the <em>x</em>-axis to form a solid of revolution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the volume of this solid.</span></p>
<div class="marks">[17]</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">prove that \(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">for <em>n</em> = 1</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{LHS}} = 1,{\text{ RHS}} = 4 - \frac{{1 + 2}}{{{2^0}}} = 4 - 3 = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so true for <em>n</em> = 1 <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">assume true for <em>n</em> = <em>k</em> <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">so \(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + k{\left( {\frac{1}{2}} \right)^{k - 1}} = 4 - \frac{{k + 2}}{{{2^{k - 1}}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">now for <em>n</em> = <em>k</em> +1</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">LHS: \(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + ... + k{\left( {\frac{1}{2}} \right)^{k - 1}} + (k + 1){\left( {\frac{1}{2}} \right)^k}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4 - \frac{{k + 2}}{{{2^{k - 1}}}} + (k + 1){\left( {\frac{1}{2}} \right)^k}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4 - \frac{{2(k + 2)}}{{{2^k}}} + \frac{{k + 1}}{{{2^k}}}\,\,\,\,\,\)(or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 4 - \frac{{(k + 1) + 2}}{{{2^{(k + 1) - 1}}}}\,\,\,\,\,\)(accept \(4 - \frac{{k + 3}}{{{2^k}}}\)) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore if it is true for <em>n</em> = <em>k</em> it is true for <em>n</em> = <em>k</em> + 1. It has been shown to be true for <em>n</em> = 1 so it is true for all \(n{\text{ }}( \in {\mathbb{Z}^ + })\). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> To obtain the final <strong><em>R</em></strong> mark, a reasonable attempt at induction must have been made.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = - \cos x{{\text{e}}^{2x}} + \int {2{{\text{e}}^{2x}}\cos x{\text{d}}x} } \) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \cos x{{\text{e}}^{2x}} + 2{{\text{e}}^{2x}}\sin x - \int {4{{\text{e}}^{2x}}\sin x{\text{d}}x} \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = - \cos x{{\text{e}}^{2x}} + 2{{\text{e}}^{2x}}\sin x} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x) + C} \) <strong><em>AG</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\sin x{{\text{e}}^{2x}}{\text{d}}x = \frac{{\sin x{{\text{e}}^{2x}}}}{2} - \int {\cos x\frac{{{{\text{e}}^{2x}}}}{2}{\text{d}}x} } \) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{\sin x{{\text{e}}^{2x}}}}{2} - \cos x\frac{{{{\text{e}}^{2x}}}}{4} - \int {\sin x\frac{{{{\text{e}}^{2x}}}}{4}{\text{d}}x} \) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{5}{4}\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{{{{\text{e}}^{2x}}\sin x}}{2} - \frac{{\cos x{{\text{e}}^{2x}}}}{4}} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {{{\text{e}}^{2x}}\sin x{\text{d}}x = \frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x) + C} \) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{{{\text{d}}y}}{{\sqrt {1 - {y^2}} }} = \int {{{\text{e}}^{2x}}\sin x{\text{d}}x} } \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arcsin y = \frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x)( + C)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">when \(x = 0,{\text{ }}y = 0 \Rightarrow C = \frac{1}{5}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y = \sin \left( {\frac{1}{5}{{\text{e}}^{2x}}(2\sin x - \cos x) + \frac{1}{5}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) </span><img 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" alt><span class="Apple-style-span" style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px; display: inline; float: none;"><span class="Apple-style-span" style="font-family: 'times new roman', times; font-size: medium; display: inline; float: none; line-height: normal;"> <strong><em>A1</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P is (1.16, 0) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1</em></strong> for 1.16 seen anywhere, <strong><em>A1</em></strong> for complete sketch.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow FT on their answer from (b)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(V = \int_0^{1.162...} {\pi {y^2}{\text{d}}x} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1.05\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow FT on their answers from (b) and (c)(i).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 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;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part A: Given that this question is at the easier end of the ‘proof by induction’ spectrum, it was disappointing that so many candidates failed to score full marks. The <em>n</em> = 1 case was generally well done. The whole point of the method is that it involves logic, so ‘let n = k’ or ‘put n = k’, instead of ‘assume ... to be true for n = k’, gains no marks. The algebraic steps need to be more convincing than some candidates were able to show. It is astonishing that the R1 mark for the final statement was so often not awarded.</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: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part B: Part (a) was often well done, although some faltered after the first integration. Part (b) was also generally well done, although there were some errors with the constant of integration. In (c) the graph was often attempted, but errors in (b) usually led to manifestly incorrect plots. Many attempted the volume of integration and some obtained the correct value.</span></p>
<div class="question_part_label">B.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves in a straight line with velocity <em>v </em>metres per second. At any time <em>t </em>seconds, \(0 \leqslant t < \frac{{3\pi }}{4}\), the velocity is given by the differential equation \(\frac{{{\text{d}}v}}{{{\text{d}}t}} + {v^2} + 1 = 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It is also given that <em>v </em>= 1 when <em>t </em>= 0 .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for <em>v </em>in terms of <em>t </em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of <em>v </em>against <em>t </em>, clearly showing the coordinates of any intercepts, and the equations of any asymptotes.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down the time <em>T </em>at which the velocity is zero.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the distance travelled in the interval [0, <em>T</em>] .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find an expression for <em>s </em>, the displacement, in terms of <em>t </em>, given that <em>s </em>= 0 when <em>t </em>= 0 .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, show that \(s = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}v}}{{{\text{d}}t}} = - {v^2} - 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to separate the variables <strong><em>M1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{1}{{1 + {v^2}}}{\text{d}}v = \int { - 1{\text{d}}t} } \) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\arctan v = - t + k\) <strong> <em>A1A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalize the lack of constant at this stage.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><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: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">when <em>t</em> = 0, <em>v</em> = 1 <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow k = \arctan 1 = \left( {\frac{\pi }{4}} \right) = (45^\circ )\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow v = \tan \left( {\frac{\pi }{4} - t} \right)\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[7 marks]</span><br></em></strong></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: 11.0px Times;"><img 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" alt><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> A1A1A1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><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: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for general shape,</span></p>
<p style="margin: 0px 0px 0px 30px; font: 11px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> A1 </em></strong>for asymptote,</span></p>
<p style="margin: 0px 0px 0px 30px; font: 11px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1 </em></strong>for correct <em>t </em>and <em>v </em>intercept.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Do not penalise if a larger domain is used.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><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: 11.0px Times;"><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>
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<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(T = \frac{\pi }{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) area under curve \( = \int_0^{\frac{\pi }{4}} {\tan \left( {\frac{\pi }{4} - t} \right){\text{d}}t} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.347\left( { = \frac{1}{2}\ln 2} \right)\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks] </em></strong></span></p>
<div class="question_part_label">c.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v = \tan \left( {\frac{\pi }{4} - t} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = \int {\tan \left( {\frac{\pi }{4} - t} \right){\text{d}}t} \) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{{\sin \left( {\frac{\pi }{4} - t} \right)}}{{\cos \left( {\frac{\pi }{4} - t} \right)}}} {\text{ d}}t\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;"> <em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln \cos \left( {\frac{\pi }{4} - t} \right) + k\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">when \(t = 0,{\text{ }}s = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = - \ln \cos \frac{\pi }{4}\) <em><strong>A1</strong></em><strong><em><br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = \ln \cos \left( {\frac{\pi }{4} - t} \right) - \ln \cos \frac{\pi }{4}\left( { = \ln \left[ {\sqrt 2 \cos \left( {\frac{\pi }{4} - t} \right)} \right]} \right)\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
<div class="question_part_label">d.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{\pi }{4} - t = \arctan v\) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = \frac{\pi }{4} - \arctan v\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = \ln \left[ {\sqrt 2 \cos \left( {\frac{\pi }{4} - \frac{\pi }{4} + \arctan v} \right)} \right]\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = \ln \left[ {\sqrt 2 \cos (\arctan v)} \right]\) <strong> <em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><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: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = \ln \left[ {\sqrt 2 \cos \left( {\arccos \frac{1}{{\sqrt {1 + {v^2}} }}} \right)} \right]\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln \frac{{\sqrt 2 }}{{\sqrt {1 + {v^2}} }}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><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: 11.0px Times;"><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: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(s = \ln \cos \left( {\frac{\pi }{4} - t} \right) - \ln \cos \frac{\pi }{4}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \ln \sec \left( {\frac{\pi }{4} - t} \right) - \ln \cos \frac{\pi }{4}\) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \ln \sqrt {1 + {{\tan }^2}\left( {\frac{\pi }{4} - t} \right)} - \ln \cos \frac{\pi }{4}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = - \ln \sqrt {1 + {v^2}} - \ln \cos \frac{\pi }{4}\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \ln \frac{1}{{\sqrt {1 + {v^2}} }} + \ln \sqrt 2 \) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><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: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3<br></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(v\frac{{dv}}{{ds}} = - {v^2} - 1\) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {\frac{v}{{{v^2} + 1}}dv = - \int {1ds} } \) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{2}\ln ({v^2} + 1) = - s + k\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">when \(s = 0\,,{\text{ }}t = 0 \Rightarrow v = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow k = \frac{1}{2}\ln 2\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow s = \frac{1}{2}\ln \frac{2}{{1 + {v^2}}}\) <strong> <em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;"> </span></em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be the most challenging question in section B with only a very small number of candidates producing fully correct answers. Many candidates did not realise that part (a) was a differential equation that needed to be solved using a method of separating the variables. Without this, further progress with the question was difficult. For those who did succeed in part (a), parts (b) and (c) were relatively well done. For the minority of candidates who attempted parts (d) and (e) only the best recognised the correct methods.</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: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be the most challenging question in section B with only a very small number of candidates producing fully correct answers. Many candidates did not realise that part (a) was a differential equation that needed to be solved using a method of separating the variables. Without this, further progress with the question was difficult. For those who did succeed in part (a), parts (b) and (c) were relatively well done. For the minority of candidates who attempted parts (d) and (e) only the best recognised the correct methods.</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: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be the most challenging question in section B with only a very small number of candidates producing fully correct answers. Many candidates did not realise that part (a) was a differential equation that needed to be solved using a method of separating the variables. Without this, further progress with the question was difficult. For those who did succeed in part (a), parts (b) and (c) were relatively well done. For the minority of candidates who attempted parts (d) and (e) only the best recognised the correct methods.</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: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be the most challenging question in section B with only a very small number of candidates producing fully correct answers. Many candidates did not realise that part (a) was a differential equation that needed to be solved using a method of separating the variables. Without this, further progress with the question was difficult. For those who did succeed in part (a), parts (b) and (c) were relatively well done. For the minority of candidates who attempted parts (d) and (e) only the best recognised the correct methods.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This proved to be the most challenging question in section B with only a very small number of candidates producing fully correct answers. Many candidates did not realise that part (a) was a differential equation that needed to be solved using a method of separating the variables. Without this, further progress with the question was difficult. For those who did succeed in part (a), parts (b) and (c) were relatively well done. For the minority of candidates who attempted parts (d) and (e) only the best recognised the correct methods.</span></p>
<div class="question_part_label">e.</div>
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