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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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<br><hr><br>