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</div><h2>SL Paper 1</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;">Consider the differential equation \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^3} - {x^3}\) for which \(y = 1\) when \(x = 0\). Use Euler’s method with a step length of \(0.1\) to find an approximation for the value of \(y\) when \(x = 0.4\).</span></p>
</div>
<br><hr><br><div class="question">
<p>Use the integral test to determine whether or not \(\sum\limits_{n = 2}^\infty {\frac{1}{{n{{\left( {{\text{ln}}\,n} \right)}^2}}}} \) converges.</p>
</div>
<br><hr><br><div class="specification">
<p>Let</p>
<p>\({I_n} = \int_1^\infty {{x^n}{{\text{e}}^{ - x}}{\text{d}}x} \) where \(n \in \mathbb{N}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using l’Hôpital’s rule, show that</p>
<p>\(\mathop {\lim }\limits_{x \to \infty } {x^n}{{\text{e}}^{ - x}} = 0\) where \(n \in \mathbb{N}\).</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>Show that, for \(n \in {\mathbb{Z}^ + }\),</p>
<p>\[{I_n} = \alpha {{\text{e}}^{ - 1}} + \beta n{I_{n - 1}}\]</p>
<p>where \(\alpha \), \(\beta \) are constants to be determined.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of \({I_3}\), giving your answer as a multiple of \({{\text{e}}^{ - 1}}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Given that \(y\) is a function of \(x\), the function \(z\) is given by \(z = \frac{{y - x}}{{y + x}}\), where \(x \in \mathbb{R},\,\,x \ne 3,\,\,y + x \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\frac{{{\text{d}}z}}{{{\text{d}}x}} = \frac{2}{{{{\left( {y + x} \right)}^2}}}\left( {x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y} \right)\).</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>Show that the differential equation \(f\left( x \right)\left( {x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y} \right) = {y^2} - {x^2}\) can be written as \(f\left( x \right)\frac{{{\text{d}}z}}{{{\text{d}}x}} = 2z\).</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>Hence show that the solution to the differential equation \(\left( {x - 3} \right)\left( {x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y} \right) = {y^2} - {x^2}\) given that \(x = 4\) when \(y = 5\) is \(\frac{{y - x}}{{y + x}} = {\left( {\frac{{x - 3}}{3}} \right)^2}\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Find the interval of convergence of the series \(\sum\limits_{k = 1}^\infty {\frac{{{{(x - 3)}^k}}}{{{k^2}}}} \).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The function \(f:\mathbb{R} \to \mathbb{R}\) is defined by \(f:x \to \left\{ {\begin{array}{*{20}{c}} { - 3x + 1}&{{\text{for }}x < 0} \\ 1&{{\text{for }}x = 0} \\ {2{x^2} - 3x + 1}&{{\text{for }}x > 0} \end{array}} \right.\).</p>
<p class="p1">By considering limits prove that \(f\) is</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">continuous at \(x = 0\);</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">differentiable at \(x = 0\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the infinite series \(S = \sum\limits_{n = 1}^\infty {{{( - 1)}^{n + 1}}\sin } \left( {\frac{1}{n}} \right)\) .</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;">Show that the series is conditionally convergent but not absolutely convergent.</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;">Show that \(S > 0.4\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function \(f\) is defined by</p>
<p>\[f(x) = \frac{{{{\text{e}}^x} + {{\text{e}}^{ - x}} + 2\cos x}}{4},{\text{ }}x \in \mathbb{R}.\]</p>
</div>
<div class="specification">
<p>The random variable \(X\) has a Poisson distribution with mean \(\mu \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({f^{(4)}}x = f(x)\);</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering derivatives of \(f\), determine the first three non-zero terms of the Maclaurin series for \(f(x)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a series in terms of \(\mu \) for the probability \(p = {\text{P}}[X \equiv 0(\bmod 4)]\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(p = {{\text{e}}^{ - \mu }}f(\mu )\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the numerical value of \(p\) when \(\mu = 3\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</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 the general solution of the differential equation \((1 - {x^2})\frac{{{\rm{d}}y}}{{{\rm{d}}x}} = 1 + xy\) , for \(\left| x \right| < 1\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Show that the solution \(y = f(x)\) that satisfies the condition \(f(0) = \frac{\pi }{2}\) is \(f(x) = \frac{{\arcsin x + \frac{\pi }{2}}}{{\sqrt {1 - {x^2}} }}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find \(\mathop {\lim }\limits_{x \to - 1} f(x)\) .</span></p>
<p align="LEFT"> </p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Calculate the following limit</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\mathop {\lim }\limits_{x \to 0} \frac{{{2^x} - 1}}{x}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Calculate the following limit</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\mathop {\lim }\limits_{x \to 0} \frac{{{{(1 + {x^2})}^{\frac{3}{2}}} - 1}}{{\ln (1 + x) - x}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">By evaluating successive derivatives at \(x = 0\) , find the Maclaurin series for </span><span style="font-family: times new roman,times; font-size: medium;">\(\ln \cos x\) up to and including the term in \({x^4}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln \cos x}}{{{x^n}}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, where \(n \in \mathbb{R}\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Using your result from (a), determine the set of values of \(n\) for which</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) the limit does not exist;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the limit is zero;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) the limit is finite and non-zero, giving its value in this case.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the series \(\sum\limits_{n = 1}^\infty {{u_n}} \) is convergent, where \({u_n} > 0\), show that the series \(\sum\limits_{n = 1}^\infty {u_n^2} \) is also convergent.</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>State the converse proposition.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By giving a suitable example, show that it is false.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sum the series \(\sum\limits_{r = 0}^\infty {{x^r}} \) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) <strong>Hence</strong>, using sigma notation, deduce a series for</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (a) \(\frac{1}{{1 + {x^2}}}\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (b) \(\arctan x\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (c) \(\frac{\pi }{6}\) .</span></p>
<div class="marks">[11]</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 \(\sum\limits_{n = 1}^{100} {n! \equiv 3(\bmod 15)} \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Assuming the Maclaurin series for \({{\text{e}}^x}\), determine the first three non-zero terms in the Maclaurin expansion of \(\frac{{{{\text{e}}^x} - {{\text{e}}^{ - x}}}}{2}\).</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;">(b) The random variable \(X\) has a Poisson distribution with mean \(\mu \). Show that \({\text{P}}\left( {X \equiv 1(\bmod 2)} \right) = a + b{{\text{e}}^{c\mu }}\) where \(a\), \(b\) and \(c\) are constants whose values are to be found.</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Use l’Hôpital’s rule to find \(\mathop {\lim }\limits_{x \to 0} (\csc x - \cot x)\).</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>QUESTION 1</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>QUESTION 1</p>
<div class="marks">[[N/A]]</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;">Using l’Hôpital’s Rule, determine the value of\[\mathop {\lim }\limits_{x \to 0} \frac{{\tan x - x}}{{1 - \cos x}} .\]<br></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the range of values of \(n\) for which \(\int_1^\infty {{x^n}{\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> exists.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the value of \(\int_1^\infty {{x^n}{\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> in terms of \(n\) , when it does exist.</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;">Find the solution to the differential equation</span></p>
<p style="text-align: center;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\((\cos x - \sin x)\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + (\cos x + \sin x)y = \cos x + \sin x\) ,<br></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">given that \(y = - 1\) when \(x = \frac{\pi }{2}\) </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.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The function \(f\) is defined by \(f(x) = {{\rm{e}}^x}\cos x\) .</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;">Show that \(f''(x) = - 2{{\rm{e}}^x}\sin x\) .</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;">Determine the Maclaurin series for \(f(x)\) up to and including the term in \({x^4}\) .</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;">By differentiating your series, determine the Maclaurin series for \({{\rm{e}}^x}\sin x\) up to </span><span style="font-family: times new roman,times; font-size: medium;">the term in \({x^3}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Differentiate the expression \({x^2}\tan y\) with respect to \(x\), where \(y\) is a function of \(x\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence solve the differential equation \({x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + x\sin 2y = {x^3}{\cos ^2}y\) given that \(y = 0\) when \(x = 1\). Give your answer in the form \(y = f(x)\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Solve the differential equation \(x\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + 2y = \sqrt {1 + {x^2}} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">given that \(y = 1\) when \(x = \sqrt 3 \) .</span></p>
</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;">Consider the infinite series \(S = \sum\limits_{n = 1}^\infty {\frac{{{x^n}}}{{{2^{2n}}\left( {2{n^2} - 1} \right)}}} \).</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;">(a) Determine the radius of convergence.</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;">(b) Determine the interval of convergence.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve the following differential equation\[(x + 1)(x + 2)\frac{{{\rm{d}}y}}{{{\rm{d}}x}} + y = x + 1\]giving your answer in the form \(y = f(x)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(\frac{{{\rm{d}}x}}{{{\rm{d}}y}} + 2y\tan x = \sin x\)</span><span style="font-family: times new roman,times; font-size: medium;"> , and \(y = 0\) when \(x = \frac{\pi }{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">, find the maximum value of <strong><em>y</em></strong>.</span></p>
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