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</div><h2>HL Paper 1</h2><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the sum of the infinite geometric sequence 27, −9, 3, −1, ... .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use mathematical induction to prove that for \(n \in {\mathbb{Z}^ + }\) ,</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;">\[a + ar + a{r^2} + ... + a{r^{n - 1}} = \frac{{a(1 - {r^n})}}{{1 - r}}.\]</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Find integer values of \(m\) and \(n\) for which</p>
<p class="p1">\[m - n{\log _3}2 = 10{\log _9}6\]</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Expand \({(x + h)^3}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the derivative of \(f(x) = {x^3}\) from first principles.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The fifth term of an arithmetic sequence is equal to <span class="s1">6 </span>and the sum of the first <span class="s1">12 </span>terms is <span class="s1">45</span>.</p>
<p class="p1">Find the first term and the common difference.</p>
</div>
<br><hr><br><div class="specification">
<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;">The complex numbers \({z_1} = 2 - 2{\text{i}}\) and \({{\text{z}}_2} = 1 - \sqrt 3 {\text{i}}\) are represented by the points A and B respectively on an Argand diagram. Given that O is the origin,</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: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find AB, giving your answer in the form \(a\sqrt {b - \sqrt 3 } \) , where <em>a</em> , \(b \in {\mathbb{Z}^ + }\) .<br></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate \({\rm{A\hat OB}}\) in terms of \(\pi \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve \(2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus and argument of \(z\) in terms of \(\theta \). Express each answer in its simplest form.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cube roots of \(z\) in modulus-argument form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the distinct complex numbers \(z = a + {\text{i}}b,\,\,w = c + {\text{i}}d\), where \(a,\,b,\,c,\,d \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the real part of \(\frac{{z + w}}{{z - w}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the real part of \(\frac{{z + w}}{{z - w}}\) when \(\left| z \right| = \left| w \right|\).</p>
<div class="marks">[2]</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: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the complex number \(\omega = \frac{{z + {\text{i}}}}{{z + 2}}\), where \(z = x + {\text{i}}y\) and \({\text{i}} = \sqrt { - 1} \).</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;">(a) If \(\omega = {\text{i}}\), determine <em>z</em> in the form \(z = r\,{\text{cis}}\,\theta \).</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;">(b) Prove that \(\omega = \frac{{({x^2} + 2x + {y^2} + y) + {\text{i}}(x + 2y + 2)}}{{{{(x + 2)}^2} + {y^2}}}\).</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;">(c) <strong>Hence</strong> show that when \(\operatorname{Re} (\omega) = 1\) the points \((x,{\text{ }}y)\) lie on a straight line, \({l_1}\), and write down its gradient.</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;">(d) Given \(\arg (z) = \arg (\omega) = \frac{\pi }{4}\), find \(\left| z \right|\).</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find three distinct roots of the equation \(8{z^3} + 27 = 0,{\text{ }}z \in \mathbb{C}\) <span class="s1">giving your answers in modulus-argument form.</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 class="p1">The roots are represented by the vertices of a triangle in an Argand diagram.</p>
<p class="p1">Show that the area of the triangle is \(\frac{{27\sqrt 3 }}{{16}}\).</p>
<div class="marks">[3]</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;">Consider the complex numbers \(z = 1 + 2{\text{i}}\) and \(w = 2 + a{\text{i}}\) , where \(a \in \mathbb{R}\) .</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;">Find <em>a</em> when</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;">(a) \(\left| w \right| = 2\left| z \right|;\) ;</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) \({\text{Re}}(zw) = 2\operatorname{Im} (zw)\) .</span></p>
</div>
<br><hr><br><div class="question">
<p>In the following Argand diagram the point A represents the complex number \( - 1 + 4{\text{i}}\) and the point B represents the complex number \( - 3 + 0{\text{i}}\). The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_06.11.20.png" alt="M17/5/MATHL/HP1/ENG/TZ2/05"></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Use mathematical induction to prove that \(n({n^2} + 5)\) <span class="s1">is divisible by 6 </span>for \(n \in {\mathbb{Z}^ + }\).</p>
</div>
<br><hr><br><div class="specification">
<p>It is given that \({\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x\) where \(r,\,x \in {\mathbb{R}^ + }\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express \(y\) in terms of \(x\). Give your answer in the form \(y = p{x^q}\), where <em>p</em> , <em>q</em> are constants.</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>The region <em>R</em>, is bounded by the graph of the function found in part (b), the <em>x</em>-axis, and the lines \(x = 1\) and \(x = \alpha \) where \(\alpha > 1\). The area of <em>R</em> is \(\sqrt 2 \).</p>
<p>Find the value of \(\alpha \).</p>
<div class="marks">[5]</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: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An 81 metre rope is cut into <em>n</em> pieces of increasing lengths that form an arithmetic sequence with a common difference of <em>d</em> metres. Given that the lengths of the shortest and longest pieces are 1.5 metres and 7.5 metres respectively, find the values of <em>n</em> and <em>d</em> .</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \(\frac{z}{{z + 2}} = 2 - {\text{i}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , \(z \in \mathbb{C}\) , find z in the form \(a + {\text{i}}b\) .</span></p>
</div>
<br><hr><br><div class="specification">
<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;">Given that \((4 - 5{\text{i}})m + 4n = 16 + 15{\text{i}}\) , where \({{\text{i}}^2} = - 1\), find <em>m </em>and <em>n </em>if</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>m </em>and <em>n </em>are real numbers;</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>m </em>and <em>n </em>are conjugate complex numbers.</span></p>
<div class="marks">[4]</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the complex number \(z = \cos \theta + {\text{i}}\sin \theta \).</span></p>
</div>
<div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The region <em>S</em> is bounded by the curve \(y = \sin x{\cos ^2}x\) and the <em>x</em>-axis between \(x = 0\) and \(x = \frac{\pi }{2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use De Moivre’s theorem to show that \({z^n} + {z^{ - n}} = 2\cos n\theta ,{\text{ }}n \in {\mathbb{Z}^ + }\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Expand \({\left( {z + {z^{ - 1}}} \right)^4}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Hence show that \({\cos ^4}\theta = p\cos 4\theta + q\cos 2\theta + r\), where \(p,{\text{ }}q\) and \(r\) are constants to </span><span style="font-family: 'times new roman', times; font-size: medium;">be determined.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({\cos ^6}\theta = \frac{1}{{32}}\cos 6\theta + \frac{3}{{16}}\cos 4\theta + \frac{{15}}{{32}}\cos 2\theta + \frac{5}{{16}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the value of \(\int_0^{\frac{\pi }{2}} {{{\cos }^6}\theta {\text{d}}\theta } \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>S</em> is rotated through \(2\pi \) radians about the <em>x</em>-axis. Find the value of the volume generated.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</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) Write down an expression for the constant term in the expansion of \({\left( {z + {z^{ - 1}}} \right)^{2k}}\), \(k \in {\mathbb{Z}^ + }\).</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;">(ii) Hence determine an expression for \(\int_0^{\frac{\pi }{2}} {{{\cos }^{2k}}\theta {\text{d}}\theta } \) in terms of <em>k</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>The geometric sequence <em>u</em><sub>1</sub>, <em>u</em><sub>2</sub>, <em>u</em><sub>3</sub>, … has common ratio <em>r.</em></p>
<p>Consider the sequence \(A = \left\{ {{a_n} = {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|{\text{:}}\,n \in {\mathbb{Z}^ + }} \right\}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>A</em> is an arithmetic sequence, stating its common difference<em> d</em> in terms of <em>r</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A particular geometric sequence has <em>u</em><sub>1</sub> = 3 and a sum to infinity of 4.</p>
<p>Find the value of <em>d</em>.</p>
<div class="marks">[3]</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;">Consider the following equations, where <em>a </em>, \(b \in \mathbb{R}:\)</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;">\(x + 3y + (a - 1)z = 1\)</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;">\(2x + 2y + (a - 2)z = 1\)</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;">\(3x + y + (a - 3)z = b.\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If each of these equations defines a plane, show that, for any value of <em>a </em>, the planes do not intersect at a unique point.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>b </em>for which the intersection of the planes is a straight line.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the complex numbers \({z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}\) and \(w = \frac{{{z_1}}}{{{z_2}}}\).</p>
</div>
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<p>By expressing \({z_1}\) and \({z_2}\) in modulus-argument form write down the modulus of \(w\);</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
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<p>By expressing \({z_1}\) and \({z_2}\) in modulus-argument form write down the argument of \(w\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
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<p>Find the smallest positive integer value of \(n\), such that \({w^n}\) is a real number.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<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;">Consider the complex numbers</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;">\({z_1} = 2\sqrt 3 {\text{cis}}\frac{{3\pi }}{2}\) and \({z_2} = - 1 + \sqrt 3 {\text{i }}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> .</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">(i) Write down \({z_1}\) in Cartesian form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">(ii) Hence determine \({({z_1} + {z_2})^ * }\) in Cartesian form.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<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 \({z_2}\) in modulus-argument form.</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) Hence solve the equation \({z^3} = {z_2}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
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<p>Let \(z = r\,{\text{cis}}\theta \) , where \(r \in {\mathbb{R}^ + }\) <span style="font: 7.0px Times;"> </span>and \(0 \leqslant \theta < 2\pi \) . Find all possible values of <em>r </em>and \(\theta \)<span style="font: 12.5px Times;"> ,</span></p>
<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;">(i) if \({z^2} = {(1 + {z_2})^2}\);</span></p>
<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;">(ii) if \(z = - \frac{1}{{{z_2}}}\).</span></p>
<div class="marks">[6]</div>
<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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the smallest positive value of <em>n </em>for which \({\left( {\frac{{{z_1}}}{{{z_2}}}} \right)^n} \in {\mathbb{R}^ + }\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) The sum of the first six terms of an arithmetic series is 81. The sum of its first eleven terms is 231. Find the first term and the common difference.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The sum of the first two terms of a geometric series is 1 and the sum of its first four terms is 5. If all of its terms are positive, find the first term and the common ratio.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) The \({r^{{\text{th}}}}\) term of a new series is defined as the product of the \({r^{{\text{th}}}}\) term of the arithmetic series and the \({r^{{\text{th}}}}\) term of the geometric series above. Show that the \({r^{{\text{th}}}}\) term of this new series is \((r + 1){2^{r - 1}}\) .</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">.</div>
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<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;">Using mathematical induction, prove that</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;">\[\sum\limits_{r = 1}^n {(r + 1){2^{r - 1}} = n{2^n},{\text{ }}n \in {\mathbb{Z}^ + }.} \]</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
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<p>The 1st, 4th and 8th terms of an arithmetic sequence, with common difference \(d\), \(d \ne 0\), are the first three terms of a geometric sequence, with common ratio \(r\). Given that the 1st term of both sequences is 9 find</p>
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<p>the value of \(d\);</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<p>the value of \(r\);</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
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<p>Show that \(\sin \left( {\theta + \frac{\pi }{2}} \right) = \cos \theta \).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
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<p class="p1">Consider \(f(x) = \sin (ax)\) where \(a\) is a constant. Prove by mathematical induction that \({f^{(n)}}(x) = {a^n}\sin \left( {ax + \frac{{n\pi }}{2}} \right)\) where \(n \in {\mathbb{Z}^ + }\) and \({f^{(n)}}(x)\) represents the \({{\text{n}}^{{\text{th}}}}\) derivative of \(f(x)\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</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;">Solve the equation \(2 - {\log _3}(x + 7) = {\log _{\tfrac{1}{3}}}2x\) .</span></p>
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<p class="p1">Solve the equation \({4^x} + {2^{x + 2}} = 3\).</p>
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<p class="p1">Write down and simplify the expansion of \({(2 + x)^4}\) <span class="s1">in ascending powers of \(x\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p class="p1">Hence find the exact value of \({(2.1)^4}\)<span class="s1">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<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;">A geometric sequence has first term <em>a</em>, common ratio <em>r</em> and sum to infinity 76. A second geometric sequence has first term <em>a</em>, common ratio \({r^3}\) and sum to infinity 36.</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;">Find <em>r</em>.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The sum of the first two terms of a geometric series is 10 and the sum of the first four terms is 30.</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) Show that the common ratio \(r\) satisfies \({r^2} = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given \(r = \sqrt 2 \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) find the first term;</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;"> (ii) find the sum of the first ten terms.</span></p>
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<p class="p1">Let \(z = \cos \theta + i\sin \theta \).</p>
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<p class="p1">Use de Moivre’s theorem to find the value of \({\left( {\cos \left( {\frac{\pi }{3}} \right) + {\text{i}}\sin \left( {\frac{\pi }{3}} \right)} \right)^3}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p class="p1">Use mathematical induction to prove that</p>
<p class="p1">\[{(\cos \theta - {\text{i}}\sin \theta )^n} = \cos n\theta - {\text{i}}\sin n\theta {\text{ for }}n \in {\mathbb{Z}^ + }.\]</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
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<p class="p1">Find an expression in terms of \(\theta \)<span class="s1"> for \({(z)^n} + {(z{\text{*}})^n},{\text{ }}n \in {\mathbb{Z}^ + }\) </span>where \(z{\text{*}}\) is the complex conjugate of \(z\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
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<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(zz{\text{*}} = 1\)<span class="s1">.</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Write down the binomial expansion of \({(z + z{\text{*}})^3}\) <span class="s2">in terms of \(z\) and \(z{\text{*}}\).</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>Hence show that \(\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
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<p class="p1">Hence solve \(4{\cos ^3}\theta - 2{\cos ^2}\theta - 3\cos \theta + 1 = 0\) for \(0 \leqslant \theta < \pi \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
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<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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Express each of the complex numbers \({z_1} = \sqrt 3 + {\text{i, }}{z_2} = - \sqrt 3 + {\text{i}}\) and \({z_3} = - 2{\text{i}}\) in modulus-argument form.</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) Hence show that the points in the complex plane representing \({z_1}\), \({z_2}\) and \({z_3}\) form the vertices of an equilateral triangle.</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;">(iii) Show that \({\text{z}}_1^{3n} + z_2^{3n} = 2z_3^{3n}\) where \(n \in \mathbb{N}\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
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<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;">(i) State the solutions of the equation \({z^7} = 1\) for \(z \in \mathbb{C}\), giving them in modulus-argument form.</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;">(ii) If <em>w</em> is the solution to \({z^7} = 1\) with least positive argument, determine the argument of 1 + <em>w</em>. Express your answer in terms of \(\pi \).</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;">(iii) Show that \({z^2} - 2z\cos \left( {\frac{{2\pi }}{7}} \right) + 1\) is a factor of the polynomial \({z^7} - 1\). State the two other quadratic factors with real coefficients.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
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<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;">Consider the complex numbers \({z_1} = 2{\text{cis}}150^\circ \) and \({z_2} = - 1 + {\text{i}}\) .</span></p>
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<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;">Calculate \(\frac{{{z_1}}}{{{z_2}}}\) giving your answer both in modulus-argument form and Cartesian form.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
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<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;">Using your results, find the exact value of tan 75° , giving your answer in the form \(a + \sqrt b \) , a , \(b \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(\frac{1}{{\sqrt n + \sqrt {n + 1} }} = \sqrt {n + 1} - \sqrt n \) where \(n \ge 0,{\text{ }}n \in \mathbb{Z}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that \(\sqrt 2 - 1 < \frac{1}{{\sqrt 2 }}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove, by mathematical induction, that \(\sum\limits_{r = 1}^{r = n} {\frac{1}{{\sqrt r }} > \sqrt n } \) for \(n \ge 2,{\text{ }}n \in \mathbb{Z}\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
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<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;">Prove by mathematical induction \(\sum\limits_{r = 1}^n {r(r!) = (n + 1)! - 1} \), \(n \in {\mathbb{Z}^ + }\).</span></p>
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<br><hr><br><div class="specification">
<p class="p1">The cubic equation \({x^3} + p{x^2} + qx + c = 0\)<span class="s1">, has roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \)</span>. By expanding \((x - \alpha )(x - \beta )(x - \gamma )\) show that</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) \(p = - (\alpha + \beta + \gamma )\);</p>
<p>(ii) \(q = \alpha \beta + \beta \gamma + \gamma \alpha \);</p>
<p>(iii) \(c = - \alpha \beta \gamma \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is now given that \(p = - 6\) and \(q = 18\) for parts (b) and (c) below.</p>
<p>(i) In the case that the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) form an arithmetic sequence, show that one of the roots is \(2\).</p>
<p>(ii) Hence determine the value of \(c\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<p class="p1">In another case the three roots \(\alpha ,{\text{ }}\beta ,{\text{ }}\gamma \) <span class="s1">form a geometric sequence. Determine the value of \(c\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
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<p class="p1">Let \(\{ {u_n}\} ,{\text{ }}n \in {\mathbb{Z}^ + }\), be an arithmetic sequence with first term equal to \(a\) and common difference of \(d\), where \(d \ne 0\). Let another sequence \(\{ {v_n}\} ,{\text{ }}n \in {\mathbb{Z}^ + }\), be defined by \({v_n} = {2^{{u_n}}}\).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(\frac{{{v_{n + 1}}}}{{{v_n}}}\) <span class="s1">is a constant.</span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Write down the first term of the sequence \(\{ {v_n}\} \).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Write down a formula for \({v_n}\) in terms of \(a\), \(d\) and \(n\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \({S_n}\) be the sum of the first \(n\) terms of the sequence \(\{ {v_n}\} \).</p>
<p>(i) Find \({S_n}\), in terms of \(a\), \(d\) and \(n\).</p>
<p>(ii) Find the values of \(d\) for which \(\sum\limits_{i = 1}^\infty {{v_i}} \) exists.</p>
<p>You are now told that \(\sum\limits_{i = 1}^\infty {{v_i}} \) does exist and is denoted by \({S_\infty }\).</p>
<p>(iii) Write down \({S_\infty }\) in terms of \(a\) and \(d\) .</p>
<p>(iv) Given that \({S_\infty } = {2^{a + 1}}\) find the value of \(d\) .</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(\{ {w_n}\} ,{\text{ }}n \in {\mathbb{Z}^ + }\), <span class="s1">be a geometric sequence with first term equal to \(p\) and common ratio \(q\), where \(p\) and \(q\) </span>are both greater than zero. Let another sequence \(\{ {z_n}\} \) be defined by \({z_n} = \ln {w_n}\).</p>
<p class="p1">Find \(\sum\limits_{i = 1}^n {{z_i}} \) giving your answer in the form \(\ln k\) <span class="s1">with \(k\) in terms of \(n\), \(p\) and \(q\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="question">
<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;">Express \(\frac{1}{{{{(1 - {\text{i}}\sqrt 3 )}^3}}}{\text{ in the form }}\frac{a}{b}{\text{ where }}a,{\text{ }}b \in \mathbb{Z}\) .</span></p>
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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;">Find the value of <em>k </em>if \({\sum\limits_{r = 1}^\infty{k\left( {\frac{1}{3}} \right)}^r} = 7\).</span></p>
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<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;">Given that \({z_1} = 2\) and \({z_2} = 1 + \sqrt 3 {\text{i}}\) are roots of the cubic equation \({z^3} + b{z^2} + cz + d = 0\)</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;">where <em>b</em>, <em>c</em>, \(d \in \mathbb{R}\),</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;">(a) write down the third root, \({z_3}\), of the equation;</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;">(b) find the values of <em>b</em>, <em>c</em> and <em>d</em> ;</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;">(c) write \({z_2}\) and \({z_3}\) in the form \(r{{\text{e}}^{{\text{i}}\theta }}\).</span></p>
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<br><hr><br><div class="specification">
<p>Consider the function \({f_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether \({f_n}\) is an odd or even function, justifying your answer.</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>By using mathematical induction, prove that</p>
<p style="text-align: center;">\({f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}\) where \(m \in \mathbb{Z}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find an expression for the derivative of \({f_n}(x)\) with respect to \(x\).</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>Show that, for \(n > 1\), the equation of the tangent to the curve \(y = {f_n}(x)\) at \(x = \frac{\pi }{4}\) is \(4x - 2y - \pi = 0\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="question">
<p>Use the method of mathematical induction to prove that \({4^n} + 15n - 1\) is divisible by 9 for \(n \in {\mathbb{Z}^ + }\).</p>
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<br><hr><br><div class="question">
<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;">Expand and simplify \({\left( {{x^2} - \frac{2}{x}} \right)^4}\).</span></p>
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<br><hr><br><div class="question">
<p class="p1"><span class="s1">Expand \({(3 - x)^4}\) </span>in ascending powers of \(x\) and simplify your answer.</p>
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<br><hr><br><div class="question">
<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;">(a) Consider the following sequence of equations.</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;"> \(1 \times 2 = \frac{1}{3}(1 \times 2 \times 3),\)</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;"> \(1 \times 2 + 2 \times 3 = \frac{1}{3}(2 \times 3 \times 4),\)</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;"> \(1 \times 2 + 2 \times 3 + 3 \times 4 = \frac{1}{3}(3 \times 4 \times 5),\)</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;"> \( \ldots {\text{ .}}\)</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;">(i) Formulate a conjecture for the \({n^{{\text{th}}}}\) equation in the sequence.</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) Verify your conjecture for <em>n</em> = 4 .</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;">(b) A sequence of numbers has the \({n^{{\text{th}}}}\) term given by \({u_n} = {2^n} + 3,{\text{ }}n \in {\mathbb{Z}^ + }\). Bill conjectures that all members of the sequence are prime numbers. Show that Bill’s conjecture is false.</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;">(c) Use mathematical induction to prove that \(5 \times {7^n} + 1\) is divisible by 6 for all \(n \in {\mathbb{Z}^ + }\).</span></p>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Show that \({(1 + {\text{i}}\tan \theta )^n} + {(1 - {\text{i}}\tan \theta )^n} = \frac{{2\cos n\theta }}{{{{\cos }^n}\theta }},\;\;\;\cos \theta \ne 0\).</p>
<p>(ii) Hence verify that \({\text{i}}\tan \frac{{3\pi }}{8}\) is a root of the equation \({(1 + z)^4} + {(1 - z)^4} = 0,\;\;\;z \in \mathbb{C}\).</p>
<p>(iii) State another root of the equation \({(1 + z)^4} + {(1 - z)^4} = 0,\;\;\;z \in \mathbb{C}\).</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>(i) Use the double angle identity \(\tan 2\theta = \frac{{2\tan \theta }}{{1 - {{\tan }^2}\theta }}\) to show that \(\tan \frac{\pi }{8} = \sqrt 2 - 1\).</p>
<p>(ii) Show that \(\cos 4x = 8{\cos ^4}x - 8{\cos ^2}x + 1\).</p>
<p>(iii) Hence find the value of \(\int_0^{\frac{\pi }{8}} {\frac{{2\cos 4x}}{{{{\cos }^2}x}}{\text{d}}x} \).</p>
<div class="marks">[13]</div>
<div class="question_part_label">b.</div>
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<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 equation \(9{x^3} - 45{x^2} + 74x - 40 = 0\) .</span></p>
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<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;">Write down the numerical value of the sum and of the product of the roots of this equation.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The roots of this equation are three consecutive terms of an arithmetic sequence.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Taking the roots to be \(\alpha {\text{ , }}\alpha \pm \beta \) , solve the equation.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
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<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;">(a) Show that the following system of equations has an infinite number of solutions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(x + y + 2z = - 2\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(3x - y + 14z = 6\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(x + 2y = - 5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The system of equations represents three planes in space.</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) Find the parametric equations of the line of intersection of the three planes.</span></p>
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<br><hr><br><div class="question">
<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;">(a) Show that \(\sin 2nx = \sin \left( {(2n + 1)x} \right)\cos x - \cos \left( {(2n + 1)x} \right)\sin x\).</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;">(b) <strong>Hence</strong> prove, by induction, that</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;">\[\cos x + \cos 3x + \cos 5x + \ldots + \cos \left( {(2n - 1)x} \right) = \frac{{\sin 2nx}}{{2\sin x}},\]</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;">for all \(n \in {\mathbb{Z}^ + }{\text{, }}\sin x \ne 0\).</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;">(c) Solve the equation \(\cos x + \cos 3x = \frac{1}{2},{\text{ }}0 < x < \pi \).</span></p>
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<br><hr><br><div class="question">
<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;">Find the cube roots of i in the form \(a + b{\text{i}}\), where \(a,{\text{ }}b \in \mathbb{R}\).</span></p>
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<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: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Factorize \({z^3} + 1\) into a linear and quadratic factor.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \(\gamma \) is one of the cube roots of −1.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that \({\gamma ^2} = \gamma - 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence find the value of \({(1 - \gamma )^6}\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p>An arithmetic sequence \({u_1}{\text{, }}{u_2}{\text{, }}{u_3} \ldots \) has \({u_1} = 1\) and common difference \(d \ne 0\). Given that \({u_2}{\text{, }}{u_3}\) and \({u_6}\) are the first three terms of a geometric sequence</p>
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<div class="specification">
<p>Given that \({u_N} = - 15\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of \(d\).</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>determine the value of \(\sum\limits_{r = 1}^N {{u_r}} \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="question">
<p>If \({z_1} = a + a\sqrt 3 i\) and \({z_2} = 1 - i\), where a is a real constant, express \({z_1}\) and \({z_2}\) in the form \(r\,{\text{cis}}\,\theta \), and hence find an expression for \({\left( {\frac{{{z_1}}}{{{z_2}}}} \right)^6}\) in terms of a and i.</p>
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<br><hr><br><div class="specification">
<p class="p1">Let \(w = \cos \frac{{2\pi }}{7} + {\text{i}}\sin \frac{{2\pi }}{7}\).</p>
</div>
<div class="specification">
<p class="p1">Consider the quadratic equation \({z^2} + bz + c = 0\) where \(b,{\text{ }}c \in \mathbb{R},{\text{ }}z \in \mathbb{C}\). The roots of this equation are \(\alpha \) and \(\alpha *\) where \(\alpha *\) is the complex conjugate of \(\alpha \).</p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Verify that \(w\) is a root of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\).</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">(i) <span class="Apple-converted-space"> </span>Expand \((w - 1)(1 + w + {w^2} + {w^3} + {w^4} + {w^5} + {w^6})\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence deduce that \(1 + w + {w^2} + {w^3} + {w^4} + {w^5} + {w^6} = 0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the roots of the equation \({z^7} - 1 = 0,{\text{ }}z \in \mathbb{C}\) in terms of \(w\) and plot these roots on an Argand diagram.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Given that \(\alpha = w + {w^2} + {w^4}\), show that \(\alpha * = {w^6} + {w^5} + {w^3}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(b\) and the value of \(c\).</p>
<div class="marks">[10]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using the values for \(b\) and \(c\) obtained in part (d)(ii), find the imaginary part of \(\alpha \), giving your answer in surd form.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(\omega \) be one of the non-real solutions of the equation \({z^3} = 1\).</p>
</div>
<div class="specification">
<p class="p1">Consider the complex numbers \(p = 1 - 3{\text{i}}\) and \(q = x + (2x + 1){\text{i}}\), where \(x \in \mathbb{R}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the value of</p>
<p class="p1">(i) <span class="Apple-converted-space"> \(1 + \omega + {\omega ^2}\)</span>;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(1 + \omega {\text{*}} + {(\omega {\text{*}})^2}\)</span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \((\omega - 3{\omega ^2})({\omega ^2} - 3\omega ) = 13\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find the values of \(x\) </span>that satisfy the equation \(\left| p \right| = \left| q \right|\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the inequality \(\operatorname{Re} (pq) + 8 < {\left( {\operatorname{Im} (pq)} \right)^2}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</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: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Expand and simplify \({\left( {x - \frac{2}{x}} \right)^4}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence determine the constant term in the expansion \((2{x^2} + 1){\left( {x - \frac{2}{x}} \right)^4}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(\sin \frac{\pi }{4} + \sin \frac{{3\pi }}{4} + \sin \frac{{5\pi }}{4} + \sin \frac{{7\pi }}{4} + \sin \frac{{9\pi }}{4}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(\frac{{1 - \cos 2x}}{{2\sin x}} \equiv \sin x,{\text{ }}x \ne k\pi \) <span class="s1">where \(k \in \mathbb{Z}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the principle of mathematical induction to prove that</p>
<p class="p1">\(\sin x + \sin 3x + \ldots + \sin (2n - 1)x = \frac{{1 - \cos 2nx}}{{2\sin x}},{\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}x \ne k\pi \) where \(k \in \mathbb{Z}\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence or otherwise solve the equation \(\sin x + \sin 3x = \cos x\) <span class="s1">in the interval \(0 < x < \pi \).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A box contains four red balls and two white balls. Darren and Marty play a game by each taking it in turn to take a ball from the box, without replacement. The first player to take a white ball is the winner.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Darren plays first, find the probability that he wins.</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">The game is now changed so that the ball chosen is replaced after each turn.</p>
<p class="p1">Darren still plays first.</p>
<p class="p1">Show that the probability of Darren winning has not changed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The sum of the first \(n\) terms of a sequence \(\{ {u_n}\} \) is given by \({S_n} = 3{n^2} - 2n\), where \(n \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of \({u_1}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \({u_6}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Prove that \(\{ {u_n}\} \) </span>is an arithmetic sequence, stating clearly its common difference.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A given polynomial function is defined as \(f(x) = {a_0} + {a_1}x + {a_2}{x^2} + \ldots + {a_n}{x^n}\). The roots of the polynomial equation \(f(x) = 0\) are consecutive terms of a geometric sequence with a common ratio of \(\frac{1}{2}\) and first term 2.</p>
<p>Given that \({a_{n - 1}} = - 63\) and \({a_n} = 16\) find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">the degree of the polynomial;</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">the value of \({a_0}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve \({\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}\).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A set of positive integers {\(1,2,3,4,5,6,7,8,9\)} is used to form a pack of nine cards.</p>
<p class="p1">Each card displays one positive integer without repetition from this set. Grace wishes to select four cards at random from this pack of nine cards.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the number of selections Grace could make if the largest integer drawn among the four cards is either a \(5\), a \(6\) or a \(7\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the number of selections Grace could make if at least two of the four integers drawn are even.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the equation \({\log _2}(x + 3) + {\log _2}(x - 3) = 4\).</p>
</div>
<br><hr><br><div class="question">
<p>Find the solution of \({\log _2}x - {\log _2}5 = 2 + {\log _2}3\).</p>
</div>
<br><hr><br><div class="specification">
<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;">Consider the following system of equations:</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;">\[x + y + z = 1\]</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;">\[2x + 3y + z = 3\]</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;">\[x + 3y - z = \lambda \]</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;">where \(\lambda \in \mathbb{R}\) .</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: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that this system does not have a unique solution for any value of \(\lambda \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the value of \(\lambda \) for which the system is consistent.</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) For this value of \(\lambda \) , find the general solution of the system.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation \({z^3} = 8{\text{i}},{\text{ }}z \in \mathbb{C}\) giving your answers in the form \(z = r(\cos \theta + {\text{i}}\sin \theta )\) <strong>and </strong>in the form \(z = a + b{\text{i}}\) where \(a,{\text{ }}b \in \mathbb{R}\).</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>Consider the complex numbers \({z_1} = 1 + {\text{i}}\) and \({z_2} = 2\left( {\cos \left( {\frac{\pi }{2}} \right) + {\text{i}}\sin \left( {\frac{\pi }{6}} \right)} \right)\).</p>
<p>(i) Write \({z_1}\) in the form \(r(\cos \theta + {\text{i}}\sin \theta )\).</p>
<p>(ii) Calculate \({z_1}{z_2}\) and write in the form \(z = a + b{\text{i}}\) where \(a,{\text{ }}b \in \mathbb{R}\).</p>
<p>(iii) Hence find the value of \(\tan \frac{{5\pi }}{{12}}\) in the form \(c + d\sqrt 3 \), where \(c,{\text{ }}d \in \mathbb{Z}\).</p>
<p>(iv) Find the smallest value \(p > 0\) such that \({({z_2})^p}\) is a positive real number.</p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
</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: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(z = x + {\text{i}}y\) be any non-zero complex number.</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) Express \(\frac{1}{z}\) in the form \(u + {\text{i}}v\) .</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) If \(z + \frac{1}{z} = k\) , \(k \in \mathbb{R}\) , show that either <em>y</em> = 0 or \({x^2} + {y^2} = 1\).</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;">(iii) Show that if \({x^2} + {y^2} = 1\) then \(\left| k \right| \leqslant 2\) .</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: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(w = \cos \theta + {\text{i}}\sin \theta \) .</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) Show that \({w^n} + {w^{ - n}} = 2\cos n\theta \) , \(n \in \mathbb{Z}\) .</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) Solve the equation \(3{w^2} - w + 2 - {w^{ - 1}} + 3{w^{ - 2}} = 0\), giving the roots in the form \(x + {\text{i}}y\) .</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Determine the roots of the equation \({(z + 2{\text{i}})^3} = 216{\text{i}}\), \(z \in \mathbb{C}\), giving the answers in the form \(z = a\sqrt 3 + b{\text{i}}\) where \(a,{\text{ }}b \in \mathbb{Z}\).</p>
</div>
<br><hr><br><div class="question">
<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;">The complex number <em>z</em> is defined as \(z = \cos \theta + {\text{i}}\sin \theta \) .</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;">(a) State de Moivre’s theorem.</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;">(b) Show that \({z^n} - \frac{1}{{{z^n}}} = 2{\text{i}}\sin (n\theta )\) .</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;">(c) Use the binomial theorem to expand \({\left( {z - \frac{1}{z}} \right)^5}\) giving your answer in simplified form.</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;">(d) Hence show that \(16{\sin ^5}\theta = \sin 5\theta - 5\sin 3\theta + 10\sin \theta \) .</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;">(e) Check that your result in part (d) is true for \(\theta = \frac{\pi }{4}\) .</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;">(f) Find \(\int_0^{\frac{\pi }{2}} {{{\sin }^5}\theta {\text{d}}\theta } \) .</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;">(g) Hence, with reference to graphs of circular functions, find \(\int_0^{\frac{\pi }{2}} {{{\cos }^5}\theta {\text{d}}\theta } \) , explaining your reasoning.</span></p>
</div>
<br><hr><br><div class="question">
<p>Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.</p>
</div>
<br><hr><br><div class="question">
<p class="p1">The following system of equations represents three planes in space.</p>
<p class="p1">\[x + 3y + z = - 1\]</p>
<p class="p1">\[x + 2y - 2z = 15\]</p>
<p class="p1">\[2x + y - z = 6\]</p>
<p class="p1">Find the coordinates of the point of intersection of the three planes.</p>
</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;">Write down the expansion of \({\left( {\cos \theta + {\text{i}}\sin \theta } \right)^3}\) in the form \(a + {\text{i}}b\) , where \(a\) and \(b\) </span><span style="font-family: times new roman,times; font-size: medium;">are in terms of \({\sin \theta }\) and \({\cos \theta }\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence show that \(\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Similarly show that \(\cos 5\theta = 16{\cos ^5}\theta - 20{\cos ^3}\theta + 5\cos \theta \) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Hence</strong> solve the equation \(\cos 5\theta + \cos 3\theta + \cos \theta = 0\) , where \(\theta \in \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">By considering the solutions of the equation \(\cos 5\theta = 0\) , show that </span><span style="font-family: times new roman,times; font-size: medium;">\(\cos \frac{\pi }{{10}} = \sqrt {\frac{{5 + \sqrt 5 }}{8}} \)</span><span style="font-family: times new roman,times; font-size: medium;"> and state the value of \(\cos \frac{{7\pi }}{{10}}\)</span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<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;">Given that \(y = \frac{1}{{1 - x}}\), use mathematical induction to prove that \(\frac{{{{\text{d}}^n}y}}{{{\text{d}}{x^n}}} = \frac{{n!}}{{{{(1 - x)}^{n + 1}}}},{\text{ }}n \in {\mathbb{Z}^ + }\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Consider the expansion of \({(1 + x)^n}\) </span>in ascending powers of \(x\), where \(n \geqslant 3\).</p>
</div>
<div class="specification">
<p class="p1">The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the first four terms of the expansion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \({n^3} - 9{n^2} + 14n = 0\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence find the value of \(n\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.<br>Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-07_om_14.39.35.png" alt="N17/5/MATHL/HP1/ENG/TZ0/10"></p>
<p>Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.<br>Chloe wins if <strong>no</strong> matches occur; otherwise Selena wins.</p>
</div>
<div class="specification">
<p>Chloe and Selena repeat their game so that they play a total of 50 times.<br>Suppose the discrete random variable <em>X </em>represents the number of times Chloe wins.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the probability that Chloe wins the game is \(\frac{3}{8}\).</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>Determine the mean of <em>X</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the variance of <em>X</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the complex numbers \(u = 2 + 3{\text{i}}\) and \(v = 3 + 2{\text{i}}\).</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) Given that \(\frac{1}{u} + \frac{{1}}{v} = \frac{{10}}{w}\), express <em>w </em>in the form \(a + b{\text{i, }}a,{\text{ }}b \in \mathbb{R}\).</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) Find \(w\)* and express it in the form \(r{e^{{\text{i}}\theta }}\).</span></p>
</div>
<br><hr><br><div class="specification">
<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;">An arithmetic sequence has first term <em>a</em> and common difference <em>d</em>, \(d \ne 0\) . The \({{\text{3}}^{{\text{rd}}}}\), \({{\text{4}}^{{\text{th}}}}\) and \({{\text{7}}^{{\text{th}}}}\) terms of the arithmetic sequence are the first three terms of a </span><span style="font-family: 'times new roman', times; font-size: medium;">geometric sequence.</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: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(a = - \frac{3}{2}d\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the \({{\text{4}}^{{\text{th}}}}\) term of the geometric sequence is the \({\text{1}}{{\text{6}}^{{\text{th}}}}\) term of the arithmetic sequence.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Use the principle of mathematical induction to prove that</p>
<p>\(1 + 2\left( {\frac{1}{2}} \right) + 3{\left( {\frac{1}{2}} \right)^2} + 4{\left( {\frac{1}{2}} \right)^3} + \, \ldots \, + n{\left( {\frac{1}{2}} \right)^{n - 1}} = 4 - \frac{{n + 2}}{{{2^{n - 1}}}}\), where \(n \in {\mathbb{Z}^ + }\).</p>
</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;">Expand \({(2 - 3x)^5}\) in ascending powers of <em>x</em>, simplifying coefficients.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The random variable \(X\) has the Poisson distribution \({\text{Po}}(m)\). Given that \({\text{P}}(X > 0) = \frac{3}{4}\), find the value of \(m\) in the form \(\ln a\) where \(a\) is an integer.</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>The random variable \(Y\) has the Poisson distribution \({\text{Po}}(2m)\). Find \({\text{P}}(Y > 1)\) in the form \(\frac{{b - \ln c}}{c}\) where \(b\) and \(c\) are integers.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Use mathematical induction to prove that \((2n)! \ge {2^n}{(n!)^2},{\text{ }}n \in {\mathbb{Z}^ + }\).</p>
</div>
<br><hr><br><div class="question">
<p>Prove by mathematical induction that \(\left( {\begin{array}{*{20}{c}} 2 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 3 \\ 2 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 4 \\ 2 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right)\), where \(n \in \mathbb{Z},n \geqslant 3\).</p>
</div>
<br><hr><br><div class="question">
<p>Find the coefficient of \({x^8}\) in the expansion of \({\left( {{x^2} - \frac{2}{x}} \right)^7}\).</p>
</div>
<br><hr><br><div class="question">
<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;">Solve the equation \({4^{x - 1}} = {2^x} + 8\).</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;">A geometric sequence \(\left\{ {{u_n}} \right\}\), with complex terms, is defined by \({u_{n + 1}} = (1 + {\text{i}}){u_n}\) and \({u_1} = 3\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the fourth term of the sequence, giving your answer in the form \(x + y{\text{i, }}x,{\text{ }}y \in \mathbb{R}\).</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) Find the sum of the first 20 terms of \(\left\{ {{u_n}} \right\}\), giving your answer in the form \(a \times (1 + {2^m})\) where \(a \in \mathbb{C}\) and \(m \in \mathbb{Z}\) are to be determined.</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 second sequence \(\left\{ {{v_n}} \right\}\) is defined by \({v_n} = {u_n}{u_{n + k}},{\text{ }}k \in \mathbb{N}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Show that \(\left\{ {{v_n}} \right\}\) is a geometric sequence.</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;"> (ii) State the first term.</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;"> (iii) Show that the common ratio is independent of <em>k</em>.</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 third sequence \(\left\{ {{w_n}} \right\}\) is defined by \({w_n} = \left| {{u_n} - {u_{n + 1}}} \right|\).</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;">(d) (i) Show that \(\left\{ {{w_n}} \right\}\) is a geometric sequence.</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;"> (ii) State the geometrical significance of this result with reference to points on the complex plane.</span></p>
</div>
<br><hr><br><div class="specification">
<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;">On Saturday, Alfred and Beatrice play 6 different games against each other. In each game, one of the two wins. The probability that Alfred wins any one of these games is \(\frac{2}{3}\).</span></p>
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<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;">Show that the probability that Alfred wins exactly 4 of the games is \(\frac{{80}}{{243}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<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) Explain why the total number of possible outcomes for the results of the 6 games is 64.</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) By expanding \({(1 + x)^6}\) and choosing a suitable value for <em>x</em>, prove</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;">\[64 = \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 0 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 1 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 2 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 3 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 4 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 5 <br>\end{array}} \right) + \left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 6 <br>\end{array}} \right)\]</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) State the meaning of this equality in the context of the 6 games played.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<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;">The following day Alfred and Beatrice play the 6 games again. Assume that the probability that Alfred wins any one of these games is still \(\frac{2}{3}\).</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;">(i) Find an expression for the probability Alfred wins 4 games on the first day and 2 on the second day. Give your answer in the form \({\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> r <br>\end{array}} \right)^2}{\left( {\frac{2}{3}} \right)^s}{\left( {\frac{1}{3}} \right)^t}\) where the values of <em>r</em>, <em>s</em> and <em>t</em> are to be found.</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) Using your answer to (c) (i) and 6 similar expressions write down the probability that Alfred wins a total of 6 games over the two days as the sum of 7 probabilities.</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;">(iii) Hence prove that \(\left( {\begin{array}{*{20}{c}}<br> {12} \\ <br> 6 <br>\end{array}} \right) = {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 0 <br>\end{array}} \right)^2} + {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 1 <br>\end{array}} \right)^2} + {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 2 <br>\end{array}} \right)^2} + {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 3 <br>\end{array}} \right)^2} + {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 4 <br>\end{array}} \right)^2} + {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 5 <br>\end{array}} \right)^2} + {\left( {\begin{array}{*{20}{c}}<br> 6 \\ <br> 6 <br>\end{array}} \right)^2}\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
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<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;">Alfred and Beatrice play <em>n</em> games. Let <em>A</em> denote the number of games Alfred wins. The expected value of <em>A</em> can be written as \({\text{E}}(A) = \sum\limits_{r = 0}^n {r\left( {\begin{array}{*{20}{c}}<br> n \\ <br> r <br>\end{array}} \right)} \frac{{{a^r}}}{{{b^n}}}\).</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;">(i) Find the values of <em>a</em> and <em>b</em>.</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;">(ii) By differentiating the expansion of \({(1 + x)^n}\), prove that the expected number of games Alfred wins is \(\frac{{2n}}{3}\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The function <em>f</em> is defined by \(f(x) = {{\text{e}}^x}\sin x\) .</span></p>
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<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;">Show that \(f''(x) = 2{{\text{e}}^x}\sin \left( {x + \frac{\pi }{2}} \right)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<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;">Obtain a similar expression for \({f^{(4)}}(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<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;">Suggest an expression for \({f^{(2n)}}(x)\), \(n \in {\mathbb{Z}^ + }\), and prove your conjecture using mathematical induction.</span></p>
<div class="marks">[8]</div>
<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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Expand and simplify \({\left( {\frac{x}{y} - \frac{y}{x}} \right)^4}\)<span style="font: 7.0px Helvetica;">.</span></span></p>
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<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;">If <em>w</em> = 2 + 2i , find the modulus and argument of <em>w</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<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;">Given \(z = \cos \left( {\frac{{5\pi }}{6}} \right) + {\text{i}}\sin \left( {\frac{{5\pi }}{6}} \right)\), find in its simplest form \({w^4}{z^6}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<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;">Given the complex numbers \({z_1} = 1 + 3{\text{i}}\) and \({z_2} = - 1 - {\text{i}}\).</span></p>
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<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;">Write down the exact values of \(\left| {{z_1}} \right|\) and \(\arg ({z_2})\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the minimum value of \(\left| {{z_1} + \alpha{z_2}} \right|\), where \(\alpha \in \mathbb{R}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(w = \cos \frac{{2\pi }}{5} + {\text{i}}\sin \frac{{2\pi }}{5}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>w</em> is a root of the equation \({z^5} - 1 = 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that \((w - 1)({w^4} + {w^3} + {w^2} + w + 1) = {w^5} - 1\) and deduce that \({w^4} + {w^3} + {w^2} + w + 1 = 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <strong>Hence</strong> show that \(\cos \frac{{2\pi }}{5} + \cos \frac{{4\pi }}{5} = - \frac{1}{2}\).</span></p>
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<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;">The mean of the first ten terms of an arithmetic sequence is 6. The mean of the first twenty terms of the arithmetic sequence is 16. Find the value of the \({15^{{\text{th}}}}\) term of the sequence.</span></p>
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<p><span style="font-family: times new roman,times; font-size: medium;">A geometric sequence \({u_1}\) , \({u_2}\) , \({u_3}\) , \(...\) has \({u_1} = 27\) and a sum to infinity of \(\frac{{81}}{2}\).</span></p>
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<p><span style="font-family: times new roman,times; font-size: medium;">Find the common ratio of the geometric sequence.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">An arithmetic sequence </span><span style="font-family: times new roman,times; font-size: medium;">\({v_1}\) , \({v_2}\) , \({v_3}\) , \(...\) is such that \({v_2} = {u_2}\) and \({v_4} = {u_4}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the greatest value of \(N\) such that \(\sum\limits_{n = 1}^N {{v_n}} > 0\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
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<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: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the definition of a derivative as \(f'(x) = \mathop {\lim }\limits_{h \to 0} \left( {\frac{{f(x + h) - f(x)}}{h}} \right)\) , show that the derivative of \(\frac{1}{{2x + 1}}{\text{ is }}\frac{{ - 2}}{{{{(2x + 1)}^2}}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</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;">Prove by induction that the \({n^{{\text{th}}}}\) derivative of \({(2x + 1)^{ - 1}}\) is \({( - 1)^n}\frac{{{2^n}n!}}{{{{(2x + 1)}^{n + 1}}}}\).</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
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<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;">Prove by mathematical induction that \({n^3} + 11n\) is divisible by 3 for all \(n \in {\mathbb{Z}^ + }\).</span></p>
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<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;">The sum, \({S_n}\), of the first <em>n</em> terms of a geometric sequence, whose \({n^{{\text{th}}}}\) term is \({u_n}\), is given by</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;">\[{S_n} = \frac{{{7^n} - {a^n}}}{{{7^n}}},{\text{ where }}a > 0.\]</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;">(a) Find an expression for \({u_n}\).</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;">(b) Find the first term and common ratio of the sequence.</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;">(c) Consider the sum to infinity of the sequence.</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;">(i) Determine the values of <em>a</em> such that the sum to infinity exists.</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;">(ii) Find the sum to infinity when it exists.</span></p>
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<p class="p1">Let \(y(x) = x{e^{3x}},{\text{ }}x \in \mathbb{R}\).</p>
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<p class="p1">Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove by induction that \(\frac{{{{\text{d}}^n}y}}{{{\text{d}}{x^n}}} = n{3^{n - 1}}{{\text{e}}^{3x}} + x{3^n}{{\text{e}}^{3x}}\) for \(n \in {\mathbb{Z}^ + }\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
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<p class="p1">Find the coordinates of any local maximum and minimum points on the graph of \(y(x)\).</p>
<p class="p1">Justify whether any such point is a maximum or a minimum.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
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<p class="p1">Find the coordinates of any points of inflexion on the graph of \(y(x)\). Justify whether any such point is a point of inflexion.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
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<p class="p1">Hence sketch the graph of \(y(x)\), indicating clearly the points found in parts (c) and (d) and any intercepts with the axes.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The first three terms of a geometric sequence are \(\sin x,{\text{ }}\sin 2x\) and \(4\sin x{\cos ^2}x,{\text{ }} - \frac{\pi }{2} < x < \frac{\pi }{2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the common ratio <em>r</em>.</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) Find the set of values of <em>x </em>for which the geometric series \(\sin x + \sin 2x + 4\sin x{\cos ^2}x + \ldots \) converges.</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;">Consider \(x = \arccos \left( {\frac{1}{4}} \right),{\text{ }}x > 0\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that the sum to infinity of this series is \(\frac{{\sqrt {15} }}{2}\).</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider \(a = {\log _2}3 \times {\log _3}4 \times {\log _4}5 \times \ldots \times {\log _{31}}32\). Given that \(a \in \mathbb{Z}\), find the value of <em>a</em>.</span></p>
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<p>Find the term independent of \(x\) in the binomial expansion of \({\left( {2{x^2} + \frac{1}{{2{x^3}}}} \right)^{10}}\).</p>
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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;">Given that <em>z </em>is the complex number \(x + {\text{i}}y\) and that \(\left| {\,z\,} \right| + z = 6 - 2{\text{i}}\) , find the value of <em>x</em></span></p>
<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;">and the value of <em>y </em>.</span></p>
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<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;">Consider a function <em>f </em>, defined by \(f(x) = \frac{x}{{2 - x}}{\text{ for }}0 \leqslant x \leqslant 1\) .</span></p>
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<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 \((f \circ f)(x)\) .</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;"> </span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<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;">Let \({F_n}(x) = \frac{x}{{{2^n} - ({2^n} - 1)x}}\), where \(0 \leqslant x \leqslant 1\).</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;">Use mathematical induction to show that for any \(n \in {\mathbb{Z}^ + }\)</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;">\[\underbrace {(f \circ f \circ ... \circ f)}_{n{\text{ times}}}(x) = {F_n}(x)\] .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({F_{ - n}}(x)\) is an expression for the inverse of \({F_n}\) .</span></p>
<div class="marks">[6]</div>
<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 Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State \({F_n}(0){\text{ and }}{F_n}(1)\) .</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) Show that \({F_n}(x) < x\) , given 0 < <em>x </em>< 1, \(n \in {\mathbb{Z}^ + }\) .</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;">(iii) For \(n \in {\mathbb{Z}^ + }\) , let \({A_n}\) be the area of the region enclosed by the graph of \(F_n^{ - 1}\) , the <em>x</em>-axis and the line <em>x </em>= 1. Find the area \({B_n}\) of the region enclosed by \({F_n}\) and \(F_n^{ - 1}\) in terms of \({A_n}\) .<span style="font: 7.0px Helvetica;"><br></span></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
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<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;">The common ratio of the terms in a geometric series is \({2^x}\) .</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) State the set of values of <em>x</em> for which the sum to infinity of the series exists.</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;">(b) If the first term of the series is 35, find the value of <em>x</em> for which the sum to infinity is 40.</span></p>
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<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;">If <em>z</em> is a non-zero complex number, we define \(L(z)\) by the equation</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;">\[L(z) = \ln \left| z \right| + {\text{i}}\arg (z),{\text{ }}0 \leqslant \arg (z) < 2\pi .\]</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;">(a) Show that when <em>z</em> is a positive real number, \(L(z) = \ln z\) .</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;">(b) Use the equation to calculate</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;">(i) \(L( - 1)\) ;</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;">(ii) \(L(1 - {\text{i}})\) ;</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;">(iii) \(L( - 1 + {\text{i}})\) .</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;">(c) Hence show that the property \(L({z_1}{z_2}) = L({z_1}) + L({z_2})\) does not hold for all values of \({z_1}\) and \({z_2}\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">Part A.</div>
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<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;">Let <em>f</em> be a function with domain \(\mathbb{R}\) that satisfies the conditions,</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;">\(f(x + y) = f(x)f(y)\) , for all <em>x</em> and <em>y</em> and \(f(0) \ne 0\) .</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;">(a) Show that \(f(0) = 1\).</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) Prove that \(f(x) \ne 0\) , for all \(x \in \mathbb{R}\) .</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) Assuming that \(f'(x)\) exists for all \(x \in \mathbb{R}\) , use the definition of derivative to show that \(f(x)\) satisfies the differential equation \(f'(x) = k{\text{ }}f(x)\) , where \(k = f'(0)\) .</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;">(d) Solve the differential equation to find an expression for \(f(x)\) .</span></p>
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<div class="question_part_label">Part B.</div>
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<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;">The first terms of an arithmetic sequence are \(\frac{1}{{{{\log }_2}x}},{\text{ }}\frac{1}{{{{\log }_8}x}},{\text{ }}\frac{1}{{{{\log }_{32}}x}},{\text{ }}\frac{1}{{{{\log }_{128}}x}},{\text{ }} \ldots \)</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;">Find <em>x</em> if the sum of the first 20 terms of the sequence is equal to 100.</span></p>
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<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;">Solve the equation \({8^{x - 1}} = {6^{3x}}\). Express your answer in terms of \(\ln 2\) and \(\ln 3\).</span></p>
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<p>Consider \(w = 2\left( {{\text{cos}}\frac{\pi }{3} + {\text{i}}\,{\text{sin}}\frac{\pi }{3}} \right)\)</p>
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<p>These four points form the vertices of a quadrilateral, <em>Q</em>.</p>
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<p>Express <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup> in modulus-argument form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
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<p>Sketch on an Argand diagram the points represented by <em>w</em><sup>0</sup> , <em>w</em><sup>1</sup> , <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
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<p>Show that the area of the quadrilateral <em>Q</em> is \(\frac{{21\sqrt 3 }}{2}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<p>Let \(z = 2\left( {{\text{cos}}\frac{\pi }{n} + {\text{i}}\,{\text{sin}}\frac{\pi }{n}} \right),\,\,n \in {\mathbb{Z}^ + }\). The points represented on an Argand diagram by \({z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}\) form the vertices of a polygon \({P_n}\).</p>
<p>Show that the area of the polygon \({P_n}\) can be expressed in the form \(a\left( {{b^n} - 1} \right){\text{sin}}\frac{\pi }{n}\), where \(a,\,\,b\, \in \mathbb{R}\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
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<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;">Find the values of <em>n</em> such that \({\left( {1 + \sqrt 3 {\text{i}}} \right)^n}\) is a real number.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Use de Moivre’s theorem to find the roots of the equation \({z^4} = 1 - {\text{i}}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Draw these roots on an Argand diagram.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) If \({{\text{z}}_1}\) is the root in the first quadrant and \({{\text{z}}_2}\) is the root in the second quadrant, find \(\frac{{{{\text{z}}_2}}}{{{{\text{z}}_1}}}\) in the form <em>a</em> + i<em>b</em> .</span></p>
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<div class="question_part_label">Part A.</div>
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<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) Expand and simplify \((x - 1)({x^4} + {x^3} + {x^2} + x + 1)\) .</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;">(b) Given that <em>b</em> is a root of the equation \({z^5} - 1 = 0\) which does not lie on the real axis in the Argand diagram, show that \(1 + b + {b^2} + {b^3} + {b^4} = 0\) .</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;">(c) If \(u = b + {b^4}\) and \(v = {b^2} + {b^3}\) show that</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) <em>u</em> + <em>v</em> = <em>uv</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;">(ii) \(u - v = \sqrt 5 \) , given that \(u - v > 0\) .</span></p>
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<div class="question_part_label">Part B.</div>
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<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;">Consider \(\omega= \cos \left( {\frac{{2\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{2\pi }}{3}} \right)\).</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;">(a) Show that</span></p>
<p style="margin: 0px 0px 0px 30px; font: 31px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) \({\omega^3} = 1;\)</span></p>
<p style="margin: 0px 0px 0px 30px; font: 31px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \(1 + \omega+ {\omega^2} = 0\)</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;">(b) (i) Deduce that \({{\text{e}}^{{\text{i}}\theta }} + {{\text{e}}^{{\text{i}}\left( {\theta + \frac{{2\pi }}{3}} \right)}} + {{\text{e}}^{{\text{i}}\left( {\theta + \frac{{4\pi }}{3}} \right)}} = 0\).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 31px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Illustrate this result for \(\theta = \frac{\pi }{2}\) on an Argand diagram.</span><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;">(c) (i) Expand and simplify \(F(z) = (z - 1)(z - \omega)(z - {\omega^2})\) where <em>z</em> is a complex number.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 31px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Solve \(F(z) = 7\), giving your answers in terms of \(\omega\).</span></p>
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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;">The graph of a polynomial function <em>f </em>of degree 4 is shown below.</span></p>
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<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><img 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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \({(x + {\text{i}}y)^2} = - 5 + 12{\text{i}},{\text{ }}x,{\text{ }}y \in \mathbb{R}\) . Show that</span></p>
<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;">(i) \({x^2} - {y^2} = - 5\) ;</span></p>
<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;">(ii) \(xy = 6\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A.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;">Hence find the two square roots of \( - 5 + 12{\text{i}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">A.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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For any complex number <em>z </em>, show that \({(z^*)^2} = ({z^2})^*\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">A.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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence write down the two square roots of \( - 5 - 12{\text{i}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A.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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why, of the four roots of the equation \(f(x) = 0\) , two are real and two are complex.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The curve passes through the point \(( - 1,\, - 18)\) . Find \(f(x)\) in the form</span></p>
<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;">\(f(x) = (x - a)(x - b)({x^2} + cx + d),{\text{ where }}a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\)<em> </em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">B.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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the two complex roots of the equation \(f(x) = 0\) in Cartesian form.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw the four roots on the complex plane (the Argand diagram).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.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 Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Express each of the four roots of the equation in the form \(r{{\text{e}}^{{\text{i}}\theta }}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">B.e.</div>
</div>
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