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</div><h2>SL Paper 1</h2><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({\log _2}32\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\log _2}\left( {\frac{{{{32}^x}}}{{{8^y}}}} \right)\) can be written as \(px + qy\) , find the value of <em>p</em> and of <em>q</em>.</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;">Find the value of each of the following, giving your answer as an integer.</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;">\({\log _6}36\)</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\log _6}4 + {\log _6}9\)</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 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;">\({\log _6}2 - {\log _6}12\)</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Expand \({(2 + x)^4}\) and simplify your result.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence, find the term in \({x^2}\) in \({(2 + x)^4}\left( {1 + \frac{1}{{{x^2}}}} \right)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">An arithmetic sequence has the first term \(\ln a\) and a common difference \(\ln 3\).</p>
<p class="p1">The 13th term in the sequence is \(8\ln 9\)<span class="s1">. Find the value of \(a\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the following sequence of figures.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-01-31_om_17.56.15.png" alt="M16/5/MATME/SP1/ENG/TZ1/04"></p>
<p class="p1">Figure <span class="s1">1 </span>contains <span class="s1">5 </span>line segments.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that Figure \(n\) <span class="s1">contains 801 </span>line segments, show that \(n = 200\).</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 total number of line segments in the first <span class="s1">200 </span>figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = k{\log _2}x\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({f^{ - 1}}\left( {\frac{2}{3}} \right)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, the first term is 3 and the second term is 7.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the tenth term.</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>Find the sum of the first ten terms of the sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The first three terms of a infinite geometric sequence are \(m - 1,{\text{ 6, }}m + 4\), where \(m \in \mathbb{Z}\).</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;">Write down an expression for the common ratio, \(r\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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, show that \(m\) satisfies the equation \({m^2} + 3m - 40 = 0\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the two possible values of \(m\).</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the possible values of \(r\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The sequence has a finite sum.</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;">State which value of \(r\) leads to this sum <strong>and </strong>justify your answer.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The sequence has a finite sum.</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;">Calculate the sum of the sequence.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c(ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In an arithmetic sequence, the first term is \(2\) and the second term is \(5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the common difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the eighth term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the sum of the first eight terms of the sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write the expression \(3\ln 2 - \ln 4\) in the form \(\ln k\), where \(k \in \mathbb{Z}\).</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>Hence or otherwise, solve \(3\ln 2 - \ln 4 = - \ln x\).</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: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The sums of the terms of a sequence follow the pattern</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;">\({S_1} = 1 + k,{\text{ }}{S_2} = 5 + 3k,{\text{ }}{S_3} = 12 + 7k,{\text{ }}{S_4} = 22 + 15k,{\text{ }} \ldots ,{\text{ where }}k \in \mathbb{Z}.\)</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;">Given that \({u_1} = 1 + k\), find \({u_2},{\text{ }}{u_3}\) and \({u_4}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find a general expression for \({u_n}\).</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><span style="font-family: times new roman,times; font-size: medium;">Consider the arithmetic sequence \(2{\text{, }}5{\text{, }}8{\text{, }}11{\text{,}} \ldots \) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({u_{101}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the arithmetic sequence \(2{\text{, }}5{\text{, }}8{\text{, }}11{\text{,}} \ldots \) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>n</em> so that \({u_n} = 152\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first two terms of an infinite geometric sequence, in order, are</p>
<p class="p1" style="text-align: center;">\(2{\log _2}x,{\text{ }}{\log _2}x\), where \(x > 0\).</p>
</div>
<div class="specification">
<p class="p1">The first three terms of an arithmetic sequence, in order, are</p>
<p class="p1" style="text-align: center;">\({\log _2}x,{\text{ }}{\log _2}\left( {\frac{x}{2}} \right),{\text{ }}{\log _2}\left( {\frac{x}{4}} \right)\), where \(x > 0\).</p>
</div>
<div class="specification">
<p class="p1">Let \({S_{12}}\) be the sum of the first <span class="s1">12 </span>terms of the arithmetic sequence.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(r\).</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 class="p1">Show that the sum of the infinite sequence is \(4{\log _2}x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Find \(d\)</span>, giving your answer as an integer.</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 class="p1">Show that \({S_{12}} = 12{\log _2}x - 66\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given that \({S_{12}}\) </span>is equal to half the sum of the infinite geometric sequence, find \(x\), giving your answer in the form \({2^p}\), where \(p \in \mathbb{Q}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The fifth term in the expansion of the binomial \({(a + b)^n}\) is given by \(\left( {\begin{array}{*{20}{c}}<br>{10}\\<br>4<br>\end{array}} \right){p^6}{(2q)^4}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(n\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down <em>a</em> and <em>b</em>, in terms of <em>p</em> and/or <em>q</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down an expression for the sixth term in the expansion.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>An arithmetic sequence has \({u_1} = {\text{lo}}{{\text{g}}_c}\left( p \right)\) and \({u_2} = {\text{lo}}{{\text{g}}_c}\left( {pq} \right)\), where \(c > 1\) and \(p,\,\,q > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(d = {\text{lo}}{{\text{g}}_c}\left( q \right)\).</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>Let \(p = {c^2}\) and \(q = {c^3}\). Find the value of \(\sum\limits_{n = 1}^{20} {{u_n}} \).</p>
<div class="marks">[6]</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 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">In an arithmetic sequence, the third term is 10 and the fifth term is 16.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the common difference.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the first term.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the sum of the first 20 terms of the sequence.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the infinite geometric sequence \(3{\text{, }}3(0.9){\text{, }}3{(0.9)^2}{\text{, }}3{(0.9)^3}{\text{, }} \ldots \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the 10th term of the sequence. Do not simplify your answer.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the infinite geometric sequence \(3{\text{, }}3(0.9){\text{, }}3{(0.9)^2}{\text{, }}3{(0.9)^3}{\text{, }} \ldots \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the sum of the infinite sequence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = lo{g_3}\sqrt x \) , for \(x > 0\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \({f^{ - 1}}(x) = {3^{2x}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of \({f^{ - 1}}\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {\log _3}x\) , for \(x > 0\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(({f^{ - 1}} \circ g)(2)\) , giving your answer as an integer.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The first three terms of an infinite geometric sequence are 32, 16 and 8.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({u_6}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the sum to infinity of this sequence.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The first two terms of an infinite geometric sequence are <em>u</em><sub>1</sub> = 18 and <em>u</em><sub>2</sub> = 12sin<sup>2</sup> <em>θ</em> , where 0 < <em>θ</em> < 2\(\pi \) , and <em>θ</em> ≠ \(\pi \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <em>r</em> in terms of <em>θ</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the possible values of <em>r</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the sum of the infinite sequence is \(\frac{{54}}{{2 + {\text{cos}}\,\left( {2\theta } \right)}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <em>θ</em> which give the greatest value of the sum.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Three consecutive terms of a geometric sequence are \(x - 3\)<span class="s1">, 6 </span>and \(x + 2\).</p>
<p class="p1">Find the possible values of \(x\).</p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The values in the fourth row of Pascal’s triangle are shown in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_05.27.42.png" alt="N16/5/MATME/SP1/ENG/TZ0/03"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the values in the fifth row of Pascal’s triangle.</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 or otherwise, find the term in \({x^3}\) in the expansion of \({(2x + 3)^5}\).</p>
<div class="marks">[5]</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'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles (diagram 1). The process is repeated twice, as shown in diagrams 2 and 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/maths_10.png" alt></span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({x_n}\) denote the length of one of the equal sides of each new triangle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({A_n}\) denote the area of each new triangle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table gives the values of \({x_n}\) and \({A_n}\), for \(1 \leqslant n \leqslant 3\). <strong>Copy </strong>and complete the table. <em>(Do </em><strong><em>not </em></strong><em>write on this page.)</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em> </em></span></p>
<table class="block_black_border" style="height: 96px; width: 504px;" border="0">
<tbody>
<tr>
<td style="text-align: center;">\(n\)</td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">1</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">2</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">3</span></td>
</tr>
<tr>
<td style="text-align: center;"><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal; text-align: start;">\({x_n}\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">8</span></td>
<td><span style="font-family: 'times new roman', times; font-size: medium;"> </span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">4</span></td>
</tr>
<tr>
<td style="text-align: center;"><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal; text-align: start;">\({A_n}\)</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">32</span></td>
<td style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">16</span></td>
<td><span style="font-family: 'times new roman', times; font-size: medium;"> </span></td>
</tr>
</tbody>
</table>
<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: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The process described above is repeated. Find \({A_6}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider an initial square of side length \(k {\text{ cm}}\). The process described above is repeated indefinitely. The total area of the shaded regions is \(k {\text{ c}}{{\text{m}}^2}\). Find the value of \(k\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \({\log _2}40 - {\log _2}5\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \({8^{{{\log }_2}5}}\) .</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 class="p1">Given that \({2^m} = 8\) and \({2^n} = 16\), write down the value of \(m\) and of \(n\).</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 or otherwise solve \({8^{2x + 1}} = {16^{2x - 3}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The first three terms of a geometric sequence are \(\ln {x^{16}}\), \(\ln {x^8}\), \(\ln {x^4}\), for \(x > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio.</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>Solve \(\sum\limits_{k = 1}^\infty {{2^{5 - k}}\ln x = 64} \).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ann and Bob play a game where they each have an eight-sided die. Ann’s die has three green faces and five red faces; Bob’s die has four green faces and four red faces. They take turns rolling their own die and note what colour faces up. The first player to roll green wins. Ann rolls first. Part of a tree diagram of the game is shown below.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-13_om_11.04.17.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ann wins on her first roll.</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>The probability that Ann wins on her third roll is \(\frac{5}{8} \times \frac{4}{8} \times p \times q\ \times \frac{3}{8}\).</p>
<p class="p1">Write down the value of \(p\) and of \(q\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>The probability that Ann wins on her tenth roll is \(\frac{3}{8}{r^k}\) where \(r \in \mathbb{Q},{\text{ }}k \in \mathbb{Z}\).</p>
<p class="p1">Find the value of \(r\) and of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ann wins the game.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Solve \({\log _2}(2\sin x) + {\log _2}(\cos x) = - 1\), for \(2\pi < x < \frac{{5\pi }}{2}\).</p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\log _3}27\);</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\log _8}\frac{1}{8}\);</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\log _{16}}4\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, solve \({\log _3}27 + {\log _8}\frac{1}{8} - {\log _{16}}4 = {\log _4}x\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \({\log _2}x + {\log _2}(x - 2) = 3\) , for \(x > 2\) .</span></p>
</div>
<br><hr><br><div class="question">
<p class="p1"><span class="s1">In the expansion of \({(3x + 1)^n}\)</span>, the coefficient of the term in \({x^2}\) <span class="s1">is \(135n\)</span>, where \(n \in {\mathbb{Z}^ + }\). Find \(n\).</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3\ln x\) and \(g(x) = \ln 5{x^3}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Express \(g(x)\) in the form \(f(x) + \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is a transformation of the graph of <em>f</em> . Give a full geometric </span><span style="font-family: times new roman,times; font-size: medium;">description of this transformation.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In an arithmetic sequence, \({u_1} = 2\) and \({u_3} = 8\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>d</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({u_{20}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({S_{20}}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Given that \({\left( {1 + \frac{2}{3}x} \right)^n}{(3 + nx)^2} = 9 + 84</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">x</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> + \ldots \) , find the value of </span><em style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">n</em><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The following diagram shows [AB], with length 2 cm. The line is divided into an infinite number of line segments. The diagram shows the first three segments.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_09.33.46.png" alt="N17/5/MATME/SP1/ENG/TZ0/10.a"></p>
<p>The length of the line segments are \(p{\text{ cm}},{\text{ }}{p^2}{\text{ cm}},{\text{ }}{p^3}{\text{ cm}},{\text{ }} \ldots \), where \(0 < p < 1\).</p>
<p>Show that \(p = \frac{2}{3}\).</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>The following diagram shows [CD], with length \(b{\text{ cm}}\), where \(b > 1\). Squares with side lengths \(k{\text{ cm}},{\text{ }}{k^2}{\text{ cm}},{\text{ }}{k^3}{\text{ cm}},{\text{ }} \ldots \), where \(0 < k < 1\), are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_09.48.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/10.b"></p>
<p>The <strong>total</strong> sum of the areas of all the squares is \(\frac{9}{{16}}\). Find the value of \(b\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{1}{4}{x^2} + 2\) </span><span style="font-family: times new roman,times; font-size: medium;"> . The line <em>L</em> is the tangent to the curve of <em>f</em> at (4, 6) .</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \frac{{90}}{{3x + 4}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, for \(2 \le x \le 12\) . The following diagram shows the graph of <em>g</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/mickey.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the equation of <em>L</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the area of the region enclosed by the curve of <em>g</em> , the <em>x</em>-axis, and </span><span style="font-family: times new roman,times; font-size: medium;">the lines \(x = 2\) and \(x = 12\) . Give your answer in the form \(a\ln b\) , where \(a,b \in \mathbb{Z}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is reflected in the<em> x</em>-axis to give the graph of <em>h</em> . The area of the </span><span style="font-family: times new roman,times; font-size: medium;">region enclosed by the lines <em>L</em> , \(x = 2\) , \(x = 12\) and the <em>x</em>-axis is 120 \(120{\text{ c}}{{\text{m}}^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area enclosed by the lines<em> L</em> , \(x = 2\) , \(x = 12\) and the graph of <em>h</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(x = \ln 3\) and \(y = \ln 5\). Write the following expressions in terms of \(x\) and \(y\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>\(\ln \left( {\frac{5}{3}} \right)\).</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>\(\ln 45\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \({\log _3}p = 6\) and \({\log _3}q = 7\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \({\log _3}{p^2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \({\log _3}\left( {\frac{p}{q}} \right)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \({\log _3}(9p)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^{x + 3}}\) .</span></p>
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<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Show that \({f^{ - 1}}(x) = \ln x - 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the domain of \({f^{ - 1}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p><span style="font-family: times new roman,times; font-size: medium;">Solve the equation \({f^{ - 1}}(x) = \ln \frac{1}{x}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<p>In an arithmetic sequence, the first term is 8 and the second term is 5.</p>
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<p>Find the common difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
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<p>Find the tenth term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
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<p>Find the sum of the first ten terms.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
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<p class="p1">Let \(f'(x) = \frac{{6 - 2x}}{{6x - {x^2}}}\), for \(0 < x < 6\).</p>
<p class="p1"><span class="s1">The graph of \(f\) </span>has a maximum point at P<span class="s1">.</span></p>
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<p class="p1"><span class="s1">The \(y\)</span>-coordinate of P is \(\ln 27\).</p>
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<p class="p1">Find the \(x\)-coordinate of <span class="s1">P</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
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<p class="p1">Find \(f(x)\), expressing your answer as a single logarithm.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
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<p class="p1"><span class="s1">The graph of \(f\) </span>is transformed by a vertical stretch with scale factor \(\frac{1}{{\ln 3}}\). The image of P under this transformation has coordinates \((a,{\text{ }}b)\).</p>
<p class="p1">Find the value of \(a\) and of \(b\), where \(a,{\text{ }}b \in \mathbb{N}\).</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">c.</div>
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