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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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">5 <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _2}\left( {\frac{{{{32}^x}}}{{{8^y}}}} \right) = {\log _2}{32^x} - {\log _2}{8^y}\) <strong><em>(A1)</em></strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = x{\log _2}32 - y{\log _2}8\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _2}8 = 3\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 5\) , \(q = - 3\) (accept \(5x - 3y\) ) <em><strong>A1 N3 </strong></em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{{32}^x}}}{{{8^y}}} = \frac{{{{({2^5})}^x}}}{{{{({2^3})}^y}}}\) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{{2^5}^x}}{{{2^3}^y}}\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = {2^{5x - 3y}}\) <em><strong>(A1)</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _2}({2^{5x - 3y}}) = 5x - 3y\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(p = 5\) , \(q = - 3\) (accept \(5x - 3y\) ) </span> <em><strong>A1 N3</strong></em> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) proved very accessible.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although many found part (b) accessible as well, a good number of candidates could not complete their way to a final result. Many gave <em>q</em> as a positive value. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct approach <strong><em>(A1)</em></strong></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;"><em>eg</em> \({6^x} = 36,{\text{ }}{6^2}\)</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;">\(2\) <strong><em>A1 N2</em></strong></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;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct simplification <strong><em>(A1)</em></strong></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;"><em>eg</em> \({\log _6}36,{\text{ }}\log (4 \times 9)\)</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;">\(2\) <strong><em>A1 N2</em></strong></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;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct simplification <strong><em>(A1)</em></strong></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;"><em>eg</em> \({\log _6}\frac{2}{{12}},{\text{ }}\log (2 \div 12)\)</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;">correct working <strong><em>(A1)</em></strong></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;"><em>eg</em> \({\log _6}\frac{1}{6},{\text{ }}{6^{ - 1}} = \frac{1}{6}{,6^x} = \frac{1}{6}\)</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;">\(-1\) <strong><em>A1 N2</em></strong></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;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of expanding <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({2^4} + 4({2^3})x + 6({2^2}){x^2} + 4(2){x^3} + {x^4}\) , \((4 + 4x + {x^2})(4 + 4x + {x^2})\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({(2 + x)^4} = 16 + 32x + 24{x^2} + 8{x^3} + {x^4}\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A2 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">finding coefficients 24 and 1 <em><strong>(A1)(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">term is \(25{x^2}\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Surprisingly few candidates employed the binomial theorem, choosing instead to expand by </span><span style="font-family: times new roman,times; font-size: medium;">repeated use of the distributive property. This earned full marks if done correctly, but often </span><span style="font-family: times new roman,times; font-size: medium;">proved prone to error.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates often expanded the entire expression in part (b). Few recognized that only two distributions are required to answer the question. Some gave the coefficient as the final answer.</span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>There are many approaches to this question, and the steps may be done in any order. There are <span class="s1">3 </span>relationships they may need to apply at some stage, for the 3rd, 4th and 5th marks. These are</p>
<p class="p1">equating bases <em>eg </em>recognising 9 is \({{\text{3}}^2}\)</p>
<p class="p1">log rules: \(\ln b + \ln c = \ln (bc),{\text{ }}\ln b - \ln c = \ln \left( {\frac{b}{c}} \right)\),</p>
<p class="p1">exponent rule: \(\ln {b^n} = n\ln b\).</p>
<p class="p1">The exception to the <strong><em>FT </em></strong>rule applies here, so that if they demonstrate correct application of the 3 relationships, they may be awarded the <strong><em>A </em></strong>marks, even if they have made a previous error. However all applications of a relationship need to be correct. Once an error has been made, do not award <strong><em>A1FT </em></strong>for their final answer, even if it follows from their working<span class="s1">.</span></p>
<p class="p1">Please check working and award marks in line with the markscheme.</p>
<p class="p2"> </p>
<p class="p1">correct substitution into \({u_{13}}\) formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln a + (13 - 1)\ln 3\)</p>
<p class="p1">set up equation for \({u_{13}}\) in any form (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln a + 12\ln 3 = 8\ln 9\)</p>
<p class="p1">correct application of relationships <span class="Apple-converted-space"> </span><strong><em>(A1)(A1)(A1)</em></strong></p>
<p class="p1">\(a = 81\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<p class="p1"><strong>Examples of application of relationships</strong></p>
<p class="p1"><strong>Example 1</strong></p>
<p class="p1">correct application of exponent rule for logs <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln a + \ln {3^{12}} = \ln {9^8}\)</p>
<p class="p1">correct application of addition rule for logs <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln (a{3^{12}}) = \ln {9^8}\)</p>
<p class="p1">substituting for 9 or 3 in ln expression in equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln (a{3^{12}}) = \ln {3^{16}},{\text{ }}\ln (a{9^6}) = \ln {9^8}\)</p>
<p class="p1"><strong>Example 2</strong></p>
<p class="p1">recognising \(9 = {3^2}\)<span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln a + 12\ln 3 = 8\ln {3^2},{\text{ }}\ln a + 12\ln {9^{\frac{1}{2}}} = 8\ln 9\)</p>
<p class="p1">one correct application of exponent rule for logs relating \(\ln 9\) to \(\ln 3\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln a + 12\ln 3 = 16\ln 3,{\text{ }}\ln a + 6\ln 9 = 8\ln 9\)</p>
<p class="p1">another correct application of exponent rule for logs <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln a = \ln {3^4},{\text{ }}\ln a = \ln {9^2}\)</p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing that it is an arithmetic sequence <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(5,{\text{ }}5 + 4,{\text{ }}5 + 4 + 4,{\text{ }} \ldots ,{\text{ }}d = 4,{\text{ }}{u_n} = {u_1} + (n - 1)d,{\text{ }}4n + 1\)</p>
<p class="p1">correct equation <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(5 + 4(n - 1) = 801\)</p>
<p class="p1">correct working (do not accept substituting \(n = 200\)) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(4n - 4 = 796,{\text{ }}n - 1 = \frac{{796}}{4}\)</p>
<p class="p1"><span class="Apple-converted-space">\(n = 200\) </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognition of sum <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({S_{200}},{\text{ }}{u_1} + {u_2} + \ldots + {u_{200}},{\text{ }}5 + 9 + 13 + \ldots + 801\)</p>
<p class="p1">correct working for AP <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{200}}{2}(5 + 801),{\text{ }}\frac{{200}}{2}{\text{ }}\left( {2(5) + 199(4)} \right)\)</p>
<p class="p1"><span class="s1">\(80\,600\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates recognized that the series was arithmetic but many worked backwards using \(n = 200\) rather than creating and solving an equation of their own to produce the given answer.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Almost all students answered (b) correctly.</p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \(f(8) = 1\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 = k{\log _2}8\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \({\log _2}8 = 3\) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 = 3k\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{1}{3}\) <em><strong>A1 N2</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find the inverse of \(f(x) = k{\log _2}x\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = k{\log _2}y\) , \(y = {2^{\frac{x}{k}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting 1 and 8 <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 = k{\log _2}8\) , \({2^{\frac{1}{k}}} = 8\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{1}{{{{\log }_2}8}}\) \(\left( {k = \frac{1}{3}} \right)\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times;"><strong><span style="font-size: medium;">METHOD 1</span></strong></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">recognizing that </span><span style="font-size: medium;">\(f(x) = \frac{2}{3}\) <em><strong>(M1)</strong></em></span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{2}{3} = \frac{1}{3}{\log _2}x\)</span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">\({\log _2}x = 2\) </span><em><strong><span style="font-size: medium;">(A1)</span></strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}\left( {\frac{2}{3}} \right) = 4\) (accept \(x = 4\)) <em><strong>A2 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><strong> <span style="font-size: medium;">METHOD 2</span></strong></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;">attempt to find inverse of \(f(x) = \frac{1}{3}{\log _2}x\) </span><em><strong><span style="font-size: medium;">(M1)</span></strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. interchanging <em>x</em> and <em>y</em> , substituting \(k = \frac{1}{3}\) into \(y = {2^{\frac{x}{k}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct inverse <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({f^{ - 1}}(x) = {2^{3x}}\) , \({2^{3x}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}\left( {\frac{2}{3}} \right) = 4\) <em><strong>A2 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;"> [4 marks]</span></strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A very poorly done question. Most candidates attempted to find the inverse function for \(f\) and used that to answer parts (a) and (b). Few recognized that the explicit inverse function was not necessary to answer the question. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Although many candidates seem to know that they can find an inverse function by interchanging <em>x</em> and <em>y</em>, very few were able to actually get the correct inverse. Almost none recognized that if \({f^{ - 1}}(1) = 8\) , then \(f(8) = 1\) . Many thought that the letters "log" could be simply "cancelled out", leaving the \(2\) and the \(8\). </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A very poorly done question. Most candidates attempted to find the inverse function for \(f\) and used that to answer parts (a) and (b). Few recognized that the explicit inverse function was not necessary to answer the question. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Although many candidates seem to know that they can find an inverse function by interchanging <em>x</em> and <em>y</em>, very few were able to actually get the correct inverse. Almost none recognized that if \({f^{ - 1}}(1) = 8\) , then \(f(8) = 1\) . Many thought that the letters "log" could be simply "cancelled out", leaving the \(2\) and the \(8\). </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to subtract terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(d = {u_2} - {u_1},{\text{ }}7 - 3\)</p>
<p>\(d = 4\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct approach <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({u_{10}} = 3 + 9(4)\)</p>
<p>\({u_{10}} = 39\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into sum <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({S_{10}} = 5(3 + 39),{\text{ }}{S_{10}} = \frac{{10}}{2}(2 \times 3 + 9 \times 4)\)</p>
<p>\({S_{10}} = 210\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct expression for \(r\) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(r = \frac{6}{{m - 1}},{\text{ }}\frac{{m + 4}}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct equation <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{6}{{m - 1}} = \frac{{m + 4}}{6},{\text{ }}\frac{6}{{m + 4}} = \frac{{m - 1}}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \((m + 4)(m - 1) = 36\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({m^2} - m + 4m - 4 = 36,{\text{ }}{m^2} + 3m - 4 = 36\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({m^2} + 3m - 40 = 0\) <strong><em>AG N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks] </em></strong></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid attempt to solve <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \((m + 8)(m - 5) = 0,{\text{ }}m = \frac{{ - 3 \pm \sqrt {9 + 4 \times 40} }}{2}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = - 8,{\text{ }}m = 5\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute <strong>any </strong>value of \(m\) to find \(r\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{6}{{ - 8 - 1}},{\text{ }}\frac{{5 + 4}}{6}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = \frac{3}{2},{\text{ }}r = - \frac{2}{3}\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(r = - \frac{2}{3}\) (may be seen in justification) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid reason <strong><em>R1 N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\left| r \right| < 1,{\text{ }} - 1 < \frac{{ - 2}}{3} < 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"><strong style="color: #000000; font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"><strong style="color: #000000; font-family: 'times new roman', times; font-size: medium;">Notes: </strong><span style="color: #000000; font-family: 'times new roman', times; font-size: medium;">Award </span><strong style="color: #000000; font-family: 'times new roman', times; font-size: medium;"><em>R1 </em></strong><span style="color: #000000; font-family: 'times new roman', times; font-size: medium;">for \(\left| r \right| < 1\) only if </span><strong style="color: #000000; font-family: 'times new roman', times; font-size: medium;"><em>A1 </em></strong><span style="color: #000000; font-family: 'times new roman', times; font-size: medium;">awarded.<br><br></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[2 marks]</strong></em></span></p>
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<div>
<p><span style="font-family: 'times new roman', times; font-size: medium;">finding the first term of the sequence which has \(\left| r \right| < 1\) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">eg \( - 8 - 1,{\text{ }}6 \div \frac{{ - 2}}{3}\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({u_1} = - 9\) (may be seen in formula) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution of \({u_1}\) and their \(r\) into \(\frac{{{u_1}}}{{1 - r}}\), as long as \(\left| r \right| < 1\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">eg \({S_\infty } = \frac{{ - 9}}{{1 - \left( { - \frac{2}{3}} \right)}},{\text{ }}\frac{{ - 9}}{{\frac{5}{3}}}\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({S_\infty } = - \frac{{27}}{5}{\text{ }}( = - 5.4)\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks] </span></strong></em></p>
</div>
<div class="question_part_label">c(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct approach <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;d = {u_2} - {u_1},{\text{ }}5 - 2\)</p>
<p class="p1">\(d = 3\) <span class="Apple-converted-space"> </span><strong><em>A1 N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct approach <strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{u_8} = 2 + 7 \times 3\), listing terms</p>
<p class="p1">\({u_8} = 23\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct approach <strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{S_8} = \frac{8}{2}(2 + 23)\), listing terms, \(\frac{8}{2}\left( {2(2) + 7(3)} \right)\)</p>
<p class="p1">\({S_8} = 100\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">All three parts of this question were very well done by the candidates. The occasional mistakes that were seen tended to be arithmetic errors which happened after the candidates had substituted correctly into the formulas given in the formula booklet.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">All three parts of this question were very well done by the candidates. The occasional mistakes that were seen tended to be arithmetic errors which happened after the candidates had substituted correctly into the formulas given in the formula booklet.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">All three parts of this question were very well done by the candidates. The occasional mistakes that were seen tended to be arithmetic errors which happened after the candidates had substituted correctly into the formulas given in the formula booklet.</p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct application of \(\ln {a^b} = b\ln a\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\ln 4 = 2\ln 2,{\text{ }}3\ln 2 = \ln {2^3},{\text{ }}3\log 2 = \log 8\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;3\ln 2 - 2\ln 2,{\text{ }}\ln 8 - \ln 4\)</p>
<p class="p1">\(\ln 2\;\;\;{\text{(accept }}k = 2{\text{)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to substitute <strong>their </strong>answer into the equation <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln 2 = - \ln x\)</p>
<p>correct application of a log rule <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln \frac{1}{x},{\text{ }}\ln \frac{1}{2} = \ln x,{\text{ }}\ln 2 + \ln x = \ln 2x\;\;\;( = 0)\)</p>
<p>\(x = \frac{1}{2}\) <strong><em>A1 N2</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to rearrange equation, with \(3\ln 2\) written as \(\ln {2^3}\) or \(\ln 8\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln x = \ln 4 - \ln {2^3},{\text{ }}\ln 8 + \ln x = \ln 4,{\text{ }}\ln {2^3} = \ln 4 - \ln x\)</p>
<p>correct working applying \(\ln a \pm \ln b\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{4}{8},{\text{ }}8x = 4,{\text{ }}\ln {2^3} = \ln \frac{4}{x}\)</p>
<p>\(x = \frac{1}{2}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was answered correctly by a large number of candidates, though there were quite a few who applied the rules of logarithms in the wrong order.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (b), many candidates knew to set their answer from part (a) equal to \( - \ln x\), but then a good number incorrectly said that \(\ln 2 = - \ln x\) led to \(2 = - x\).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid method <strong><em>(M1)</em></strong></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;"><em>eg</em> \({u_2} = {S_2} - {S_1},{\text{ }}1 + k + {u_2} = 5 + 3k\)</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;">\({u_2} = 4 + 2k,{\text{ }}{u_3} = 7 + 4k,{\text{ }}{u_4} = 10 + 8k\) <strong><em>A1A1A1 N4</em></strong></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;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct AP <strong>or </strong>GP <strong><em>(A1)</em></strong></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;"><em>eg</em> finding common difference is \(3\), common ratio is \(2\)</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;">valid approach using arithmetic <strong>and </strong>geometric formulas <strong><em>(M1)</em></strong></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;"><em>eg</em> \(1 + 3(n - 1)\) <strong>and</strong> \({r^{n - 1}}k\)</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;">\({u_n} = 3n - 2 + {2^{n - 1}}k\) <strong><em>A1A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </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;"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for \(3n - 2\), <strong><em>A1 </em></strong>for \({2^{n - 1}}k\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </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;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(d = 3\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substitution into \({u_n} = a + (n - 1)d\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{101}} = 2 + 100 \times 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_{101}} = 302\) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct approach <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(152 = 2 + (n - 1) \times 3\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct simplification <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(150 = (n - 1) \times 3\) , \(50 = n - 1\) , \(152 = - 1 + 3n\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 51\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-size: medium;"><span style="font-family: times new roman,times;">Candidates probably had the most success with this question with many good solutions which </span><span style="font-family: times new roman,times;">were written with the working clearly shown. Many used the alternate approach of \({u_n} = 3n - 1\) .</span></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Candidates probably had the most success with this question with many good solutions which </span><span style="font-family: times new roman,times; font-size: medium;">were written with the working clearly shown. Many used the alternate approach of \({u_n} = 3n - 1\) .</span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of dividing terms (in any order) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{\mu _2}}}{{{\mu _1}}},{\text{ }}\frac{{2{{\log }_2}x}}{{{{\log }_2}x}}\)</p>
<p class="p2"><span class="Apple-converted-space">\(r = \frac{1}{2}\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{2{{\log }_2}x}}{{1 - \frac{1}{2}}}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{2{{\log }_2}x}}{{\frac{1}{2}}}\)</p>
<p class="p2">\({S_\infty } = 4{\log _2}x\) <strong><em>AG N0</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of subtracting two terms (in any order) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({u_3} - {u_2},{\text{ }}{\log _2}x - {\log _2}\frac{x}{2}\)</p>
<p class="p1">correct application of the properties of logs <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({\log _2}\left( {\frac{{\frac{x}{2}}}{x}} \right),{\text{ }}{\log _2}\left( {\frac{x}{2} \times \frac{1}{x}} \right),{\text{ }}({\log _2}x - {\log _2}2) - {\log _2}x\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({\log _2}\frac{1}{2},{\text{ }} - {\log _2}2\)</p>
<p class="p2"><span class="Apple-converted-space">\(d = - 1\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct substitution into the formula for the sum of an arithmetic sequence <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{12}}{2}\left( {2{{\log }_2}x + (12 - 1)( - 1)} \right)\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(6(2{\log _2}x - 11),{\text{ }}\frac{{12}}{2}(2{\log _2}x - 11)\)</p>
<p class="p2"><span class="Apple-converted-space">\(12{\log _2}x - 66\) </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct equation <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(12{\log _2}x - 66 = 2{\log _2}x\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(10{\log _2}x = 66,{\text{ }}{\log _2}x = 6.6,{\text{ }}{2^{66}} = {x^{10}},{\text{ }}{\log _2}\left( {\frac{{{x^{12}}}}{{{x^2}}}} \right) = 66\)</p>
<p class="p2"><span class="s2">\(x = {2^{6.6}}\) (accept \(p = \frac{{66}}{{10}}\)</span><span class="s3">) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 10\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = p\) , \(b = 2q\) (or \(a = 2q\) , \(b = p\) ) <em><strong>A1A1 N1N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br>{10}\\<br>5<br>\end{array}} \right){p^5}{(2q)^5}\) <em><strong> A1A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The most common error was in (c) where many candidates interpreted the "sixth" term as using \(\left( {\begin{array}{*{20}{c}}<br>{10}\\<br>6<br>\end{array}} \right)\) , with accompanying powers of 4 and 6 in the expression. </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach involving addition or subtraction    <em><strong>M1</strong></em><br><em>eg</em>  \({u_2} = {\text{lo}}{{\text{g}}_c}\,p + d,\,\,{u_1} - {u_2}\)</p>
<p>correct application of log law   <strong><em>A1</em></strong><br><em>eg</em>  \({\text{lo}}{{\text{g}}_c}\left( {pq} \right) = {\text{lo}}{{\text{g}}_c}\,p + {\text{lo}}{{\text{g}}_c}\,q,\,\,{\text{lo}}{{\text{g}}_c}\left( {\frac{{pq}}{p}} \right)\)</p>
<p>\(d = {\text{lo}}{{\text{g}}_c}\,q\)Â Â Â <em><strong>AG N0<br></strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong> (finding \({u_1}\) and <em>d</em>)</p>
<p>recognizing \(\sum { = {S_{20}}} \) (seen anywhere)   <em><strong>(A1)</strong></em></p>
<p>attempt to find \({u_1}\) or <em>d</em> using \({\text{lo}}{{\text{g}}_c}\,{c^k} = k\)   <em><strong>(M1)</strong></em><br>eg  \({\text{lo}}{{\text{g}}_c}\,c\), \({\text{3}}\,{\text{lo}}{{\text{g}}_c}\,c\), correct value of \({u_1}\) or <em>d</em></p>
<p>\({u_1}\) = 2, <em>d </em>= 3 (seen anywhere)   <em><strong> (A1)(A1)</strong></em></p>
<p>correct working  <em><strong> (A1)</strong></em><br><em>eg</em>  \({S_{20}} = \frac{{20}}{2}\left( {2 \times 2 + 19 \times 3} \right),\,\,{S_{20}} = \frac{{20}}{2}\left( {2 + 59} \right),\,\,10\left( {61} \right)\)</p>
<p>\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610Â Â Â <em><strong>A1 N2</strong></em></p>
<p>Â </p>
<p><strong>METHOD 2</strong> (expressing <em>S</em> in terms of <em>c</em>)</p>
<p>recognizing \(\sum { = {S_{20}}} \) (seen anywhere)    <em><strong>(A1)</strong></em></p>
<p>correct expression for <em>S</em> in terms of <em>c</em>Â Â Â <em><strong>(A1)</strong></em><br><em>eg</em>Â Â \(10\left( {2\,{\text{lo}}{{\text{g}}_c}\,{c^2} + 19\,{\text{lo}}{{\text{g}}_c}\,{c^3}} \right)\)</p>
<p>\({\text{lo}}{{\text{g}}_c}\,{c^2} = 2,\,\,\,{\text{lo}}{{\text{g}}_c}\,{c^3} = 3\)Â Â (seen anywhere)Â Â Â <em><strong>(A1)</strong><strong>(A1)</strong></em></p>
<p>correct working   <em><strong>(A1)</strong></em></p>
<p><em>eg</em>Â Â \({S_{20}} = \frac{{20}}{2}\left( {2 \times 2 + 19 \times 3} \right),\,\,{S_{20}} = \frac{{20}}{2}\left( {2 + 59} \right),\,\,10\left( {61} \right)\)</p>
<p>\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610Â Â Â <em><strong>A1 N2</strong></em></p>
<p>Â </p>
<p><strong>METHOD 3</strong> (expressing <em>S</em> in terms of <em>c</em>)</p>
<p>recognizing \(\sum { = {S_{20}}} \) (seen anywhere)    <em><strong>(A1)</strong></em></p>
<p>correct expression for <em>S</em> in terms of <em>c</em>    <em><strong>(A1)</strong></em><br><em>eg</em>  \(10\left( {2\,{\text{lo}}{{\text{g}}_c}\,{c^2} + 19\,{\text{lo}}{{\text{g}}_c}\,{c^3}} \right)\)</p>
<p>correct application of log law   <em><strong>(A1)</strong></em><br>eg  \(2\,{\text{lo}}{{\text{g}}_c}\,{c^2} = \,\,{\text{lo}}{{\text{g}}_c}\,{c^4},\,\,19\,{\text{lo}}{{\text{g}}_c}\,{c^3} = \,\,{\text{lo}}{{\text{g}}_c}\,{c^{57}},\,\,10\,\left( {{\text{lo}}{{\text{g}}_c}\,{{\left( {{c^2}} \right)}^2} + \,\,{\text{lo}}{{\text{g}}_c}\,{{\left( {{c^3}} \right)}^{19}}} \right),\,\,10\,\left( {{\text{lo}}{{\text{g}}_c}\,{c^4} + \,{\text{lo}}{{\text{g}}_c}\,{c^{57}}} \right),\,\,10\left( {{\text{lo}}{{\text{g}}_c}\,{c^{61}}} \right)\)</p>
<p>correct application of definition of log   <em><strong>(A1)</strong></em><br>eg  \({\text{lo}}{{\text{g}}_c}\,{c^{61}} = 61,\,\,{\text{lo}}{{\text{g}}_c}\,{c^4} = 4,\,\,{\text{lo}}{{\text{g}}_c}\,{c^{57}} = 57\)</p>
<p>correct working   <em><strong>(A1)</strong></em><br>eg  \({S_{20}} = \frac{{20}}{2}\left( {4 + 57} \right),\,\,10\left( {61} \right)\)</p>
<p>\(\sum\limits_{n = 1}^{20} {{u_n}} \) = 610Â Â Â <em><strong>A1 N2</strong></em></p>
<p><strong><em>[6 marks]</em></strong></p>
<p>Â </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find \(d\)<em> </em><strong><em>(M1)</em></strong></span></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;"><em>eg \(\frac{{16 - 10}}{2}{\text{, }}10 - 2d = 16 - 4d{\text{, }}2d = 6{\text{, }}d = 6\)</em></span></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;">\(d = 3\) <strong><em>A1 N2</em></strong></span></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;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct approach <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(10 = {u_1} + 2 \times 3{\text{, }}10 - 3 - 3\)</span></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;">\({u_1} = 4\) <strong><em>A1 N2</em></strong></span></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;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into sum or term formula <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(\frac{{20}}{2}(2 \times 4 + 19 \times 3){\text{, }}{u_{20}} = 4 + 19 \times 3\)</span></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;">correct simplification <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(8 + 57{\text{, }}4 + 61\)</span></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;">\({S_{20}} = 650\) <strong><em>A1 N2</em></strong></span></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;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_{10}} = 3{(0.9)^9}\) <strong><em>A1 N1</em> </strong></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[1 mark]</strong></span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing \(r = 0.9\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <strong><em>A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(S = \frac{3}{{1 - 0.9}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(S = \frac{3}{{0.1}}\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(S = 30\) <em><strong>A1 N3</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[4 marks]</em></span></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by most candidates. There were a surprising number of candidates who lost a mark for not simplifying \(\frac{3}{{0.1}}\) to 30 , and there were a few candidates who used the formula for the finite sum unsuccessfully. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by most candidates. There were a surprising number of candidates who lost a mark for not simplifying \(\frac{3}{{0.1}}\) to 30, and there were a few candidates who used the formula for the finite sum unsuccessfully. </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">interchanging <em>x</em> and <em>y</em> (seen anywhere) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = \log \sqrt y \) (accept any base)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct manipulation <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3^x = \sqrt y \) , \({3^y} = {x^{\frac{1}{2}}}\) , \(x = \frac{1}{2}{\log _3}y\) , \(2y = {\log _3}x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}(x) = {3^{2x}}\) <em><strong>AG N0</strong></em> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(y > 0\) , \({f^{ - 1}}(x) > 0\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(g(2) = lo{g_3}2\) (seen anywhere) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(({f^{ - 1}} \circ g)(2) = {3^{2\log {_3}2}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using log or index rule <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(({f^{ - 1}} \circ g)(2) = {3^{\log {_3}4}}\) , \({3^{{{\log }_3}2^2}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(({f^{ - 1}} \circ g)(2) = 4\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to form composite (in any order) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(({f^{ - 1}} \circ g)(x) = {3^{2{{\log }_3}x}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using log or index rule <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(({f^{ - 1}} \circ g)(x) = {3^{{{\log }_3}{x^2}}}\) , \({3^{{{\log }_3}{x^{}}}}^2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(({f^{ - 1}} \circ g)(x) = {x^2}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(({f^{ - 1}} \circ g)(2) = 4\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates were generally skilled at finding the inverse of a logarithmic function.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few correctly gave the range of this function, often stating “all real numbers” or “ \(y \ge 0\) ”, missing the idea that the range of an inverse is the domain of the original function.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Some candidates </span><span style="font-family: times new roman,times; font-size: medium;">answered part (c) correctly, although many did not get beyond \({3^{2{{\log }_3}2}}\) . Some attempted to form </span><span style="font-family: times new roman,times; font-size: medium;">the composite in the incorrect order. Others interpreted \(({f^{ - 1}} \circ g)(2)\) as multiplication by 2.</span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(r = \frac{{16}}{{32}}\left( { = \frac{1}{2}} \right)\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct calculation or listing terms <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(32 \times {\left( {\frac{1}{2}} \right)^{6 - 1}}\) , \(8 \times {\left( {\frac{1}{2}} \right)^3}\) , 32, \(\ldots \) 4, 2, 1</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> \({u_6} = 1\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct substitution in \({S_\infty }\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{32}}{{1 - \frac{1}{2}}}\) , \(\frac{{32}}{{\frac{1}{2}}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({S_\infty } = 64\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was very well done by the majority of candidates. There were some who used a value of <em>r</em> greater than one, with the most common error being \(r = 2\) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was very well done by the majority of candidates. There were some who used a value of <em>r</em> greater than one, with the most common error being \(r = 2\) . </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A handful of candidates struggled with the basic computation involved in part (c). </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach   <em><strong>(M1)</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{{u_2}}}{{{u_1}}},\,\,\frac{{{u_1}}}{{{u_2}}}\)</p>
<p>\(r = \frac{{12\,{{\sin }^2}\,\theta }}{{18}}\left( { = \frac{{2\,{{\sin }^2}\,\theta }}{3}} \right)\)Â Â Â Â <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that sin<em>θ</em> is bounded   <strong><em> (M1)</em></strong></p>
<p><em>eg</em>  0 <em>≤</em> sin<sup>2</sup> <em>θ </em>≤ 1, −1 ≤<em> sinθ </em>≤ 1, −1 <<em> sinθ </em>< 1</p>
<p>0 < r ≤ \(\frac{2}{3}\)   <em><strong> A2 N3</strong></em></p>
<p><strong>Note:</strong> If working shown, award <em><strong>M1A1</strong></em> for correct values with incorrect inequality sign(s).<br>If no working shown, award <em><strong>N1</strong></em> for correct values with incorrect inequality sign(s).</p>
<p><em><strong>[</strong></em><em><strong>3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula for infinite sum    <em><strong>A1</strong></em></p>
<p><em>eg </em>\(\frac{{18}}{{1 - \frac{{2\,{\text{si}}{{\text{n}}^2}\,\theta }}{3}}}\)</p>
<p>evidence of choosing an appropriate rule for cos 2<em>θ</em> (seen anywhere)     <em><strong>(M1)</strong></em></p>
<p><em>eg  </em>cos 2<em>θ </em>= 1<em> </em>− 2 sin<sup>2</sup> <em>θ</em></p>
<p>correct substitution of identity/working (seen anywhere)Â Â Â <em><strong>(A1)</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{18}}{{1 - \frac{2}{3}\left( {\frac{{1 - {\text{cos}}\,2\theta }}{2}} \right)}},\,\,\frac{{54}}{{3 - 2\left( {\frac{{1 - {\text{cos}}\,2\theta }}{2}} \right)}},\,\,\frac{{18}}{{\frac{{3 - 2\,{\text{si}}{{\text{n}}^2}\,\theta }}{3}}}\)</p>
<p>correct working that clearly leads to the given answer    <em><strong>A1</strong></em></p>
<p><em>eg</em>Â Â \(\frac{{18 \times 3}}{{2 + \left( {1 - 2\,{\text{si}}{{\text{n}}^2}\,\theta } \right)}},\,\,\frac{{54}}{{3 - \left( {1 - {\text{cos}}\,2\theta } \right)}}\)</p>
<p>\(\frac{{54}}{{2 + {\text{cos}}\left( {2\theta } \right)}}\)Â Â Â <em><strong>AG N0</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Â </p>
<p><strong>METHOD 1 </strong>(using differentiation)</p>
<p>recognizing \(\frac{{{\text{d}}{S_\infty }}}{{{\text{d}}\theta }} = 0\) (seen anywhere)    <em><strong>(M1)</strong></em></p>
<p>finding any correct expression for \(\frac{{{\text{d}}{S_\infty }}}{{{\text{d}}\theta }}\)    <em><strong>(A1)</strong></em></p>
<p><em>eg  </em>\(\frac{{0 - 54 \times \left( { - 2\,{\text{sin}}\,2\,\theta } \right)}}{{{{\left( {2 + {\text{cos}}\,2\,\theta } \right)}^2}}},\,\, - 54{\left( {2 + {\text{cos}}\,2\,\theta } \right)^{ - 2}}\,\left( { - 2\,{\text{sin}}\,2\,\theta } \right)\)</p>
<p>correct working   <em><strong> (A1)</strong></em></p>
<p><em>eg </em> sin 2<em>θ</em> = 0</p>
<p>any correct value for sin<sup>−1</sup>(0) (seen anywhere)    <em><strong>(A1)</strong></em></p>
<p><em>eg </em> 0, \(\pi \), … , sketch of sine curve with <em>x</em>-intercept(s) marked both correct values for 2<em>θ</em> (ignore additional values)   <em><strong>(A1)</strong></em></p>
<p>2<em>θ </em>= \(\pi \), 3\(\pi \) (accept values in degrees)</p>
<p>both correct answers \(\theta = \frac{\pi }{2},\,\frac{{3\pi }}{2}\)    <em><strong>A1 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if either or both correct answers are given in degrees.<br>Award <em><strong>A0Â </strong></em>if additional values are given.</p>
<p>Â </p>
<p><strong>METHOD 2 </strong>(using denominator)</p>
<p>recognizing when S<sub>∞</sub> is greatest   <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 2 + cos 2<em>θ </em>is a minimum, 1−<em>r</em> is smallest<br>correct working   <em><strong>(A1)</strong></em></p>
<p><em>eg  </em>minimum value of 2 + cos 2<em>θ </em>is 1, minimum <em>r</em> = \(\frac{2}{3}\)</p>
<p>correct working   <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{cos}}\,2\,\theta = - 1,\,\,\frac{2}{3}\,{\text{si}}{{\text{n}}^2}\,\theta = \frac{2}{3},\,\,{\text{si}}{{\text{n}}^2}\theta = 1\)</p>
<p><strong>EITHER</strong> (using cos 2<em>θ</em>)</p>
<p>any correct value for cos<sup>−1</sup>(−1) (seen anywhere)   <em><strong>(A1)</strong></em></p>
<p><em>eg  </em>\(\pi \), 3\(\pi \), … (accept values in degrees), sketch of cosine curve with <em>x</em>-intercept(s) marked</p>
<p>both correct values for 2<em>θ  </em>(ignore additional values)   <em><strong>(A1)</strong></em></p>
<p>2<em>θ </em>= \(\pi \), 3\(\pi \) (accept values in degrees)</p>
<p><strong>OR</strong> (using sin<em>θ</em>)</p>
<p>sin<em>θ = </em>±1   (A1)</p>
<p>sin<sup>−1</sup>(1) = \(\frac{\pi }{2}\) (accept values in degrees) (seen anywhere)   <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>both correct answers \(\theta = \frac{\pi }{2},\,\frac{{3\pi }}{2}\)     <em><strong>A1 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if either or both correct answers are given in degrees.<br>Award <em><strong>A0 </strong></em>if additional values are given.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(r = \frac{6}{{x - 3}},{\text{ }}(x - 3) \times r = 6,{\text{ }}(x - 3){r^2} = x + 2\)</p>
<p class="p1">correct equation in terms of \(x\) only <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{6}{{x - 3}} = \frac{{x + 2}}{6},{\text{ }}(x - 3)(x + 2) = {6^2},{\text{ }}36 = {x^2} - x - 6\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({x^2} - x - 42,{\text{ }}{x^2} - x = 42\)</p>
<p class="p1">valid attempt to solve <strong>their </strong>quadratic equation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)factorizing, formula, completing the square</p>
<p class="p1">evidence of correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\((x - 7)(x + 6),{\text{ }}\frac{{1 \pm \sqrt {169} }}{2}\)</p>
<p class="p1">\(x = 7,{\text{ }}x = - 6\) <strong><em>A1 N4</em></strong></p>
<p class="p1"><strong>METHOD 2 (finding </strong><span class="s1"><strong><em>r </em></strong></span><strong>first)</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(r = \frac{6}{{x - 3}},{\text{ }}6r = x + 2,{\text{ }}(x - 3){r^2} = x + 2\)</p>
<p class="p1">correct equation in terms of \(r\) only <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{6}{r} + 3 = 6r - 2,{\text{ }}6 + 3r = 6{r^2} - 2r,{\text{ }}6{r^2} - 5r - 6 = 0\)</p>
<p class="p1">evidence of correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\((3r + 2)(2r - 3),{\text{ }}\frac{{5 \pm \sqrt {25 + 144} }}{{12}}\)</p>
<p class="p1"><span class="Apple-converted-space">\(r = - \frac{2}{3},{\text{ }}r = \frac{3}{2}\) </span><strong><em>A1</em></strong></p>
<p class="p1">substituting their values of \(r\) to find \(x\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\((x - 3)\left( {\frac{2}{3}} \right) = 6,{\text{ }}x = 6\left( {\frac{3}{2}} \right) - 2\)</p>
<p class="p1"><span class="Apple-converted-space">\(x = 7,{\text{ }}x = - 6\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Nearly all candidates attempted to set up an expression, or pair of expressions, for the common ratio of the geometric sequence. When done correctly, these expressions led to a quadratic equation which was solved correctly by many candidates.</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">1, 5, 10, 10, 5, 1 <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of binomial expansion with binomial coefficient <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){a^{n - r}}{b^r}\)</span>, selecting correct term, \({(2x)^5}{(3)^0} + 5{(2x)^4}{(3)^1} + 10{(2x)^3}{(3)^2} + \ldots \)</p>
<p class="p1">correct substitution into correct term <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)(A1)(A1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(10{(2)^3}{(3)^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 5 \\ 3 \end{array}} \right){(2x)^3}{(3)^2}\)</p>
<p class="p4"> </p>
<p class="p2"><span class="s2"><strong>Note: </strong>Award </span><span class="s1"><strong><em>A1 </em></strong></span>for each factor.</p>
<p class="p5"> </p>
<p class="p3"><span class="s3">\(720{x^3}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p6"> </p>
<p class="p1"><strong>Notes: </strong>Do not award any marks if there is clear evidence of adding instead of multiplying.</p>
<p class="p2"><span class="s2">Do not award final </span><span class="s1"><strong><em>A1 </em></strong></span>for a final answer of 720, even if \(720{x^3}\) <span class="s2">is seen previously.</span></p>
<p class="p6"> </p>
<p class="p3"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;">valid method for finding side length </span><strong style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><em>(M1)</em></strong></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;"><em>eg</em> \({8^2} + {8^2} = {c^2},{\text{ }}45 - 45 - 90{\text{ side ratios, }}8\sqrt 2 ,{\text{ }}\frac{1}{2}{s^2} = 16,{\text{ }}{x^2} + {x^2} = {8^2}\)</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;">correct working for area <strong><em>(A1)</em></strong></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;"><em>eg</em> \(\frac{1}{2} \times 4 \times 4\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px 'Times New Roman';"> </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 style="text-align: center;"><span style="color: #3f3f3f; font-family: 'times new roman', times; font-size: medium; line-height: normal; text-align: start;">\(\sqrt {32}\)</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 style="text-align: center;"><span style="font-family: 'times new roman', times; font-size: medium;">8</span></td>
</tr>
</tbody>
</table>
<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;"><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>A1A1 N2N2</em></strong></span></span></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;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognize geometric progression for \({A_n}\) <strong><em>(R1)</em></strong></span></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;">eg \({u_n} = {u_1}{r^{n - 1}}\)</span></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;">\(r = \frac{1}{2}\) <em> </em><strong><em>(A1)</em></strong></span></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;">correct working <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(32{\left( {\frac{1}{2}} \right)^5};{\text{ 4, 2, 1, }}\frac{1}{2},{\text{ }}\frac{1}{4},{\text{ }} \ldots \)</span></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;">\({A_6} = 1\) <strong><em>A1 N3</em></strong></span></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;"><strong><em> </em></strong></span></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;"><strong>METHOD 2</strong></span></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;">attempt to find \({x_6}\) <strong><em>(M1)</em></strong></span></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;"><em>eg</em> \(8{\left( {\frac{1}{{\sqrt 2 }}} \right)^5},{\text{ }}2\sqrt 2 ,{\text{ 2, }}\sqrt 2 ,{\text{ 1, }} \ldots \)</span></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;">\({x_6} = \sqrt 2 \) <strong><em>(A1)</em></strong></span></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;">correct working <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(\frac{1}{2}{\left( {\sqrt 2 } \right)^2}\)</span></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;">\({A_6} = 1\) <strong><em>A1 N3</em></strong></span></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;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognize infinite geometric series <strong><em>(R1)</em></strong></span></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;"><em>eg</em> \({S_n} = \frac{a}{{1 - r}},{\text{ }}\left| r \right| < 1\)</span></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;">area of first triangle in terms of \(k\) <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(\frac{1}{2}{\left( {\frac{k}{2}} \right)^2}\)</span></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;">attempt to substitute into sum of infinite geometric series (must have \(k\)) <strong><em>(M1)</em></strong></span></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;"><em>eg</em> \(\frac{{\frac{1}{2}{{\left( {\frac{k}{2}} \right)}^2}}}{{1 - \frac{1}{2}}},{\text{ }}\frac{k}{{1 - \frac{1}{2}}}\)</span></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;">correct equation <strong><em>A1</em></strong></span></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;"><em>eg</em> \(\frac{{\frac{1}{2}{{\left( {\frac{k}{2}} \right)}^2}}}{{1 - \frac{1}{2}}} = k,{\text{ }}k = \frac{{\frac{{{k^2}}}{8}}}{{\frac{1}{2}}}\)</span></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;">correct working <strong><em>(A1)</em></strong></span></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;">eg \({k^2} = 4k\)</span></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;">valid attempt to solve <strong>their</strong> quadratic <strong><em>(M1)</em></strong></span></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;">eg \(k(k - 4),{\text{ }}k = 4{\text{ or }}k = 0\)</span></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;">\(k = 4\) <strong><em>A1 N2</em></strong></span></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;"><strong>METHOD 2</strong></span></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;">recognizing that there are four sets of infinitely shaded regions with equal area <strong><em>R1</em></strong></span></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;">area of original square is \({k^2}\) <strong><em>(A1)</em></strong></span></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;">so total shaded area is \(\frac{{{k^2}}}{4}\) <strong><em>(A1)</em></strong></span></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;">correct equation \(\frac{{{k^2}}}{4} = k\) <strong><em>A1</em></strong></span></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;">\({k^2} = 4k\) <strong><em>(A1)</em></strong></span></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;">valid attempt to solve their quadratic <strong><em>(M1)</em></strong></span></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;">eg \(k(k - 4),{\text{ }}k = 4{\text{ or }}k = 0\)</span></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;">\(k = 4\) <strong><em>A1 N2</em></strong></span></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;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct formula <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\log a - \log b = \log \frac{a}{b}\) , \(\log \left( {\frac{{40}}{5}} \right)\) , \(\log 8 + \log 5 - \log 5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Ignore missing or incorrect base. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\log _2}8\) , \({2^3} = 8\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _2}40 - {\log _2}5 = 3\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to write \(8\) as a power of \(2\) (seen anywhere) <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({({2^3})^{{{\log }_2}5}}\) , \({2^3} = 8\) , \({2^a}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">multiplying powers <em><strong> (M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({2^{3{{\log }_2}5}}\) , \(a{\log _2}5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({2^{{{\log }_2}125}}\) , \({\log _2}{5^3}\) , \({\left( {{2^{{{\log }_2}5}}} \right)^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({8^{{{\log }_2}5}} = 125\) <em><strong>A1 N3 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates readily earned marks in part (a). Some interpreted \({\log _2}40 - {\log _2}5\) to mean \(\frac{{{{\log }_2}40}}{{{{\log }_2}5}}\) , an error which led to no further marks. Others left the answer as \({\log _2}5\) where an integer answer is expected.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) proved challenging for most candidates, with few recognizing that changing \(8\) to base \(2\) is a helpful move. Some made it as far as \({2^{3{{\log }_2}5}}\) yet could not make that final leap to an integer.</span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(m = 3,{\text{ }}n = 4\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to apply \({({2^a})^b} = {2^{ab}}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;6x + 3,{\text{ }}4(2x - 3)\)</p>
<p class="p1">equating <strong>their </strong>powers of \(2\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;3(2x + 1) = 8x - 12\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;8x - 12 = 6x + 3,{\text{ }}2x = 15\)</p>
<p class="p1">\(x = \frac{{15}}{2}\;\;\;(7.5)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Indices laws were well understood with many candidates solving the equation correctly. Some candidates used logs, which took longer, and errors crept in.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Indices laws were well understood with many candidates solving the equation correctly. Some candidates used logs, which took longer, and errors crept in.</p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>correct use \(\log {x^n} = n\log x\) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(16\ln x\)</p>
<p>valid approach to find \(r\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{u_{n + 1}}}}{{{u_n}}},{\text{ }}\frac{{\ln {x^8}}}{{\ln {x^{16}}}},{\text{ }}\frac{{4\ln x}}{{8\ln x}},{\text{ }}\ln {x^4} = \ln {x^{16}} \times {r^2}\)</p>
<p>\(r = \frac{1}{2}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing a sum (finite or infinite) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({2^4}\ln x + {2^3}\ln x,{\text{ }}\frac{a}{{1 - r}},{\text{ }}{S_\infty },{\text{ }}16\ln x + \ldots \)</p>
<p>valid approach (seen anywhere) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)recognizing GP is the same as part (a), using <strong>their</strong> \(r\) value from part (a), \(r = \frac{1}{2}\)</p>
<p>correct substitution into infinite sum (only if \(\left| r \right|\) is a constant and less than 1) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{2^4}\ln x}}{{1 - \frac{1}{2}}},{\text{ }}\frac{{\ln {x^{16}}}}{{\frac{1}{2}}},{\text{ }}32\ln x\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\ln x = 2\)</p>
<p>\(x = {{\text{e}}^2}\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing Ann rolls green <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{\text{P(G)}}\)</p>
<p class="p1">\(\frac{3}{8}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(p = \frac{4}{8},{\text{ }}q = \frac{5}{8}\) or \(q = \frac{4}{8},{\text{ }}p = \frac{5}{8}\) <strong><em>A1A1 N2</em></strong></p>
<p>(ii) recognizes Ann and Bob lose \(9\) times <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\(\overbrace {{A_L}{B_\(\overbrace {{A_L}{B_ \ldots \(\overbrace {{A_L}{B_{\text{ 9 times, }}\underbrace {\left( {\frac{5}{8} \times \frac{4}{8}} \right) \times \ldots \times \left( {\frac{5}{8} \times \frac{4}{8}} \right)}_{{\text{9 times}}}\)</p>
<p>\(k = 9\;\;\;\)(seen anywhere) <strong><em>A1 N2</em></strong></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{\left( {\frac{5}{8} \times \frac{4}{8}} \right)^9} \times \frac{3}{8},{\text{ }}\left( {\frac{5}{8} \times \frac{4}{8}} \right) \times \ldots \times \left( {\frac{5}{8} \times \frac{4}{8}} \right) \times \frac{3}{8}\)</p>
<p>\(r = \frac{{20}}{{64}}\;\;\;\left( { = \frac{5}{{16}}} \right)\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognize the probability is an infinite sum <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)Ann wins on her \({{\text{1}}^{{\text{st}}}}\) roll or \({{\text{2}}^{{\text{nd}}}}\) roll or \({{\text{3}}^{{\text{rd}}}}\) roll…, \({S_\infty }\)</p>
<p class="p1">recognizing GP <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({u_1} = \frac{3}{8}\;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(r = \frac{{20}}{{64}}\;\;\;\)(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">correct substitution into infinite sum of GP <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{\frac{3}{8}}}{{1 - \frac{5}{{16}}}},{\text{ }}\frac{3}{8}\left( {\frac{1}{{1 - \left( {\frac{5}{8} \times \frac{4}{8}} \right)}}} \right),{\text{ }}\frac{1}{{1 - \frac{5}{{16}}}}\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\;\;\;\frac{{\frac{3}{8}}}{{\frac{{11}}{{16}}}},{\text{ }}\frac{3}{8} \times \frac{{16}}{{11}}\)</p>
<p class="p1">\({\text{P (Ann wins)}} = \frac{{48}}{{88}}\;\;\;\left( { = \frac{6}{{11}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
<p class="p1"><strong><em>Total [15 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some teachers’ comments suggested that the word ‘loses’ in the diagram was misleading. But candidate scripts did not indicate any adverse effect.</p>
<p class="p1">a) <span class="Apple-converted-space"> </span>Very well answered.</p>
<p class="p1">b) <span class="Apple-converted-space"> </span>i) Probabilities \(p\) and \(q\) were typically found correctly. ii) Fewer candidates identified the common ratio and number of rolls correctly.</p>
<p class="p1">Few candidates recognized that this was an infinite geometric sum although some did see that a geometric progression was involved.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some teachers’ comments suggested that the word ‘loses’ in the diagram was misleading, But candidate scripts did not indicate any adverse effect.</p>
<p class="p1">a) <span class="Apple-converted-space"> </span>Very well answered.</p>
<p class="p1">b) <span class="Apple-converted-space"> </span>i) Probabilities \(p\) and \(q\) were typically found correctly. ii) Fewer candidates identified the common ratio and number of rolls correctly.</p>
<p class="p1">Few candidates recognized that this was an infinite geometric sum although some did see that a geometric progression was involved.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Some teachers’ comments suggested that the word ‘loses’ in the diagram was misleading, But candidate scripts did not indicate any adverse effect.</p>
<p class="p1">a) <span class="Apple-converted-space"> </span>Very well answered.</p>
<p class="p1">b) <span class="Apple-converted-space"> </span>i) Probabilities \(p\) and \(q\) were typically found correctly. ii) Fewer candidates identified the common ratio and number of rolls correctly.</p>
<p class="p1">Few candidates recognized that this was an infinite geometric sum although some did see that a geometric progression was involved.</p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>correct application of \(\log a + \log b = \log ab\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\log _2}(2\sin x\cos x),{\text{ }}\log 2 + \log (\sin x) + \log (\cos x)\)</p>
<p>correct equation without logs <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(2\sin x\cos x = {2^{ - 1}},{\text{ }}\sin x\cos x = \frac{1}{4},{\text{ }}\sin 2x = \frac{1}{2}\)</p>
<p>recognizing double-angle identity (seen anywhere) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\log (\sin 2x),{\text{ }}2\sin x\cos x = \sin 2x,{\text{ }}\sin 2x = \frac{1}{2}\)</p>
<p>evaluating \({\sin ^{ - 1}}\left( {\frac{1}{2}} \right) = \frac{\pi }{6}{\text{ }}(30^\circ )\) <strong><em>(A1)</em></strong></p>
<p>correct working <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(x = \frac{\pi }{{12}} + 2\pi ,{\text{ }}2x = \frac{{25\pi }}{6},{\text{ }}\frac{{29\pi }}{6},{\text{ }}750^\circ ,{\text{ }}870^\circ ,{\text{ }}x = \frac{\pi }{{12}}\)<strong>and</strong> \(x = \frac{{5\pi }}{{12}}\), one correct final answer</p>
<p>\(x = \frac{{25\pi }}{{12}},{\text{ }}\frac{{29\pi }}{{12}}\) (do not accept additional values) <strong><em>A2</em></strong> <strong><em>N0</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\log _3}27 = 3\) <strong><em>A1 N1</em></strong></span></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;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\log _8}\frac{1}{8} = - 1\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[1 mark]</strong></em></span></p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\log _{16}}4 = \frac{1}{2}\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[1 mark]</strong></em></span></p>
<div class="question_part_label">a(iii).</div>
</div>
<div class="question" style="padding-left: 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;">correct equation with <strong>their </strong>three values <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(\frac{3}{2} = {\log _4}x{\text{, }}3 + ( - 1) - \frac{1}{2} = {\log _4}x\)</span></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;">correct working involving powers <strong><em>(A1)</em></strong></span></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;"><em>eg</em> \(x = {4^{\frac{3}{2}}}{\text{, }}{4^{\frac{3}{2}}} = {4^{{{\log }_4}x}}\)</span></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;">\(x = 8\) <strong><em>A1 N2</em></strong></span></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;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing \(\log a + \log b = \log ab\) (seen anywhere) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _2}(x(x - 2))\) , \({x^2} - 2x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing \({\log _a}b = x \Leftrightarrow {a^x} = b\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({2^3} = 8\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct simplification <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x(x - 2) = {2^3}\) , \({x^2} - 2x - 8\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct approach to solve <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. factorizing, quadratic formula</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \((x - 4)(x + 2)\) , \(\frac{{2 \pm \sqrt {36} }}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 4\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Candidates secure in their understanding of logarithm properties usually had success with this </span><span style="font-family: times new roman,times; font-size: medium;">problem, solving the resulting quadratic either by factoring or using the quadratic formula. </span><span style="font-family: times new roman,times; font-size: medium;">The majority of successful candidates correctly rejected the solution that was not in the </span><span style="font-family: times new roman,times; font-size: medium;">domain. A number of candidates, however, were unclear on logarithm properties. Some </span><span style="font-family: times new roman,times; font-size: medium;">unsuccessful candidates were able to demonstrate understanding of one property but without </span><span style="font-family: times new roman,times; font-size: medium;">both were not able to make much progress. A few candidates employed a “guess and check” </span><span style="font-family: times new roman,times; font-size: medium;">strategy, but this did not earn full marks.</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>Note: </strong>Accept sloppy notation (such as missing brackets, or binomial coefficient which includes \({x^2}\)).</p>
<p> </p>
<p>evidence of valid binomial expansion with binomial coefficients <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){(3x)^r}{(1)^{n - r}},{\text{ }}{(3x)^n} + n{(3x)^{n - 1}} + \left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right){(3x)^{n - 2}} + \ldots ,{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){(1)^{n - r}}{(3x)^r}\)</p>
<p>attempt to identify correct term <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} n \\ {n - 2} \end{array}} \right),{\text{ }}{(3x)^2},{\text{ }}n - r = 2\)</p>
<p>setting <strong>correct </strong>coefficient or term equal to \(135n\) (may be seen later) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\;\;\;9\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) = 135n,{\text{ }}\left( {\begin{array}{*{20}{c}} n \\ {n - 2} \end{array}} \right){(3x)^2} = 135n,{\text{ }}\frac{{9n(n - 1)}}{2} = 135n{x^2}\)</p>
<p>correct working for binomial coefficient (using \(_n{C_r}\) formula) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{n(n - 1)(n - 2)(n - 3) \ldots }}{{2 \times 1 \times (n - 2)(n - 3)(n - 4) \ldots }},{\text{ }}\frac{{n(n - 1)}}{2}\)</p>
<p><strong>EITHER</strong></p>
<p>evidence of correct working (with linear equation in \(n\)) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{9(n - 1)}}{2} = 135,{\text{ }}\frac{{9(n - 1)}}{2}{x^2} = 135{x^2}\)</p>
<p>correct simplification <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;n - 1 = \frac{{135 \times 2}}{9},{\text{ }}\frac{{(n - 1)}}{2} = 15\)</p>
<p>\(n = 31\) <strong><em>A1 N2</em></strong></p>
<p><strong>OR</strong></p>
<p>evidence of correct working (with quadratic equation in \(n\)) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;9{n^2} - 279n = 0,{\text{ }}{n^2} - n = 30n,{\text{ (9}}{{\text{n}}^2} - 9n){x^2} = 270n{x^2}\)</p>
<p>evidence of solving <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;9n(n - 31) = 0,{\text{ }}9{n^2} = 279n\)</p>
<p>\(n = 31\) <strong><em>A1 N2</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A0 </em></strong>for additional answers.</p>
<p> </p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to apply rules of logarithms <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln {a^b} = b\ln a\) , \(\ln ab = \ln a + \ln b\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct application of \(\ln {a^b} = b\ln a\) (seen anywhere) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3\ln x = \ln {x^3}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct application of \(\ln ab = \ln a + \ln b\) (seen anywhere) <strong><em>A1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln 5{x^3} = \ln 5 + \ln {x^3}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">so \(\ln 5{x^3} = \ln 5 + 3\ln x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(g(x) = f(x) + \ln 5\) (accept \(g(x) = 3\ln x + \ln 5\) ) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">transformation with correct name, direction, and value <strong><em>A3</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. translation by \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>{\ln 5}<br>\end{array}} \right)\) </span><span style="font-family: times new roman,times; font-size: medium;">, shift up by \(\ln 5\) , vertical translation of \(\ln 5\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was very poorly done by the majority of candidates. While candidates seemed to have a vague idea of how to apply the rules of logarithms in part (a), very few did so successfully. The most common error in part (a) was to begin incorrectly with \(\ln 5{x^3} = 3\ln 5x\) . This error was often followed by other errors. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), very few candidates were able to describe the transformation as a vertical translation (or shift). Many candidates attempted to describe numerous incorrect transformations, and some left part (b) entirely blank. </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find <em>d</em> <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{u_3} - {u_1}}}{2}\) , \(8 = 2 + 2d\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(d = 3\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{20}} = 2 + (20 - 1)3\) , \({u_{20}} = 3 \times 20 - 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({u_{20}} = 59\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{20}} = \frac{{20}}{2}(2 + 59)\) , \({S_{20}} = \frac{{20}}{2}(2 \times 2 + 19 \times 3)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({S_{20}} = 610\) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was answered correctly by the large majority of candidates. The few mistakes seen were due to either incorrect substitution into the formula or simple arithmetic errors. Even where candidates made mistakes, they were usually able to earn full follow-through marks in the subsequent parts of the question. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was answered correctly by the large majority of candidates. The few mistakes seen were due to either incorrect substitution into the formula or simple arithmetic errors. Even where candidates made mistakes, they were usually able to earn full follow-through marks in the subsequent parts of the question. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was answered correctly by the large majority of candidates. The few mistakes seen were due to either incorrect substitution into the formula or simple arithmetic errors. Even where candidates made mistakes, they were usually able to earn full follow-through marks in the subsequent parts of the question. </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to expand \({\left( {1 + \frac{2}{3}x} \right)^n}\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. Pascal's triangle, \({\left( {1 + \frac{2}{3}x} \right)^n} = 1 + \frac{2}{3}nx + \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct first two terms of \({\left( {1 + \frac{2}{3}x} \right)^n}\) (seen anywhere) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(1 + \frac{2}{3}nx\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct first two terms of quadratic (seen anywhere) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 9 , \(6nx\) , \((9 + 6nx + {n^2}{x^2})\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct calculation for the <em>x</em>-term <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{2}{3}nx \times 9 + 6nx\) , \(6n + 6n\) , \(12n\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6n + 6n = 84\) , \(12nx = 84x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 7\) <em><strong>A1</strong></em> <em><strong>N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks] </span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This question proved quite challenging for the majority of candidates, although there were a small number who were able to find the correct value of <em>n</em> using algebraic and investigative methods. While most candidates recognized the need to apply the binomial theorem, the majority seemed to have no idea how to do so when the exponent was a variable, <em>n</em>, rather than a known integer. Most candidates who attempted this question did expand the quadratic correctly, but many went no further, or simply set the<em> x</em>-term of the quadratic equal to \(84x\), ignoring the expansion of the first binomial altogether. </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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>infinite sum of segments is 2 (seen anywhere) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(p + {p^2} + {p^3} + \ldots = 2,{\text{ }}\frac{{{u_1}}}{{1 - r}} = 2\)</p>
<p>recognizing GP <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)ratio is \(p,{\text{ }}\frac{{{u_1}}}{{1 - r}},{\text{ }}{u_n} = {u_1} \times {r^{n - 1}},{\text{ }}\frac{{{u_1}({r^n} - 1)}}{{r - 1}}\)</p>
<p>correct substitution into \({S_\infty }\) formula (may be seen in equation) <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{p}{{1 - p}}\)</p>
<p>correct equation <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{p}{{1 - p}} = 2,{\text{ }}p = 2 - 2p\)</p>
<p>correct working leading to answer <strong><em>A1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3p = 2,{\text{ }}2 - 3p = 0\)</p>
<p>\(p = \frac{2}{3}{\text{ (cm)}}\) <strong><em>AG N0</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing infinite geometric series with squares <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({k^2} + {k^4} + {k^6} + \ldots ,{\text{ }}\frac{{{k^2}}}{{1 - {k^2}}}\)</p>
<p>correct substitution into \({S_\infty } = \frac{9}{{16}}\) (must substitute into formula) <strong><em>(A2)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{k^2}}}{{1 - {k^2}}} = \frac{9}{{16}}\)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(16{k^2} = 9 - 9{k^2},{\text{ }}25{k^2} = 9,{\text{ }}{k^2} = \frac{9}{{25}}\)</p>
<p>\(k = \frac{3}{5}\) (seen anywhere) <strong><em>A1</em></strong></p>
<p>valid approach with segments and CD (may be seen earlier) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(r = k,{\text{ }}{S_\infty } = b\)</p>
<p>correct expression for \(b\) in terms of \(k\) (may be seen earlier) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(b = \frac{k}{{1 - k}},{\text{ }}b = \sum\limits_{n = 1}^\infty {{k^n},{\text{ }}b = k + {k^2} + {k^3} + \ldots } \)</p>
<p>substituting <strong>their</strong> value of \(k\) into <strong>their</strong> formula for \(b\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\frac{3}{5}}}{{1 - \frac{3}{5}}},{\text{ }}\frac{{\left( {\frac{3}{5}} \right)}}{{\left( {\frac{2}{5}} \right)}}\)</p>
<p>\(b = \frac{3}{2}\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding \(f'(x) = \frac{1}{2}x\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find \(f'(4)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct value \(f'(4) = 2\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation in any form <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. </span><span style="font-family: times new roman,times; font-size: medium;">\(y - 6 = 2(x - 4)\) , \(y = 2x - 2\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = \int_2^{12} {\frac{{90}}{{3x + 4}}} {\rm{d}}x\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct integral <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30\ln (3x + 4)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting limits and subtracting <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30\ln (3 \times 12 + 4) - 30\ln (3 \times 2 + 4)\) , \(30\ln 40 - 30\ln 10\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30(\ln 40 - \ln 10)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct application of \(\ln b - \ln a\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(30\ln \frac{{40}}{{10}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 30\ln 4\) <em><strong>A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, area <em>h</em> = area <em>g</em> , 120 + <strong>their</strong> answer from (b)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 120 + 30\ln 4\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While most candidates answered part (a) correctly, finding the equation of the tangent, there were some who did not consider the value of their derivative when \(x = 4\) . </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), most candidates knew that they needed to integrate to find the area, but errors in integration, and misapplication of the rules of logarithms kept many from finding the correct area. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), it was clear that a significant number of candidates understood the idea of the reflected function, and some recognized that the integral was the negative of the integral from part (b), but only a few recognized the relationship between the areas. Many thought the area between <em>h</em> and the <em>x</em>-axis was 120. </span></p>
<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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct approach <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln 5 - \ln 3\)</p>
<p class="p1"><span class="Apple-converted-space">\(\ln \left( {\frac{5}{3}} \right) = y - x\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">recognizing factors of 45 </span>(may be seen in log expansion) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln (9 \times 5),{\text{ }}3 \times 3 \times 5,{\text{ }}\log {3^2} \times \log 5\)</p>
<p class="p2">correct application of \(\log (ab) = \log a + \log b\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln 9 + \ln 5,{\text{ }}\ln 3 + \ln 3 + \ln 5,{\text{ }}\ln {3^2} + \ln 5\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(2\ln 3 + \ln 5,{\text{ }}x + x + y\)</p>
<p class="p1"><span class="Apple-converted-space">\(\ln 45 = 2x + y\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates were able to earn some or all the marks on this question. Part (a) was answered correctly by nearly all candidates.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates were able to earn some or all the marks on this question. In part (b), the majority of candidates knew they needed to factor 45, though some did not apply the log rules correctly to earn all the available marks here.</p>
<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>
</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 \({\log _3}\left( {\frac{p}{q}} \right)\) .</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 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>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct formula <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\log {u^n} = n\log u\) , \(2{\log _3}p\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\log _3}({p^2}) = 12\) </span><strong><span style="font-family: times new roman,times; font-size: medium;"><em>A1 N2</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid method using \(p = {3^6}\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\log _3}{({3^6})^2}\) , \(\log {3^{12}}\) , \(12{\log _3}3\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({\log _3}({p^2}) = 12\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2 </span></strong></em></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong></span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct formula <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(\log \left( {\frac{p}{q}} \right) = \log p - \log q\) , \(6 - 7\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _3}\left( {\frac{p}{q}} \right) = - 1\) <strong><em>A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid method using \(p = {3^6}\) and \(q = {3^7}\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\log _3}\left( {\frac{{{3^6}}}{{{3^7}}}} \right)\) , \(\log {3^{ - 1}}\) , \( - {\log _3}3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _3}\left( {\frac{p}{q}} \right) = - 1\) <strong><em>A1 N2 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></em></strong></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct formula <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\log _3}uv = {\log _3}u + {\log _3}v\) , \(\log 9 + \log p\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _3}9 = 2\) (may be seen in expression) <strong><em> A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \(2 + \log p\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _3}(9p) = 8\) <strong><em>A1 N2</em> </strong></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid method using \(p = {3^6}\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\log _3}(9 \times {3^6})\) , \({\log _3}({3^2} \times {3^6})\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> \({\log _3}9 + {\log _3}{3^6}\) , \({\log _3}{3^8}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\log _3}(9p) = 8\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [7 marks]<br></span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<div style="font-size: 13.28px; font-family: sans-serif; left: 381.347px; top: 884.773px; transform: scale(1.03143, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="319.7425695290709"><span style="font-size: medium; font-family: times new roman,times;">This question proved to be surprisingly challenging for many candidates. A common misunderstanding was to set \(p\) equal to \(6\) and \(q\) equal to \(7\). A large number of candidates had trouble applying the rules of logarithms, and made multiple errors in each part of the question. Common types of errors included incorrect working such as \({\log _3}{p^2} = 36\) in part (a), \({\log _3}\left( {\frac{p}{q}} \right) = \frac{{{{\log }_3}6}}{{{{\log }_3}7}}\) or \({\log _3}\left( {\frac{p}{q}} \right) = {\log _3}6 - {\log _3}7\) in part (b), and \({\log _3}(9p) = 54\) in part (c). </span></div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<div style="font-size: 13.28px; font-family: sans-serif; left: 381.347px; top: 884.773px; transform: scale(1.03143, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="319.7425695290709"><span style="font-size: medium; font-family: times new roman,times;">This question proved to be surprisingly challenging for many candidates. A common misunderstanding was to set \(p\) equal to \(6\) and \(q\) equal to \(7\). A large number of candidates had trouble applying the rules of logarithms, and made multiple errors in each part of the question. Common types of errors included incorrect working such as \({\log _3}{p^2} = 36\) in part (a), \({\log _3}\left( {\frac{p}{q}} \right) = \frac{{{{\log }_3}6}}{{{{\log }_3}7}}\) or \({\log _3}\left( {\frac{p}{q}} \right) = {\log _3}6 - {\log _3}7\) in part (b), and \({\log _3}(9p) = 54\) in part (c). </span></div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<div style="font-size: 13.28px; font-family: sans-serif; left: 381.347px; top: 884.773px; transform: scale(1.03143, 1); transform-origin: 0% 0% 0px;" dir="ltr" data-font-name="Helvetica" data-canvas-width="319.7425695290709"><span style="font-size: medium; font-family: times new roman,times;">This question proved to be surprisingly challenging for many candidates. A common misunderstanding was to set \(p\) equal to \(6\) and \(q\) equal to \(7\). A large number of candidates had trouble applying the rules of logarithms, and made multiple errors in each part of the question. Common types of errors included incorrect working such as \({\log _3}{p^2} = 36\) in part (a), \({\log _3}\left( {\frac{p}{q}} \right) = \frac{{{{\log }_3}6}}{{{{\log }_3}7}}\) or \({\log _3}\left( {\frac{p}{q}} \right) = {\log _3}6 - {\log _3}7\) in part (b), and \({\log _3}(9p) = 54\) in part (c). </span></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;">Let \(f(x) = {{\rm{e}}^{x + 3}}\) .</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;">(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>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) interchanging <em>x</em> and <em>y</em> (seen anywhere) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = {{\rm{e}}^{y + 3}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct manipulation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln x = y + 3\) , \(\ln y = x + 3\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}(x) = \ln x - 3\) <em><strong>AG N0</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(x > 0\) <em><strong>A1 N1</strong></em> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">collecting like terms; using laws of logs <em><strong>(A1)(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln x - \ln \left( {\frac{1}{x}} \right) = 3\) , \(\ln x + \ln x = 3\) , \(\ln \left( {\frac{x}{{\frac{1}{x}}}} \right) = 3\) , \(\ln {x^2} = 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplify <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\ln x = \frac{3}{2}\) , \({x^2} = {{\rm{e}}^3}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = {{\rm{e}}^{\frac{3}{2}}}\left( { = \sqrt {{{\rm{e}}^3}} } \right)\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates interchanged the \(x\) and \(y\) to find the inverse function, but very few could write down the correct domain of the inverse, often giving \(x \ge 0\) , \(x > 3\) and "all real numbers" as responses. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Where students attempted to solve the equation in (b), most treated \(\ln x - 3\) as \(\ln (x - 3)\) and created an incorrect equation from the outset. The few who applied laws of logarithms often carried the algebra through to completion. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, the first term is 8 and the second term is 5.</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.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>subtracting terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(5 - 8,{\text{ }}{u_2} - {u_1}\)</p>
<p>\(d = - 3\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({u_{10}} = 8 + (10 - 1)( - 3),{\text{ }}8 - 27,{\text{ }} - 3(10) + 11\)</p>
<p>\({u_{10}} = - 19\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula for sum <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({S_{10}} = \frac{{10}}{2}(8 - 19),{\text{ 5}}\left( {2(8) + (10 - 1)( - 3)} \right)\)</p>
<p>\({S_{10}} = - 55\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<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>
</div>
<div class="specification">
<p class="p1"><span class="s1">The \(y\)</span>-coordinate of P is \(\ln 27\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing \(f'(x) = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(6 - 2x = 0\)</p>
<p class="p1"><span class="Apple-converted-space">\(x = 3\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">evidence of integration <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {f',{\text{ }}\int {\frac{{6 - 2x}}{{6x - {x^2}}}{\text{d}}x} } \)</p>
<p class="p1">using substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {\frac{1}{u}{\text{d}}u} \) where \(u = 6x - {x^2}\)</p>
<p class="p1">correct integral <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln (u) + c,{\text{ }}\ln (6x - {x^2})\)</p>
<p class="p1">substituting \((3,{\text{ }}\ln 27)\) into <strong>their </strong>integrated expression (must have \(c\)) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln (6 \times 3 - {3^2}) + c = \ln 27,{\text{ }}\ln (18 - 9) + \ln k = \ln 27\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em></span>\(\,\,\,\,\,\)\(c = \ln 27 - \ln 9\)</p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p2"><span class="Apple-converted-space">\(c = \ln 3\) </span><span class="s2"><strong><em>(A1)</em></strong></span></p>
<p class="p1">attempt to substitute <strong>their </strong>value of \(c\) into \(f(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = \ln (6x - {x^2}) + \ln 3\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">attempt to substitute <strong>their </strong>value of \(c\) into \(f(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = \ln (6x - {x^2}) + \ln 27 - \ln 9\)</p>
<p class="p1">correct use of a log law <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f(x) = \ln (6x - {x^2}) + \ln \left( {\frac{{27}}{9}} \right),{\text{ }}f(x) = \ln \left( {27(6x - {x^2})} \right) - \ln 9\)</p>
<p class="p1"><span class="Apple-converted-space">\(f(x) = \ln \left( {3(6x - {x^2})} \right)\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(a = 3\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\ln 27}}{{\ln 3}}\)</p>
<p class="p1">correct use of log law <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{3\ln 3}}{{\ln 3}},{\text{ }}{\log _3}27\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = 3\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part a) was well answered.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part b) most candidates realised that integration was required but fewer recognised the need to use integration by substitution. Quite a number of candidates who integrated correctly omitted finding the constant of integration.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part c) many candidates showed good understanding of transformations and could apply them correctly, however, correct use of the laws of logarithms was challenging for many. In particular, a common error was \(\frac{{\ln 27}}{{\ln 3}} = \ln 9\).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br>