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</div><h2>SL Paper 2</h2><div class="question">
<p class="p1">Consider the expansion of \({\left( {\frac{{{x^3}}}{2} + \frac{p}{x}} \right)^8}\). The constant term is \(5103\). Find the possible values of \(p\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach to find the required term <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^{8 - r}}{\left( {\frac{p}{x}} \right)^r},{\text{ }}{\left( {\frac{{{x^3}}}{2}} \right)^8}{\left( {\frac{p}{x}} \right)^0} + \left( {\begin{array}{*{20}{c}} 8 \\ 1 \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^7}{\left( {\frac{p}{x}} \right)^1} + \ldots \), Pascal’s triangle to required value</p>
<p>identifying constant term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\({7^{{\text{th}}}}{\text{ term, }}r = 6,{\text{ }}{\left( {\frac{1}{2}} \right)^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right),{\text{ }}{\left( {\frac{{{x^3}}}{2}} \right)^2}{\left( {\frac{p}{x}} \right)^6}\)</p>
<p>correct calculation (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^2}{\left( {\frac{p}{x}} \right)^6},{\text{ }}\frac{{8 \times 7}}{2} \times \frac{{{p^6}}}{{{2^2}}}\)</p>
<p>setting up equation with <strong>their </strong>constant term equal to \(5103\) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;\)\(\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){\left( {\frac{{{x^3}}}{2}} \right)^2}{\left( {\frac{p}{x}} \right)^6} = 5103,{\text{ }}{p^6} = \frac{{5103}}{7}\)</p>
<p>\(p = \pm 3\) <strong><em>A1A1 N3</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Candidates tended to either do very well or very poorly in this question. Some had difficulty understanding what the constant term was, while others were unable to find the value of \(r\) that led to the constant term. Many algebraic errors were seen in the calculation of the term, mostly having to do with forgetting to square \(\frac{1}{2}\). Some missed the negative solution for <em>p</em>, despite the fact that the question asked for the “values” of \(p\).</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Expand \(\sum\limits_{r = 4}^7 {{2^r}} \)</span><span style="font-family: times new roman,times; font-size: medium;"> as the sum of four terms.</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;">(i) Find the value of \(\sum\limits_{r = 4}^{30} {{2^r}} \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Explain why \(\sum\limits_{r = 4}^\infty {{2^r}} \) cannot be evaluated.</span></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><span style="font-family: times new roman,times; font-size: medium;">\(\sum\limits_{r = 4}^7 {{2^r}} = {2^4} + {2^5} + {2^6} + {2^7}\) (accept \(16 + 32 + 64 + 128\) ) <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;">(i) <strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing a GP <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({u_1} = {2^4}\) , \(r = 2\) , \(n = 27\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>(A1)</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for sum <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{27}} = \frac{{{2^4}({2^{27}} - 1)}}{{2 - 1}}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({S_{27}} = 2147483632\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N4</strong> </span></em></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;">recognizing \(\sum\limits_{r = 4}^{30} { = \sum\limits_{r = 1}^{30} { - \sum\limits_{r = 1}^3 {} } } \) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing GP with \({u_1} = 2\) , \(r = 2\) , \(n = 30\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for sum </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_{30}} = \frac{{2({2^{30}} - 1)}}{{2 - 1}}\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 214783646\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sum\limits_{r = 4}^{30} {{2^r}} = 2147483646 - (2 + 4 + 8)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 2147483632\) <em><strong> A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) valid reason (e.g. <strong>infinite</strong> GP, diverging series), <strong>and</strong> \(r \ge 1\) (accept \(r > 1\) ) <em><strong>R1R1 N2</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>
<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 proved difficult for many candidates. A number of students seemed unfamiliar with sigma notation. Many were successful with part (a), although some listed terms or found an overall sum with no working. </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;">The results for part (b) were much more varied. Many candidates did not realize that \(n\) was \(27\) and used \(30\) instead. Very few candidates gave a complete explanation why the infinite series could not be evaluated; candidates often claimed that the value could not be found because there were an infinite number of terms. </span></p>
<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;"><span>Expand \({(x - 2)^4}\)</span><span> and simplify your result.</span></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 term in \({x^3}\) in \((3x + 4){(x - 2)^4}\) .</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. \({(x - 2)^4} = {x^4} + 4{x^3}( - 2) + 6{x^2}{( - 2)^2} + 4x{( - 2)^3} + {( - 2)^4}\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({(x - 2)^4} = {x^4} - 8{x^3} + 24{x^2} - 32x + 16\)</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;">finding coefficients, \(3 \times 24( = 72)\) , \(4 \times( - 8)( = - 32)\) <strong><em>(A1)(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">t</span><span style="font-family: times new roman,times; font-size: medium;">erm is \(40{x^3}\) <em><strong>A1 N3</strong> </em></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;"><em>[3 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;">Where candidates recognized the binomial nature of the expression, many completed the expansion successfully, although some omitted the negative signs. </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 candidates recognized the binomial nature of the expression, many completed the expansion successfully, although some omitted the negative signs. Few recognized that only the multiplications that achieve an index of 3 are required in part (b) and distributed over the entire expression. Others did not recognize that two terms in the expansion must be combined. </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 \(f(x) = {\log _3}\frac{x}{2} + {\log _3}16 - {\log _3}4\) , 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 \(f(x) = {\log _3}2x\) .</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;">Find the value of \(f(0.5)\) and of \(f(4.5)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function <em>f</em> can also be written in the form \(f(x) = \frac{{\ln ax}}{{\ln b}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the value of <em>a</em> and of <em>b</em> .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence on graph paper, <strong>sketch</strong> the graph of <em>f</em> , for \( - 5 \le x \le 5\) , \( - 5 \le y \le 5\) , </span><span style="font-family: times new roman,times; font-size: medium;">using a scale of 1 cm to 1 unit on each axis.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the equation of the asymptote.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i), (ii) and (iii).</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 value of \({f^{ - 1}}(0)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</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 point A lies on the graph of <em>f</em> . At A, \(x = 4.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">On your diagram, sketch the graph of \({f^{ - 1}}\) , noting clearly the image of point A.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</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;">combining 2 terms <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _3}8x - {\log _3}4\) , \({\log _3}\frac{1}{2}x + {\log _3}4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">expression which clearly leads to answer given <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _3}\frac{{8x}}{4}\) , \({\log _3}\frac{{4x}}{2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = {\log _3}2x\) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute either value into <em>f</em> <strong> <em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({\log _3}1\) , \({\log _3}9\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(0.5) = 0\) , \(f(4.5) = 2\) <em><strong>A1A1 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>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(a = 2\) , \(b = 3\) <em><strong>A1A1 N1N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii)<br><img src="images/doc.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for sketch approximately through \((0.5 \pm 0.1{\text{, }}0 \pm 0.1)\) , </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for approximately correct shape, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for sketch asymptotic to the <em>y</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \(x = 0\) (must be an equation) <em><strong>A1 N1</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">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({f^{ - 1}}(0) = 0.5\) <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">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/home.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1A1 N4</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for sketch approximately through \((0 \pm 0.1{\text{, }}0.5 \pm 0.1)\) , </span><span style="font-family: times new roman,times; font-size: medium;"><strong><em>A1</em></strong> for approximately correct shape of the graph reflected over \(y = x\) , </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for sketch asymptotic to <em>x</em>-axis, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for point \((2 \pm 0.1{\text{, }}4.5 \pm 0.1)\) clearly marked and on curve.</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">e.</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;">Few candidates had difficulty with part (a) although it was often communicated using some very sloppy applications of the rules of logarithm, writing \(\frac{{\log 16}}{{\log 4}}\) instead of \(\log \left( {\frac{{16}}{4}} \right)\) . </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) was generally done well.</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;">Part (c) (i) was generally done well; candidates seemed quite comfortable changing bases. There were some very good sketches in (c) (ii), but there were also some very poor ones with candidates only considering shape and not the location of the <em>x</em>-intercept or the asymptote. A surprising number of candidates did not use the scale required by the question and/or did not use graph paper to sketch the graph. In some cases, it was evident that students simply transposed their graphs from their GDC without any analytical consideration. </span></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was poorly done as candidates did not consider the command term, “write down” and often proceeded to find the inverse function before making the appropriate substitution.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) eluded a great many candidates as most preferred to attempt to find the inverse analytically rather than simply reflecting the graph of <em>f</em> in the line \(y = x\) . This graph also suffered from the same sort of problems as the graph in (c) (ii). Some students did not have their curve passing through \((2{\text{, }}4.5)\) nor did they clearly indicate its position as instructed. This point was often mislabelled on the graph of <em>f</em>. The efforts in this question demonstrated that students often work tenuously from one question to the next, without considering the "big picture", thereby failing to make important links with earlier parts of the question.</span></p>
<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;">Consider the expansion of \({\left( {2{x^3} + \frac{b}{x}} \right)^8} = 256{x^{24}} + 3072{x^{20}} + \ldots + k{x^0} + \ldots \) .</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>b</em>.</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 <em>k</em>.</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;">valid attempt to find term in \({x^{20}}\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>8\\<br>1<br>\end{array}} \right)({2^7})(b)\) , \({(2{x^3})^7}\left( {\frac{b}{x}} \right) = 3072\)</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. \(\left( {\begin{array}{*{20}{c}}<br>8\\<br>1<br>\end{array}} \right)({2^7})(b) = 3072\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(b = 3\) <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;">evidence of choosing correct term <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 7th term, \(r = 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>8\\<br>6<br>\end{array}} \right){(2{x^3})^2}{\left( {\frac{3}{x}} \right)^6}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 81648\) (accept \(81600\) ) <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><span style="font-family: times new roman,times; font-size: medium;">An unfamiliar presentation confused a number of candidates who attempted to set up an equation with the wrong term in part (a). Time and again, candidates omitted the binomial coefficient in their set up leading to an incorrect result. In part (b) it was common to see the constant term treated as the last term of the expansion rather than the 7th term. </span></p>
<p> </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;">An unfamiliar presentation confused a number of candidates who attempted to set up an equation with the wrong term in part (a). Time and again, candidates omitted the binomial coefficient in their set up leading to an incorrect result. In part (b) it was common to see the constant term treated as the last term of the expansion rather than the 7th term. </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 series, the first term is −7 and the sum of the first 20 terms is 620.</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 common difference.</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 the 78<sup>th</sup> term.</span></p>
<div class="marks">[2]</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 substitute into sum formula for AP <em><strong> M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{20}} = \frac{{20}}{2}(2( - 7) + 19d)\) , \(\frac{{20}}{2}( - 7 + {u_{20}})\) </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting up correct equation using sum formula <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{20}}{2}(2( - 7) + 19d = 620\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(d = 4\) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution \( - 7 + 77(4)\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({u_{78}} = 301\) <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>
<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>The following table shows values of ln <em>x</em> and ln <em>y</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The relationship between ln <em>x</em> and ln <em>y</em> can be modelled by the regression equation ln <em>y</em> = <em>a</em> ln <em>x</em> + <em>b</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>a</em> and of <em>b</em>.</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>Use the regression equation to estimate the value of <em>y</em> when<em> x</em> = 3.57.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The relationship between <em>x</em> and <em>y</em> can be modelled using the formula <em>y</em> = <em>kx<sup>n</sup></em>, where <em>k</em> ≠ 0 , <em>n</em> ≠ 0 , <em>n</em> ≠ 1.</p>
<p>By expressing ln <em>y</em> in terms of ln <em>x</em>, find the value of <em>n</em> and of <em>k</em>.</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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em> one correct value</p>
<p>−0.453620, 6.14210</p>
<p><em>a</em> = −0.454, <em>b</em> = 6.14 <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong> (A1)</strong></em></p>
<p><em>eg </em>−0.454 ln 3.57 + 6.14</p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em> ln <em>y</em> = 5.56484</p>
<p>261.083 (260.409 from 3 sf)</p>
<p><em>y</em> = 261, (<em>y</em> = 260 from 3sf) <em><strong>A1 N3</strong></em></p>
<p><strong>Note:</strong> If no working shown, award <em><strong>N1</strong></em> for 5.56484.<br>If no working shown, award <em><strong>N2</strong> </em>for ln <em>y</em> = 5.56484.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>valid approach for expressing ln <em>y</em> in terms of ln <em>x</em> <em><strong>(M1)</strong></em></p>
<p><em>eg </em> \({\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b\)</p>
<p>correct application of addition rule for logs <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,k + {\text{ln}}\,\left( {{x^n}} \right)\)</p>
<p>correct application of exponent rule for logs <em><strong>A1</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,k + n\,{\text{ln}}\,x\)</p>
<p>comparing one term with regression equation (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(n = a,\,\,b = {\text{ln}}\,k\)</p>
<p>correct working for <em>k</em> <strong>(A1)</strong></p>
<p><em>eg </em>\({\text{ln}}\,k = 6.14210,\,\,\,k = {e^{6.14210}}\)</p>
<p>465.030</p>
<p>\(n = - 0.454,\,\,k = 465\) (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\({e^{{\text{ln}}\,y}} = {e^{a\,{\text{ln}}\,x + b}}\)</p>
<p>correct use of exponent laws for \({e^{a\,{\text{ln}}\,x + b}}\) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({e^{a\,{\text{ln}}\,x}} \times {e^b}\)</p>
<p>correct application of exponent rule for \(a\,{\text{ln}}\,x\) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,{x^a}\)</p>
<p>correct equation in<em> y</em> <em><strong>A1</strong></em></p>
<p><em>eg </em>\(y = {x^a} \times {e^b}\)</p>
<p>comparing one term with equation of model (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(k = {e^b},\,\,n = a\)</p>
<p>465.030</p>
<p>\(n = - 0.454,\,\,k = 465\) (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>valid approach for expressing ln <em>y</em> in terms of ln <em>x</em> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,y = {\text{ln}}\,\left( {k{x^n}} \right),\,\,{\text{ln}}\,\left( {k{x^n}} \right) = a\,{\text{ln}}\,x + b\)</p>
<p>correct application of exponent rule for logs (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,\left( {{x^a}} \right) + b\)</p>
<p>correct working for <em>b</em> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em>\(b = {\text{ln}}\,\left( {{e^b}} \right)\)</p>
<p>correct application of addition rule for logs <em><strong>A1</strong></em></p>
<p><em>eg </em>\({\text{ln}}\,\left( {{e^b}{x^a}} \right)\)</p>
<p>comparing one term with equation of model (check <em><strong>FT</strong></em>) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(k = {e^b},\,\,n = a\)</p>
<p>465.030</p>
<p>\(n = - 0.454,\,\,k = 465\) (464 from 3sf) <em><strong>A1A1 N2N2</strong></em></p>
<p><em><strong>[7 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.</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><span style="font-family: times new roman,times; font-size: medium;">Consider the arithmetic sequence 3, 9, 15, \(\ldots \) , 1353 .</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 common difference.</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 the number of terms in the sequence.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the sum of the 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><span style="font-family: times new roman,times; font-size: medium;">common difference is 6 <strong><em>A1 N1</em> </strong></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;">evidence of appropriate approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_n} = 1353\)</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;">e.g. \(1353 = 3 + (n - 1)6\) , \(\frac{{1353 + 3}}{6}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 226\) <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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{226}} = \frac{{226(3 + 1353)}}{2}\) , \(\frac{{226}}{2}(2 \times 3 + 225 \times 6)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_{226}} = 153228\) (accept 153000) <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates did well on this question. Any errors were usually arithmetic in nature but </span><span style="font-family: times new roman,times; font-size: medium;">candidates were able to obtain follow-through marks on errors made in earlier parts.</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;">Most candidates did well on this question. Any errors were usually arithmetic in nature but </span><span style="font-family: times new roman,times; font-size: medium;">candidates were able to obtain follow-through marks on errors made in earlier parts.</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;">Most candidates did well on this question. Any errors were usually arithmetic in nature but </span><span style="font-family: times new roman,times; font-size: medium;">candidates were able to obtain follow-through marks on errors made in earlier parts.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The first term of an infinite geometric sequence is 4. The sum of the infinite sequence is 200.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio.</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 sum of the first 8 terms.</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 least value of <em>n</em> for which <em>S</em><sub><em>n</em></sub> > 163.</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>correct substitution into infinite sum <em><strong>(A1)</strong></em><br><em>eg</em> \(200 = \frac{4}{{1 - r}}\)</p>
<p><em>r </em>= 0.98 (exact) <em><strong>A1 N2</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>correct substitution <em><strong>(A1)</strong></em></p>
<p>\(\frac{{4\left( {1 - {{0.98}^8}} \right)}}{{1 - 0.98}}\)</p>
<p>29.8473</p>
<p>29.8 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to set up inequality (accept equation) <em><strong>(M1)</strong></em><br><em>eg</em> \(\frac{{4\left( {1 - {{0.98}^n}} \right)}}{{1 - 0.98}} > 163,\,\,\frac{{4\left( {1 - {{0.98}^n}} \right)}}{{1 - 0.98}} = 163\)</p>
<p>correct inequality for <em>n</em> (accept equation) or crossover values <em><strong>(A1)</strong></em><br><em>eg n</em> > 83.5234, <em>n</em> = 83.5234, <em>S</em><sub><em>83</em></sub> = 162.606 <strong>and </strong><em>S</em><sub><em>84</em></sub> = 163.354</p>
<p><em>n</em> = 84 <em><strong>A1 N1</strong></em></p>
<p><strong><em>[3 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><span style="font-family: times new roman,times; font-size: medium;">An arithmetic sequence is given by \(5\), \(8\), \(11\), ….</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;">(a) Write down the value of \(d\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">Find</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \({u_{100}}\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \({S_{100}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) </span><span style="font-family: times new roman,times; font-size: medium;">Given that \({u_n} = 1502\) , find the value of \(n\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(d\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) \({u_{100}}\) ;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) \({S_{100}}\) .</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><span style="font-family: times new roman,times; font-size: medium;">Given that \({u_n} = 1502\) , find the value of \(n\) .</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><span style="font-family: times new roman,times; font-size: medium;">(a) \(d = 3\) <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>
<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;">(b) (i) correct substitution into term formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{100}} = 5 + 3(99)\) , \(5 + 3(100 - 1)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_{100}} = 302\) <strong><em>A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into sum formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({S_{100}} = \frac{{100}}{2}(2(5) + 99(3))\) , \({S_{100}} = \frac{{100}}{2}(5 + 302)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_{100}} = 15350\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></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;">(c) </span><span style="font-family: times new roman,times; font-size: medium;">correct substitution into term formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(1502 = 5 + 3(n - 1)\) , \(1502 = 3n + 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 500\) <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><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">Total [7 marks]<br></span></em></strong></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(d = 3\) <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;">(i) correct substitution into term formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{100}} = 5 + 3(99)\) , \(5 + 3(100 - 1)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_{100}} = 302\) <strong><em>A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into sum formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({S_{100}} = \frac{{100}}{2}(2(5) + 99(3))\) , \({S_{100}} = \frac{{100}}{2}(5 + 302)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_{100}} = 15350\) <em><strong>A1 N2 </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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into term formula <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(1502 = 5 + 3(n - 1)\) , \(1502 = 3n + 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 500\) <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><strong><em><span style="font-family: times new roman,times; font-size: medium;"> </span></em></strong></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">Total [7 marks]<br></span></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><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates had little difficulty with this question. If errors were made, they were normally made out of carelessness. A very few candidates mistakenly used the formulas for geometric sequences and series.</span></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The majority of candidates had little difficulty with this question. If errors were made, they were normally made out of carelessness. A very few candidates mistakenly used the formulas for geometric sequences and series.</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;">The majority of candidates had little difficulty with this question. If errors were made, they were normally made out of carelessness. A very few candidates mistakenly used the formulas for geometric sequences and series.</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;">The majority of candidates had little difficulty with this question. If errors were made, they were normally made out of carelessness. A very few candidates mistakenly used the formulas for geometric sequences and series.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the expansion of \({\left( {{x^2} + \frac{2}{x}} \right)^{10}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the number of terms of this expansion.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coefficient of \({x^8}\).</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"><span class="s1">11 </span>terms <strong><em>A1 N1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {10} \\ r \end{array}} \right){({x^2})^{10 - r}}{\left( {\frac{2}{x}} \right)^r},{\text{ }}{a^{10}}{b^0} + \left( {\begin{array}{*{20}{c}} {10} \\ 1 \end{array}} \right){a^9}{b^1}\left( {\begin{array}{*{20}{c}} {10} \\ 2 \end{array}} \right){a^8}{b^2} + \ldots \)</p>
<p class="p2">Pascal’s triangle to \({11^{th}}\) <span class="s1">row</span></p>
<p class="p1">valid attempt to find value of \(r\) which gives term in \({x^8}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({({x^2})^{10 - r}}\left( {\frac{1}{{{x^r}}}} \right) = {x^8},{\text{ }}{x^{2r}}{\left( {\frac{2}{x}} \right)^{10 - r}} = {x^8}\)</p>
<p class="p1">identifying required term (may be indicated in expansion) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(r = 6,{\text{ }}{{\text{5}}^{{\text{th}}}}{\text{ term, }}{{\text{7}}^{{\text{th}}}}{\text{ term}}\)</p>
<p class="p1">correct working (may be seen in expansion) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {10} \\ 6 \end{array}} \right){({x^2})^6}{\left( {\frac{2}{x}} \right)^4},{\text{ }}210 \times 16\)</p>
<p class="p1"><span class="s2">3360 <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><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;">
<p class="p1">Although slightly challenging, this question aimed at assessing candidates’ fluency at using the binomial theorem to find the coefficient of a term.</p>
<p class="p1">In part a), most candidates realized that the expansion had 11 terms, although a few answered 10.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part b), many candidates attempted to answer and knew what they needed to find. However, the execution of the plan was not always successful. A fair amount of students had difficulties with the powers of the factors of the required term and could only earn the first method mark for a valid approach. Some candidates gave the term instead of the coefficient as the answer. A few of them attempted to expand the binomial algebraically and very few added instead of multiplied, losing all marks.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider an infinite geometric sequence with \({u_1} = 40\) and \(r = \frac{1}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \({u_4}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the sum of the infinite sequence.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider an arithmetic sequence with <em>n</em> terms, with first term (\( - 36\)) and eighth </span><span style="font-family: times new roman,times; font-size: medium;">term (\( - 8\)) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the common difference.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \({S_n} = 2{n^2} - 38n\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</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 sum of the infinite geometric sequence is equal to twice the sum of the </span><span style="font-family: times new roman,times; font-size: medium;">arithmetic sequence. Find <em>n</em> .</span></p>
<div class="marks">[5]</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;">(i) correct approach <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_4} = (40){\frac{1}{2}^{(4 - 1)}}\) </span><span style="font-family: times new roman,times; font-size: medium;">, listing terms</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_4} = 5\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into formula for infinite sum <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_\infty } = \frac{{40}}{{1 - 0.5}}\) , \({S_\infty } = \frac{{40}}{{0.5}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_\infty } = 80\) <em><strong>A1 N2</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(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) attempt to set up expression for \({u_8}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 36 + (8 - 1)d\)</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;">e.g. \( - 8 = - 36 + (8 - 1)d\) , \(\frac{{ - 8 - ( - 36)}}{7}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(d = 4\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into formula for sum <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_n} = \frac{n}{2}(2( - 36) + (n - 1)4)\)</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;">e.g. \({S_n} = \frac{n}{2}(4n - 76)\) , \( - 36n + 2{n^2} - 2n\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_n} = 2{n^2} - 38n\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">multiplying \({S_n}\) (AP) by 2 or dividing <em>S</em> (infinite GP) by 2 <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2{S_n}\) , \(\frac{{{S_\infty }}}{2}\) , 40</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into \(2{S_n} = {S_\infty }\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2{n^2} - 38n = 40\) , \(4{n^2} - 76n - 80\) (\( = 0\))</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to solve <strong>their</strong> quadratic (equation) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. intersection of graphs, formula</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 20\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 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;">Most candidates found part (a) straightforward, although a common error in (a)(ii) was to calculate 40 divided by \(\frac{1}{2}\) as 20. </span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), some candidates had difficulty with the "show that" and worked backwards from the answer given. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates obtained the correct equation in part (c), although some did not reject the negative value of <em>n</em> as impossible in this context. </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;">Consider the infinite geometric sequence \(3000{\text{, }}- 1800{\text{, }}1080{\text{, }} - 648, \ldots \) . </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 common ratio.</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 the 10th 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><span style="font-family: times new roman,times; font-size: medium;">Find the <strong>exact</strong> sum of the infinite 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;">evidence of dividing two terms <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - \frac{{1800}}{{3000}}\) , \( - \frac{{1800}}{{1080}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(r = - 0.6\) <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;">evidence of substituting into the formula for the 10th term <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{10}} = 3000{( - 0.6)^9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_{10}} = 30.2\) (accept the exact value \( - 30.233088\)) <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;">evidence of substituting into the formula for the infinite sum <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(S = \frac{{3000}}{{1.6}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(S = 1875\) <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 generally well done by most candidates.</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;">This question was generally well done by most candidates, although quite a few showed </span><span style="font-family: times new roman,times; font-size: medium;">difficulty answering part (b) exactly or to three significant figures.</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;">This question was generally well done by most candidates, although quite a few showed </span><span style="font-family: times new roman,times; font-size: medium;">difficulty answering part (b) exactly or to three significant figures. Some candidates reversed </span><span style="font-family: times new roman,times; font-size: medium;">the division of terms to obtain a ratio of \( - \frac{5}{3}\). Of these, most did not recognize this ratio as an </span><span style="font-family: times new roman,times; font-size: medium;">inappropriate value when finding the sum in part (c).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">The third term in the expansion of \({(x + k)^8}\) is \(63{x^6}\). Find the possible values of \(k\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach to find the required term <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){x^{8 - r}}{k^r}\)<em>, </em>Pascal’s triangle to \({{\text{8}}^{{\text{th}}}}\) row, \({x^8} + 8{x^7}k + 28{x^6}{k^2} + \ldots \)</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ 2 \end{array}} \right){x^6}{k^2},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){x^6}{k^2},{\text{ }}r = 2\)</p>
<p>setting up equation in \(k\) with <strong>their </strong>coefficient/term <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;28{k^2}{x^6} = 63{x^6},{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 6 \end{array}} \right){k^2} = 63\)</p>
<p>\(k = \pm 1.5{\text{ (exact)}}\) <strong><em>A1A1 N3</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Candidates who recognized that the third term is required usually completed the question successfully, although some candidates only gave a single value for \(k\). A few candidates attempted to fully expand algebraically, which proved to be a fruitless enterprise.</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the expansion of \({(3{x^2} + 2)^9}\) .</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 number of terms in the expansion.</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 the term in \({x^4}\) .</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">10 terms <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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of binomial expansion <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({a^9}{b^0} + \left( \begin{array}{l}<br>9\\<br>1<br>\end{array} \right){a^8}b + \left( \begin{array}{l}<br>9\\<br>2<br>\end{array} \right){a^7}{b^2} + \ldots \), \(\left( \begin{array}{l}<br>9\\<br>r<br>\end{array} \right){(a)^{n - r}}{(b)^r}\) </span><span style="font-family: times new roman,times; font-size: medium;">, Pascal’s triangle</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct term <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 8th term, \(r = 7\) , \(\left( \begin{array}{l}<br>9\\<br>7<br>\end{array} \right)\) , \({(3{x^2})^2}{2^7}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression of complete term <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( \begin{array}{l}<br>9\\<br>7<br>\end{array} \right){(3{x^2})^2}{(2)^7}\) , \(_2^9C{(3{x^2})^2}{(2)^7}\) , \(36 \times 9 \times 128\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(41472{x^4}\) (accept \(41500{x^4}\) ) <em><strong>A1 N2</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 were familiar with the binomial expansion, although some expanded entirely which at times led to careless errors. Others attempted to use Pascal's Triangle. Common errors included misidentifying the binomial coefficient corresponding to this term and not squaring the 3 in (\(3{x^2}\)) . </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;">Many candidates were familiar with the binomial expansion, although some expanded entirely which at times led to careless errors. Others attempted to use Pascal's triangle. Common errors included misidentifying the binomial coefficient corresponding to this term and not squaring the 3 in \((3{x^2})\) .</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;">Consider the expansion of \({(x + 3)^{10}}\).</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 the number of terms in this expansion.</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 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 term containing \({x^3}\).</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;">11 terms <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';"><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.</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;">evidence of binomial expansion <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> \(\left( \begin{array}{c}n\\r\end{array} \right)\) \({a^{n - r}}{b^r}\), attempt to expand</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;">evidence of choosing correct term <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> \({8^{{\text{th}}}}{\text{ term, }}r = 7\), \(\left( \begin{array}{c}10\\7\end{array} \right)\), \({(x)^3}{(3)^7}\)</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> \(\left( \begin{array}{c}10\\7\end{array} \right)\) \({(x)^3}{(3)^7}\), \(\left( \begin{array}{c}10\\3\end{array} \right)\)\({(x)^3}{(3)^7}\),</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;">\(262440{x^3}{\text{ (accept }}262000{x^3})\) <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';"><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">
<p><span style="font-family: times new roman,times; font-size: medium;">The constant term in the expansion of \({\left( {\frac{x}{a} + \frac{{{a^2}}}{x}} \right)^6}\) , where \(a \in \mathbb{R}\) is \(1280\). Find \(a\) . </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;">evidence of binomial expansion <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> selecting correct term,\({\left( {\frac{x}{a}} \right)^6}{\left( {\frac{{{a^2}}}{x}} \right)^0} + \left( \begin{array}{l}<br>6\\<br>1<br>\end{array} \right){\left( {\frac{x}{a}} \right)^5}{\left( {\frac{{{a^2}}}{x}} \right)^1} + \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of identifying constant term in expansion for power \(6\) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(r = 3\) , 4<sup>th</sup> term </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct term (may be seen in equation) <strong><em>A2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(20\frac{{{a^6}}}{{{a^3}}}\) , \(\left( \begin{array}{l}<br>6\\<br>3<br>\end{array} \right){\left( {\frac{x}{a}} \right)^3}{\left( {\frac{{{a^2}}}{x}} \right)^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up <strong>their</strong> equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\left( \begin{array}{l}<br>6\\<br>3<br>\end{array} \right){\left( {\frac{x}{a}} \right)^3}{\left( {\frac{{{a^2}}}{x}} \right)^3} = 1280\), \({a^3} = 1280\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation in one variable \(a\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(20{a^3} = 1280\) , \({a^3} = 64\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 4\) <em><strong>A1 N4 </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;">Many candidates struggled with this question. Some had difficulty with the binomial expansion, while others did not understand that the constant term had no \(x\) , while still others were unable to simplify a ratio of exponentials with a common base. Some candidates found \(r = 3\) using algebraic methods while others found it by writing out the first several terms. In some cases, candidates just set the entire expansion equal to 1280.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ten students were surveyed about the number of hours, \(x\), they spent browsing the Internet during week <span class="s1">1 </span>of the school year. The results of the survey are given below.</p>
<p class="p1">\[\sum\limits_{i = 1}^{10} {{x_i} = 252,{\text{ }}\sigma = 5{\text{ and median}} = 27.} \]</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">During week </span><span class="s2">4</span>, the survey was extended to all <span class="s2">200 </span>students in the school. The results are shown in the cumulative frequency graph:</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_16.35.16.png" alt="N16/5/MATME/SP2/ENG/TZ0/08.d"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the mean number of hours spent browsing the Internet.</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"><span class="s1">During week </span><span class="s2">2</span>, the students worked on a major project and they each spent an additional five hours browsing the Internet. For week <span class="s2">2</span><span class="s3">, write down</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the mean;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the standard deviation.</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">During week </span><span class="s2">3 </span>each student spent <span class="s2">5% </span>less time browsing the Internet than during week <span class="s2">1</span>. For week <span class="s2">3</span><span class="s3">, find</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>the median;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>the variance.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the number of students who spent between <span class="s1">25 </span>and <span class="s1">30 </span><span class="s2">hours browsing the Internet.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Given that <span class="s1">10% </span>of the students spent more than <span class="s1"><em>k </em></span>hours browsing the Internet, find the maximum value of \(k\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute into formula for mean <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{\Sigma x}}{{10}},{\text{ }}\frac{{252}}{n},{\text{ }}\frac{{252}}{{10}}\)</p>
<p class="p1">mean \( = 25.2{\text{ (hours)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <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">(i) <span class="Apple-converted-space"> </span>mean \( = 30.2{\text{ (hours)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 N1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> \(\sigma = 5{\text{ (hours)}}\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></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">(i) <span class="Apple-converted-space"> </span>valid approach <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>\(\,\,\,\,\,\)</span>95%, 5% of 27</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(0.95 \times 27,{\text{ }}27 - (5\% {\text{ of }}27)\)</p>
<p class="p1">median \( = 25.65{\text{ (exact), }}25.7{\text{ (hours)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1">variance \( = {({\text{standard deviation}})^2}\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">valid attempt to find new standard deviation <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\({\sigma _{new}} = 0.95 \times 5,{\text{ }}4.75\)</p>
<p class="p1">variance \( = 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">variance \( = {({\text{standard deviation}})^2}\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">valid attempt to find new variance <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\({0.95^2}{\text{ }},{\text{ }}0.9025 \times {\sigma ^2}\)</p>
<p class="p1">new variance \( = 22.5625{\text{ }}({\text{exact}}),{\text{ }}22.6\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>both correct frequencies <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)</span>80, 150</p>
<p class="p1">subtracting <strong>their </strong>frequencies in either order <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)\(150 - 80,{\text{ }}80 - 150\)</p>
<p class="p1"><span class="s2">70 </span>(students) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>evidence of a valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)10% of 200, 90%</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(0.90 \times 200,{\text{ }}200 - 20\)</span><span class="s2">, 180 </span>students</p>
<p class="p3"><span class="s3">\(k = 35\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p3"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">d.</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>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The first three terms of an arithmetic sequence are 36, 40, 44,….</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) Write down the value of <em>d</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find \({u_8}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a(i) and (ii).</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;">(i) Show that \({S_n} = 2{n^2} + 34n\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, write down the value of \({S_{14}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</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) \(d = 4\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of valid approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_8} = 36 + 7(4)\) , repeated addition of <em>d</em> from 36</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_8} = 64\) <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(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) correct substitution into sum formula <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_n} = \frac{n}{2}\left\{ {2\left( {36} \right) + (n - 1)(4)} \right\}\) , \(\frac{n}{2}\left\{ {72 + 4n - 4} \right\}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of simplifying </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{n}{2}\left\{ {4n + 68} \right\}\) <em><strong> A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_n} = 2{n^2} + 34n\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(868\) <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">b(i) and (ii).</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;">The majority of candidates were successful with this question. Most had little difficulty with part (a).</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Some candidates were unable to show the required result in part (b), often substituting values for <em>n</em> rather than working with the formula for the sum of an arithmetic series. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The mass \(M\) <span class="s1">of a decaying substance is measured at one minute intervals. The points \((t,{\text{ }}\ln M)\) are plotted for \(0 \leqslant t \leqslant 10\)</span>, where \(t\) is in minutes. The line of best fit is drawn. This is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.21.06.png" alt="M16/5/MATME/SP2/ENG/TZ1/05"></p>
<p class="p1">The correlation coefficient for this linear model is \(r = - 0.998\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">State <strong>two </strong></span>words that describe the linear correlation between \(\ln M\) <span class="s1">and \(t\).</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 class="p1">The equation of the line of best fit is \(\ln M = - 0.12t + 4.67\)<span class="s1">. Given that \(M = a \times {b^t}\), find the value of \(b\).</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 class="p1">strong, negative (both required) <strong><em>A2 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"><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>\(\,\,\,\,\,\)\({{\text{e}}^{\ln M}} = {{\text{e}}^{ - 0.12t + 4.67}}\)</p>
<p class="p1">correct use of exponent laws for \({{\text{e}}^{ - 0.12t + 4.67}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({{\text{e}}^{ - 0.12t}} \times {{\text{e}}^{4.67}}\)</p>
<p class="p1">comparing coefficients/terms <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({b^t} = {{\text{e}}^{ - 0.12t}}\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = {{\text{e}}^{ - 0.12}}{\text{ (exact), }}0.887\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong>METHOD 2</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>\(\,\,\,\,\,\)\(\ln (a \times {b^t}) = - 0.12t + 4.67\)</p>
<p class="p1">correct use of log laws for \(\ln (a{b^t})\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\ln a + t\ln b\)</p>
<p class="p1">comparing coefficients <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - 0.12 = \ln b\)</p>
<p class="p1"><span class="Apple-converted-space">\(b = {{\text{e}}^{ - 0.12}}{\text{ (exact), }}0.887\) </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">This turned out to be one of the more challenging questions on the paper. Although many candidates correctly described the linear correlation in part (a), a surprisingly large number of candidates were unable to do so.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was not well done with many candidates unable to transfer their knowledge of exponentials and/or log manipulation to the question. After rewriting the line of best fit as \( = {{\text{e}}^{ - 0.12t + 4.67}}\), candidates were neither able to apply the rules for exponentials correctly nor were they familiar with the method of comparing coefficients to find the answer. There were numerous failed attempts of trying to estimate points from the graph and substitute these into the equation, while others arbitrarily chose values for \(t\) in an effort to set up a system of equations, the latter having some measure of success.</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;">An arithmetic sequence, \({u_1}{\text{, }}{u_2}{\text{, }}{u_3} \ldots ,\) has \(d = 11\) and \({u_{27}} = 263\) .</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 \({u_1}\).</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;">(i) Given that \({u_n} = 516\) , find the value of <em>n</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) For this value of <em>n</em> , find \({S_n}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</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;">evidence of equation for \({u_{27}}\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(263 = {u_1} + 26 \times 11\) , \({u_{27}} = {u_1} + (n - 1) \times 11\) , \(263 - (11 \times 26)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_1} = - 23\) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) correct equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(516 = - 23 + (n - 1) \times 11\) , \(539 = (n - 1) \times 11\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 50\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct substitution into sum formula <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{50}} = \frac{{50( - 23 + 516)}}{2}\) , \({S_{50}} = \frac{{50(2 \times ( - 23) + 49 \times 11)}}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({S_{50}} = 12325\) (accept 12300) <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">b(i) and (ii).</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;">This problem was done well by the vast majority of candidates. Most students set out their </span><span style="font-family: times new roman,times; font-size: medium;">working very neatly and logically and gained full marks.</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;">This problem was done well by the vast majority of candidates. Most students set out their </span><span style="font-family: times new roman,times; font-size: medium;">working very neatly and logically and gained full marks.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In an arithmetic sequence \({u_{10}} = 8,{\text{ }}{u_{11}} = 6.5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the value of the common difference.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the first term.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the sum of the first 50 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>\(d = - 1.5\) <strong><em>A1 N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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>\(\;\;\;{u_{10}} = {u_1} + 9d,{\text{ }}8 = {u_1} - 9( - 1.5)\)</p>
<p class="p1">correct working <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;8 = {u_1} + 9d,{\text{ }}6.5 = {u_1} + 10d,{\text{ }}{u_1} = 8 - 9( - 1.5)\)</p>
<p class="p1">\({u_1} = 21.5\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">attempt to list 3 or more terms in either direction <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;9.5,{\text{ }}11,{\text{ }}12.5,{\text{ }} \ldots ;{\text{ }}5,{\text{ }}3.5,{\text{ }}2,{\text{ }} \ldots {\text{ }} \ldots \)</p>
<p class="p1">correct list of 4 or more terms in <strong>correct </strong>direction <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;9.5,{\text{ }}11,{\text{ }}12.5,{\text{ }}14\)</p>
<p class="p1">\({u_1} = 21.5\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct expression <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{50}}{2}\left( {2(21.5) + 49( - 1.5)} \right),{\text{ }}\frac{{50}}{2}(21.5 - 52),{\text{ }}\sum\limits_{k = 1}^{50} {21.5 + (k - 1)( - 1.5)} \)</p>
<p>\({\text{sum}} = - 762.5\;\;\;{\text{(exact)}}\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p><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">In general, candidates showed confidence in this area of the syllabus. Appropriate formulae were chosen for parts (b) and (c) and many candidates were able to achieve full marks. However, many candidates found the common difference to be \( + 1.5\) in part (a) by subtracting \({u_{10}} - {u_{11}}\) or believing that the common difference should always be positive.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In general, candidates showed confidence in this area of the syllabus. Appropriate formulae were chosen for parts (b) and (c) and many candidates were able to achieve full marks. However, many candidates found the common difference to be \( + 1.5\) in part (a) by subtracting \({u_{10}} - {u_{11}}\) or believing that the common difference should always be positive.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In general, candidates showed confidence in this area of the syllabus. Appropriate formulae were chosen for parts (b) and (c) and many candidates were able to achieve full marks. However, many candidates found the common difference to be \( + 1.5\) in part (a) by subtracting \({u_{10}} - {u_{11}}\) or believing that the common difference should always be positive.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The sum of the first three terms of a geometric sequence is \(62.755\), and the sum of the </span><span style="font-family: times new roman,times; font-size: medium;">infinite sequence is \(440\). Find the common ratio.</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;">correct substitution into sum of a geometric sequence <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(62.755 = {u_1}\left( {\frac{{1 - {r^3}}}{{1 - r}}} \right)\) , \({u_1} + {u_1}r + {u_1}{r^2} = 62.755\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into sum to infinity <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{{u_1}}}{{1 - r}} = 440\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to eliminate one variable <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em>substituting \({u_1} = 440(1 - r)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation in one variable <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(62.755 = 440(1 - r)\left( {\frac{{1 - {r^3}}}{{1 - r}}} \right)\) , \(440(1 - r)(1 + r + {r^2}) = 62.755\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempting to solve the equation in a single variable <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em>sketch, setting equation equal to zero, \(62.755 = 440(1 - {r^3})\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r =0.95 = \frac{{19}}{{20}}\) <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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates were able to successfully obtain two equations in two variables, but far fewer were able to correctly solve for the value of \(r\). Some candidates had misread errors for either \(440\) or \(62.755\), with some candidates taking the French and Spanish exams mistaking the decimal comma for a thousands comma. Many candidates who attempted to solve algebraically did not cancel the \(1 - r\) from both sides and ended up with a 4<sup>th</sup> degree equation that they could not solve. Some of these candidates obtained the extraneous answer of \(r - 1\) as well. Some candidates used a minimum of algebra to eliminate the first term and then quickly solved the resulting equation on their GDC.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first two terms of a geometric sequence \({u_n}\) are \({u_1} = 4\) and \({u_2} = 4.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the common ratio.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find \({u_5}\).</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 class="p1">Another sequence \({v_n}\) is defined by \({v_n} = a{n^k}\), where \(a,{\text{ }}k \in \mathbb{R}\), and \(n \in {\mathbb{Z}^ + }\), such that \({v_1} = 0.05\) and \({v_2} = 0.25\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(a\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the value of \(k\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the smallest value of \(n\) for which \({v_n} > {u_n}\).</p>
<div class="marks">[5]</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">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(r = \frac{{{u_2}}}{{{u_1}}},{\text{ }}\frac{4}{{4.2}}\)</p>
<p class="p1">\(r = 1.05\;\;\;{\text{(exact)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>attempt to substitute into formula, with <strong>their </strong>\(r\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(4 \times {1.05^n},{\text{ }}4 \times 1.05 \times 1.05 \ldots \)</p>
<p class="p1">correct substitution <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(4 \times {1.05^4},{\text{ }}4 \times 1.05 \times 1.05 \times 1.05 \times 1.05\)</p>
<p class="p1">\({u_5} = 4.862025{\text{ (exact), }}4.86{\text{ }}[4.86,{\text{ }}4.87]{\text{ }}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>attempt to substitute \(n = 1\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.05 = a \times {1^k}\)</p>
<p class="p1">\(a = 0.05\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct substitution of \(n = 2\) into \({v_2}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.25 = a \times {2^k}\)</p>
<p class="p1">correct work <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)finding intersection point, \(k = {\log _2}\left( {\frac{{0.25}}{{0.05}}} \right),{\text{ }}\frac{{\log 5}}{{\log 2}}\)</p>
<p class="p1">\(2.32192\)</p>
<p class="p1">\(k = {\log _2}5\;\;\;{\text{(exact), }}2.32{\text{ }}[2.32,{\text{ }}2.33]\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct expression for \({u_n}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(4 \times {1.05^{n - 1}}\)</p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">correct substitution into inequality (accept equation) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.05 \times {n^k} > 4 \times {1.05^{n - 1}}\)</p>
<p class="p1">valid approach to solve inequality (accept equation) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)finding point of intersection, \(n = 7.57994{\text{ }}(7.59508{\text{ from 2.32)}}\)</p>
<p class="p1">\(n = 8\;\;\;\)(must be an integer) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">table of values</p>
<p class="p1">when \(n = 7,{\text{ }}{u_7} = 5.3604,{\text{ }}{v_7} = 4.5836\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">when \(n = 8,{\text{ }}{u_8} = 5.6284,{\text{ }}{v_8} = 6.2496\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(n = 8\;\;\;\)(must be an integer) <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 [14 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">Most candidates answered part (a) correctly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">A surprising number assumed the second sequence to be geometric as well, and thus part (b) was confusing for many. It was quite common that students did not clearly show which work was relevant to part (i) and which to part (ii), thus often losing marks.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Few students successfully completed part (c) as tried to solve algebraically instead of graphically. Those who used the table of values did not always show two sets of values and consequently lost marks.</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;">Find the term \({x^3}\) in the expansion of \({\left( {\frac{2}{3}x - 3} \right)^8}\) .</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;">evidence of using binomial expansion <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. selecting correct term, \({a^8}{b^0} + \left( {\begin{array}{*{20}{c}}<br>8\\<br>1<br>\end{array}} \right){a^7}b + \left( {\begin{array}{*{20}{c}}<br>8\\<br>2<br>\end{array}} \right){a^6}{b^2} + \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of calculating the factors, in any order <em><strong>A1A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 56 , \(\frac{{{2^2}}}{{{3^3}}}\) , \( - {3^5}\) , \(\left( {\begin{array}{*{20}{c}}<br>8\\<br>5<br>\end{array}} \right){\left( {\frac{2}{3}x} \right)^3}{( - 3)^5}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 4032{x^3}\) (accept = \( - 4030{x^3}\) to 3 s.f.) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 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 produced mixed results in this question. Many showed a binomial expansion in </span><span style="font-family: times new roman,times; font-size: medium;">some form, although simply writing rows of Pascal’s triangle is insufficient evidence. A </span><span style="font-family: times new roman,times; font-size: medium;">common error was to answer with the coefficient of the term, and many neglected the use of </span><span style="font-family: times new roman,times; font-size: medium;">brackets when showing working. Although sloppy notation was not penalized if candidates </span><span style="font-family: times new roman,times; font-size: medium;">achieved a correct result, for some the missing brackets led to a wrong answer.</span></p>
</div>
<br><hr><br><div class="question">
<p>In the expansion of \(a{x^3}{(2 + ax)^{11}}\), the coefficient of the term in \({x^5}\) is 11880. Find the value of \(a\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach for expansion (must have correct substitution for parameters, but accept an incorrect value for \(r\)) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {11} \\ r \end{array}} \right){(2)^{11 - r}}a{x^r},{\text{ }}\left( {\begin{array}{*{20}{c}} {11} \\ 3 \end{array}} \right){(2)^8}{(ax)^3},{\text{ }}{2^{11}} + \left( {\begin{array}{*{20}{c}} {11} \\ 1 \end{array}} \right){(2)^{10}}{(ax)^1} + \left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){(2)^9}(ax) + \ldots \)</p>
<p>recognizing need to find term in \({x^2}\) in binomial expansion <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(r = 2,{\text{ }}{(ax)^2}\)</p>
<p>correct term or coefficient in binomial expansion (may be seen in equation) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){(ax)^2}{(2)^9},{\text{ }}55({a^2}{x^2})(512),{\text{ }}28160{a^2}\)</p>
<p>setting up equation in \({x^5}\) with <strong>their</strong> coefficient/term (do not accept other powers of \(x\)) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(a{x^3}\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){(ax)^2}{(2)^9} = 11880{x^5}\)</p>
<p>correct equation <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(28160{a^3} = 11880\)</p>
<p>\(a = \frac{3}{4}\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[6 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 class="p1">Consider the expansion of \({(2x + 3)^8}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the number of terms in this expansion.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the term in \({x^3}\).</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">9 terms <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach to find the required term <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ r \end{array}} \right){(2x)^{8 - r}}{(3)^r},{\text{ }}{(2x)^8}{(3)^0} + {(2x)^7}{(3)^1} + \ldots \), Pascal’s triangle to \({{\text{8}}^{{\text{th}}}}\) row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{{\text{6}}^{{\text{th}}}}{\text{ term, }}r = 5,{\text{ }}\left( {\begin{array}{*{20}{c}} 8 \\ 5 \end{array}} \right),{\text{ (2x}}{{\text{)}}^3}{(3)^5}\)</p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\left( {\begin{array}{*{20}{c}} 8 \\ 5 \end{array}} \right){(2x)^3}{(3)^5},{\text{ }}56 \times {2^3} \times {3^5}\)</p>
<p>\(108864{x^3}\;\;\;\)(accept \(109000{x^3}\)) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Do not award any marks if there is clear evidence of adding instead of multiplying.</p>
<p>Do not award final <strong><em>A1 </em></strong>for a final answer of \(108864\), even if \(108864{x^3}\) is seen previously.</p>
<p>If no working shown award <strong><em>N2 </em></strong>for \(108864\).</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">This is a common question and yet it was not unusual to see candidates writing out the expansion in full or using Pascal’s triangle to find the correct binomial coefficient. Of those candidates who managed to identify the correct term, many omitted the parentheses around \(2\chi \) which led to an incorrect answer. Most candidates were able to distinguish between “the term in \({x^{3n}}\)” and the coefficient. There are still a significant number of candidates who add the parts of a term rather than multiply them and this approach gained no marks.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This is a common question and yet it was not unusual to see candidates writing out the expansion in full or using Pascal’s triangle to find the correct binomial coefficient. Of those candidates who managed to identify the correct term, many omitted the parentheses around \(2\chi \) which led to an incorrect answer. Most candidates were able to distinguish between “the term in \({x^{3n}}\)” and the coefficient. There are still a significant number of candidates who add the parts of a term rather than multiply them and this approach gained no marks.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative, \(f’(x)\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 </strong></p>
<p>derivative of \(f(x)\) <strong><em>A2</em></strong></p>
<p>\(7{({x^2} + 3)^6}(x2)\)</p>
<p>recognizing need to find \({x^4}\) term in \({({x^2} + 3)^6}\) (seen anywhere) <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(14x{\text{ (term in }}{x^4})\)</p>
<p>valid approach to find the terms in \({({x^2} + 3)^6}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 6 \\ r \end{array}} \right){({x^2})^{6 - r}}{(3)^r},{\text{ }}{({x^2})^6}{(3)^0} + {({x^2})^5}{(3)^1} + \ldots \), Pascal’s triangle to 6th row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{5th term, }}r = 2,{\text{ }}\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right),{\text{ }}{({x^2})^2}{(3)^4}\)</p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right){({x^2})^2}{(3)^4},{\text{ }}15 \times {3^4},{\text{ }}14x \times 15 \times 81{({x^2})^2}\)</p>
<p>\(17010{x^5}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>recognition of need to find \({x^6}\) in \({({x^2} + 3)^7}\) (seen anywhere) <strong><em>R1 </em></strong></p>
<p>valid approach to find the terms in \({({x^2} + 3)^7}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ r \end{array}} \right){({x^2})^{7 - r}}{(3)^r},{\text{ }}{({x^2})^7}{(3)^0} + {({x^2})^6}{(3)^1} + \ldots \), Pascal’s triangle to 7th row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)6th term, \(r = 3,{\text{ }}\left( {\begin{array}{*{20}{c}} 7 \\ 3 \end{array}} \right),{\text{ (}}{{\text{x}}^2}{)^3}{(3)^4}\)</p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 4 \end{array}} \right){{\text{(}}{{\text{x}}^2})^3}{(3)^4},{\text{ }}35 \times {3^4}\)</p>
<p>correct term <strong><em>(A1)</em></strong></p>
<p>\(2835{x^6}\)</p>
<p>differentiating their term in \({x^6}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\((2835{x^6})',{\text{ (6)(2835}}{{\text{x}}^5})\)</p>
<p>\(17010{x^5}\) <strong><em>A1</em></strong> <strong><em>N3</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><span style="font-family: times new roman,times; font-size: medium;">In an arithmetic sequence \({u_1} = 7\) , \({u_{20}} = 64\) and \({u_n} = 3709\) .</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 value of the common difference.</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 <em>n</em> .</span></p>
<div class="marks">[2]</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;">evidence of choosing the formula for 20th term <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{20}} = {u_1} + 19d\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(64 = 7 + 19d\) , \(d = \frac{{64 - 7}}{{19}}\) </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;">[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 substitution into formula for \({u_n}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3709 = 7 + 3(n - 1)\) , \(3709 = 3n + 4\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 1235\) <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">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;">The majority of candidates gained full marks in this question. However, the presentation of work was often disappointing, and there were a few incorrect trial and error approaches in part (a) resulting in division by 20 rather than 19. </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;">The majority of candidates gained full marks in this question. However, the presentation of work was often disappointing, and there were a few incorrect trial and error approaches in part (a) resulting in division by 20 rather than 19. </span></p>
<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;">Find the term in \({x^4}\) in the expansion of \({\left( {3{x^2} - \frac{2}{x}} \right)^5}\) .</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;">evidence of substituting into binomial expansion <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({a^5} + \left( {\begin{array}{*{20}{c}}<br>5\\<br>1<br>\end{array}} \right){a^4}b + \left( {\begin{array}{*{20}{c}}<br>5\\<br>2<br>\end{array}} \right){a^3}{b^2} + \ldots \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">identifying correct term for \({x^4}\) <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of calculating the factors, in any order <em><strong>A1A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>5\\<br>2<br>\end{array}} \right),27{x^6},\frac{4}{{{x^2}}}\) ; \(10{(3{x^2})^3}{\left( {\frac{{ - 2}}{x}} \right)^2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for each correct factor.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\rm{term}} = 1080{x^4}\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>M1M1A1A1A1A0</strong></em> for 1080 with working shown.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 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;"> Although a great number of students recognized they could use the binomial theorem, fewer were successful in finding the term in \({x^4}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Candidates showed various difficulties when trying to solve this problem:</span></p>
<ul>
<li><span style="font-family: times new roman,times; font-size: medium;"> choosing the incorrect term</span></li>
<li><span style="font-family: times new roman,times; font-size: medium;"> attempting to expand \({\left( {3{x^2} - \frac{2}{x}} \right)^5}\) by hand</span></li>
<li><span style="font-family: times new roman,times; font-size: medium;"> finding only the coefficient of the term</span></li>
<li><span style="font-family: times new roman,times; font-size: medium;"> not being able to determine which term would yield an \({x^4}\)</span></li>
<li><span style="font-family: times new roman,times; font-size: medium;"> errors in the calculations of the coefficient.</span></li>
</ul>
</div>
<br><hr><br><div class="specification">
<p class="p1">A population of rare birds, \({P_t}\), can be modelled by the equation \({P_t} = {P_0}{{\text{e}}^{kt}}\), where \({P_0}\) is the initial population, and \(t\) is measured in decades. After one decade, it is estimated that \(\frac{{{P_1}}}{{{P_0}}} = 0.9\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Find the value of \(k\).</p>
<p class="p1">(ii) Interpret the meaning of the value of \(k\).</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 least number of <strong>whole </strong>years for which \(\frac{{{P_t}}}{{{P_0}}} < 0.75\).</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">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(0.9 = {{\text{e}}^{k(1)}}\)</p>
<p class="p1">\(k = - 0.105360\)</p>
<p class="p1"><span class="Apple-converted-space">\(k = \ln 0.9{\text{ (exact), }} - 0.105\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>correct interpretation <span class="Apple-converted-space"> </span><strong><em>R1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)population is decreasing, growth rate is negative</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"><strong>METHOD 1</strong></p>
<p class="p2">valid approach (accept an equality, but do not accept 0.74<span class="s1">) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(P < 0.75{P_0},{\text{ }}{P_0}{{\text{e}}^{kt}} < 0.75{P_0},{\text{ }}0.75 = {{\text{e}}^{t\ln 0.9}}\)</p>
<p class="p1">valid approach to solve <strong>their </strong>inequality <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)logs, graph</p>
<p class="p1"><span class="Apple-converted-space">\(t > 2.73045{\text{ }}({\text{accept }}t = 2.73045){\text{ }}(2.73982{\text{ from }} - 0.105)\) </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="s2">28 </span>years <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p2">valid approach which gives both crossover values accurate to at least 2 <span class="s1">sf <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{{P_{2.7}}}}{{{P_0}}} = 0.75241 \ldots ,{\text{ }}\frac{{{P_{2.8}}}}{{{P_0}}} = 0.74452 \ldots \)</p>
<p class="p1"><span class="Apple-converted-space">\(t = 2.8\) </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s2">28 </span>years <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><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;">
<p class="p1">Part (a) was generally done well, with many candidates able to find the value of \(k\) correctly and to interpret its meaning. Lack of accuracy was occasionally a concern, with some candidates writing their value of \(k\) to 2 significant figures or evaluating \(\ln (0.9)\) incorrectly.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Few candidates were successful in part (b) with many unable to set up an inequality or equation which would allow them to find the condition on \(t\). Some were able to find the value of \(t\) in decades but most were unable to correctly interpret their inequality in terms of the least number of whole years. While a solution through analytic methods was readily available, very few students attempted to use their GDC to solve their initial equation or inequality.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the expansion of \({x^2}{\left( {3{x^2} + \frac{k}{x}} \right)^8}\). The constant term is \({\text{16 128}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(k\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <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> \({\text{\(\left( \begin{array}{c}8\\r\end{array} \right)\)}}{\left( {3{x^2}} \right)^{8 - r}}{\left( {\frac{k}{x}} \right)^r}\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\left( {3{x^2}} \right)^8} + {\text{\(\left( \begin{array}{c}8\\1\end{array} \right)\)}}{\left( {3{x^2}} \right)^7}\left( {\frac{k}{x}} \right) + {\text{\(\left( \begin{array}{c}8\\2\end{array} \right)\)}}{\left( {3{x^2}} \right)^6}{\left( {\frac{k}{x}} \right)^2} + \ldots \), Pascal’s triangle to \({9^{{\text{th}}}}\) line</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;">attempt to find value of <em>r </em>which gives term in \({x^0}\) <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> exponent in binomial must give \({x^{ - 2}},{\text{ }}{x^2}{\left( {{x^2}} \right)^{8 - r}}{\left( {\frac{k}{x}} \right)^r} = {x^0}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">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> \(2(8 - r) - r = - 2,{\text{ }}18 - 3r = 0,{\text{ }}2r + ( - 8 + r) = - 2\)</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;">evidence of correct term <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> \({\text{\(\left( \begin{array}{c}8\\2\end{array} \right)\), \(\left( \begin{array}{c}8\\6\end{array} \right)\)}}{\left( {3{x^2}} \right)^2}{\left( {\frac{k}{x}} \right)^2},{\text{ }}r = 6,{\text{ }}r = 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;">equating <strong>their </strong>term and 16128 to solve for</span><span style="font-family: 'times new roman', times; font-size: medium;"> \(k\) </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times;"> </span> <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> \({x^2}{\text{\(\left( \begin{array}{c}8\\6\end{array} \right)\)}}{\left( {3{x^2}} \right)^2}{\left( {\frac{k}{x}} \right)^6} = 16128,{\text{ }}{k^6} = \frac{{16128}}{{28(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;">\(k = \pm 2\) <strong><em>A1A1 N2</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>If no working shown, award <strong><em>N0 </em></strong>for \(k = 2\).</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>Total [7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In a geometric series, \({u_1} = \frac{1}{{81}}\) and \({u_4} = \frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">.</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 value of \(r\) .</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 smallest value of <em>n</em> for which \({S_n} > 40\) .</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 substituting into formula for \(n\)th term of GP <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_4} = \frac{1}{{81}}{r^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up correct equation \(\frac{1}{{81}}{r^3} = \frac{1}{3}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(r = 3\) </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;"> [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;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up an inequality (accept an equation) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{\frac{1}{{81}}({3^n} - 1)}}{2} > 40\) , \(\frac{{\frac{1}{{81}}(1 - {3^n})}}{{ - 2}} > 40\) , \({3^n} > 6481\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of solving <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. graph, taking logs </span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(n > 7.9888 \ldots \) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>(A1)</strong> </span></em></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(\therefore n = 8\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A1 N2</strong> </span></em></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;">if \(n = 7\) , sum \( = 13.49 \ldots \) ; if \(n = 8\) , sum \( = 40.49 \ldots \) <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(n = 8\) </span><span style="font-family: times new roman,times; font-size: medium;">(is the smallest value) <em><strong>A2 N2</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) was well done. </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) a good number of candidates did not realize that they could use logs to solve the problem, nor did they make good use of their GDCs. Some students did use a trial and error approach to check various values however, in many cases, they only checked one of the "crossover" values. Most candidates had difficulty with notation, opting to set up an equation rather than an inequality. </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"><span class="s1">Find the term in \({x^6}\) </span>in the expansion of \({(x + 2)^9}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, find the term in \({x^7}\) <span class="s1">in the expansion of \(5x{(x + 2)^9}\).</span></p>
<div class="marks">[2]</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">valid approach to find the required term <span class="Apple-converted-space"> </span><strong><em>(M1</em></strong>)</p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 9 \\ r \end{array}} \right){(x)^{9 - r}}{(2)^r},{\text{ }}{x^9} + 9{x^8}(2) + \left( {\begin{array}{*{20}{c}} 9 \\ 2 \end{array}} \right){x^7}{(2)^2} + \ldots \)</span>, Pascal’s triangle to the 9th <span class="s1">row</span></p>
<p class="p1">identifying correct term (may be indicated in expansion) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)<span class="s2">4th </span>term, \(r = 6,{\text{ }}\left( {\begin{array}{*{20}{c}} 9 \\ 3 \end{array}} \right),{\text{ }}{(x)^6}{(2)^3}\)</p>
<p class="p1">correct calculation (may be seen in expansion) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 9 \\ 3 \end{array}} \right){(x)^6}{(2)^3},{\text{ }}84 \times {2^3}\) <span class="Apple-converted-space"> </span></p>
<p class="p1"><span class="Apple-converted-space">672\({x^6}\) </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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)recognizing \({x^7}\) is found when multiplying \(5x \times 672{x^6}\)</p>
<p class="p1"><span class="s1">\(3360{x^7}\) <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>[2 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">Many candidates approached this question using an appropriate and efficient method to identify the required term. While many of those who were successful in (a) were also successful in (b), a significant number of candidates did not realize that multiplying their answer in (a) by \(5x\) would give them the term in \({x^7}\). This led to an attempt to find a binomial expansion, which was generally unsuccessful. Some candidates continue to be unable to distinguish between the “coefficient” and the “term” and lost a point as a result.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Many candidates approached this question using an appropriate and efficient method to identify the required term. While many of those who were successful in (a) were also successful in (b), a significant number of candidates did not realize that multiplying their answer in (a) by \(5x\) would give them the term in \({x^7}\). This led to an attempt to find a binomial expansion, which was generally unsuccessful. Some candidates continue to be unable to distinguish between the “coefficient” and the “term” and lost a point as a result.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first three terms of an arithmetic sequence are \({u_1} = 0.3,{\text{ }}{u_2} = 1.5,{\text{ }}{u_3} = 2.7\).</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 <span class="s1">30th </span>term of the sequence.</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 <span class="s1">30 </span>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 class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(1.5 - 0.3,{\text{ }}1.5 - 2.7,{\text{ }}2.7 = 0.3 + 2d\)</p>
<p class="p1"><span class="Apple-converted-space">\(d = 1.2\) </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">correct substitution into term formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(0.3 + 1.2(30 - 1),{\text{ }}{u_{30}} = 0.3 + 29(1.2)\)</p>
<p class="p1"><span class="Apple-converted-space">\({u_{30}} = 35.1\) </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 substitution into sum formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({S_{30}} = \frac{{30}}{2}(0.3 + 35.1),{\text{ }}\frac{{30}}{2}\left( {2(0.3) + 29(1.2)} \right)\)</p>
<p class="p1"><span class="Apple-converted-space">\({S_{30}} = 531\) </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">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates found this question straightforward and accessible. They could find the correct difference and substituted correctly into term and sum formula respectively.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates found this question straightforward and accessible. They could find the correct difference and substituted correctly into term and sum formula respectively.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates found this question straightforward and accessible. They could find the correct difference and substituted correctly into term and sum formula respectively.</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;">In an arithmetic sequence, \({S_{40}} = 1900\) and \({u_{40}} = 106\) . Find the value of \({u_1}\) and of <em>d</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times;"><strong><span style="font-size: medium;">METHOD 1</span></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting into formula for \({S_{40}}\) <em><strong> (M1)</strong></em></span></p>
<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. \(1900 = \frac{{40({u_1} + 106)}}{2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({u_1} = - 11\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting into formula for \({u_{40}}\) or \({S_{40}}\) <em><strong>(M1)</strong></em></span></p>
<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. \(106 = - 11 + 39d\) , \(1900 = 20( - 22 + 39d)\)</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 align="LEFT"><span style="font-family: times new roman,times;"><strong><span style="font-size: medium;">METHOD 2</span></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting into formula for \({S_{40}}\) <em><strong>(M1)</strong></em></span></p>
<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. \(20(2{u_1} + 39d) = 1900\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting into formula for \({u_{40}}\) <em><strong> (M1)</strong></em></span></p>
<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. \(106 = {u_1} + 39d\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({u_1} = - 11\) , \(d = 3\) <em><strong>A1A1 N2N2</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><strong><em><span style="font-size: medium;">[6 marks]</span></em></strong></span></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;">Most candidates answered this question correctly. Those who chose to solve with a system of equations often did so algebraically, using a fair bit of time doing so and sometimes making a careless error in the process. Few candidates took advantage of the system solving features of the GDC. </span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In an arithmetic series, the first term is –7 and the sum of the first 20 terms is 620.</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 common difference.</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 the 78th term.</span></p>
<div class="marks">[2]</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;">attempt to substitute into sum formula for AP (accept term formula) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{20}} = \frac{{20}}{2}\left\{ {2( - 7) + 19\left. d \right\}} \right.\) , \(\left( {{\rm{or}}\frac{{20}}{2}\left( { - 7 + {u_{20}}} \right)} \right)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up correct equation using sum formula <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{20}}{2}\left\{ {2( - 7) + \left. {19d} \right\}} \right. = 620\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(d = 4\) </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;"> [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;">correct substitution \({u_{78}} = - 7 + 77(4)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">= 301 <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>
<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 generally well answered. Many candidates who got part (a) wrong, recovered and received full follow through marks in part (b). There were a few candidates who confused the term and sum formula in part (a). </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 generally well answered. Many candidates who got part (a) wrong, recovered and received full follow through marks in part (b). There were a few candidates who confused the term and sum formula in part (a). </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The first term of a geometric sequence is 200 and the sum of the first four terms </span><span style="font-family: times new roman,times; font-size: medium;">is 324.8.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the common ratio.</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 first term of a geometric sequence is 200 and the sum of the first four terms </span><span style="font-family: times new roman,times; font-size: medium;">is 324.8.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the tenth term.</span></p>
<div class="marks">[2]</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;">correct substitution into sum of a geometric sequence <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(200\left( {\frac{{1 - {r^4}}}{{1 - r}}} \right)\) , \(200 + 200r + 200{r^2} + 200{r^3}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up an equation involving a sum and 324.8 <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(200\left( {\frac{{1 - {r^4}}}{{1 - r}}} \right) = 324.8\) , \(200 + 200r + 200{r^2} + 200{r^3} = 324.8\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(r = 0.4\) (exact) <em><strong>A2 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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({u_{10}} = 200 \times {0.4^9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({u_{10}} = 0.0524288\) (exact), \(0.0524\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<p> </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;">In part (a), although most candidates substituted correctly into the formula for the sum of a geometric series and set it equal to 324.8, some used the formula for the sum to infinity and a few the formula for the sum of an arithmetic series. The overwhelming error made was in attempting to solve the equation algebraically and getting nowhere, or getting a wrong answer. The great majority did not recognize the need to use the GDC to find the value of <em>r</em>. </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) many did not obtain any marks since they weren't able to find an answer to part (a). Those who were able to get a value for <em>r</em> in part (a) generally went on to gain full marks in (b). However, this was one of the most common places for rounding errors to be made. </span></p>
<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;"><span style="font-family: TimesNewRomanPSMT;">The first three terms of an arithmetic sequence are </span><span style="font-family: TimesNewRomanPSMT;">5 , 6.7 , 8.4 .</span></span></p>
<p><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><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">The first three terms of an arithmetic sequence are </span><span style="font-family: TimesNewRomanPSMT;">5 , 6.7 , 8.4 .</span></span></p>
<p><span style="font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Find the 28</span><sup><span style="font-family: TimesNewRomanPSMT;"><span style="font-family: TimesNewRomanPSMT;">th </span></span></sup><span style="font-family: TimesNewRomanPSMT;">term of the sequence.</span></span></p>
<p> </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;"><span style="font-family: TimesNewRomanPSMT;">The first three terms of an arithmetic sequence are \(5\)</span><span style="font-family: TimesNewRomanPSMT;"> , \(6.7\) , \(8.4\) .</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the sum of the first 28 terms.</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><span style="font-family: times new roman,times; font-size: medium;">valid method <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. subtracting terms, using sequence formula </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(d = 1.7\) <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><span style="font-family: times new roman,times; font-size: medium;">correct substitution into term formula <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(5 + 27(1.7)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">28<sup>th</sup> term is 50.9 (exact) <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;">correct substitution into sum formula <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{28}} = \frac{{28}}{2}(2(5) + 27(1.7))\) , \(\frac{{28}}{2}(5 + 50.9)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({S_{28}} = 782.6\) (exact) [\(782\), \(783\)] <em><strong>A1 N2</strong> </em></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></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><span style="font-family: times new roman,times; font-size: medium;">Most candidates performed well on this question. A few were confused between the term number and the value of a term. </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;">Most candidates performed well on this question. A few were confused between the term number and the value of a term. </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;">Most candidates performed well on this question. A few were confused between the term number and the value of a term. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">In a geometric sequence, the fourth term is <span class="s1">8 </span>times the first term. The sum of the first <span class="s1">10 </span>terms is <span class="s1">2557.5</span>. Find the 10th term of this sequence.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">correct equation to find \(r\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({u_1}{r^3} = 8{u_1},{\text{ }}{r^3} = 8\)</p>
<p class="p2">\(r = 2\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2">correct equation to find \({u_1}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\({u_1}({2^{10}} - 1) = 2557.5,{\text{ }}{u_1} = \frac{{2557.5}}{{{r^{10}} - 1}}(r - 1)\)</p>
<p class="p2"><span class="Apple-converted-space">\({u_1} = 2.5\) </span><strong><em>(A1)</em></strong></p>
<p class="p3"><span class="Apple-converted-space">\({u_{10}} = 2.5{(2)^9}\) </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s2">1280 <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p2"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">The majority of candidates did well on this question although identifying the common ratio was not always as easily done and some candidates lost marks as a result of using an inappropriate method such as 8/4. Other candidates correctly guessed the value of \(r\) or did so by showing that if \(r = 2\), then the ratio of the fourth term to the first term would be 8. It was also disappointing to see some candidates use an incorrect formula for the sum of the first \(n\) terms of a geometric series: a common error seen was \(2557.5 = \frac{{{u_1}({2^{10}} - 1)}}{{10 - 1}}\), which led to the wrong answer of 22.5.</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In the expansion of \({(3x - 2)^{12}}\) , the term in \({x^5}\) can be expressed as \(\left( {\begin{array}{*{20}{c}}<br>{12}\\<br>r<br>\end{array}} \right) \times {(3x)^p} \times {( - 2)^q}\) .</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;">(a) </span><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(p\) , of \(q\) and of \(r\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) </span><span style="font-family: times new roman,times; font-size: medium;">Find the coefficient of the term in \({x^5}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of \(p\) , of \(q\) and of \(r\) .</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 coefficient of the term in \({x^5}\) .</span></p>
<div class="marks">[2]</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;">(a) \(p = 5\) , \(q = 7\) , \(r = 7\) (accept \(r = 5\)) <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>
<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;">(b) </span><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> \(\left( {\begin{array}{*{20}{c}}<br>{12}\\<br>7<br>\end{array}} \right) \times {(3x)^5} \times {( - 2)^7}\) , \(792\) , \(243\) , \( - {2^7}\) , \(24634368\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">coefficient of term in \({x^5}\) is \( - 24634368\) <strong><em>A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Do not award the final <em><strong>A1</strong></em> for an answer that contains \(x\). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [5 marks]<br></span></strong></em></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 5\) , \(q = 7\) , \(r = 7\) (accept \(r = 5\)) <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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<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> \(\left( {\begin{array}{*{20}{c}}<br>{12}\\<br>7<br>\end{array}} \right) \times {(3x)^5} \times {( - 2)^7}\) , \(792\) , \(243\) , \( - {2^7}\) , \(24634368\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">coefficient of term in \({x^5}\) is \( - 24634368\) <strong><em>A1 N2</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Do not award the final <em><strong>A1</strong></em> for an answer that contains \(x\). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [5 marks]<br></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;">Candidates frequently made reasonable attempts at both parts of the question. Those who correctly stated the values in (a) were generally successful in part (b). Many candidates offered the whole term rather than the coefficient in part (b) and lost the final mark. Some candidates appeared to have misread the order of the variables, stating that \(p = 7\) (instead of \(r = 7\) ), \(q = 5\) (instead of \(p = 5\)) and \(r = 5\) or \(7\) (instead of \(q = 5\) or \(7\)). A large number of candidates did not make the connection between parts (a) and (b).</span></p>
<div class="question_part_label">.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates frequently made reasonable attempts at both parts of the question. Those who correctly stated the values in (a) were generally successful in part (b). Many candidates offered the whole term rather than the coefficient in part (b) and lost the final mark. Some candidates appeared to have misread the order of the variables, stating that \(p = 7\) (instead of \(r = 7\) ), \(q = 5\) (instead of \(p = 5\)) and \(r = 5\) or \(7\) (instead of \(q = 5\) or \(7\)). A large number of candidates did not make the connection between parts (a) and (b).</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 frequently made reasonable attempts at both parts of the question. Those who correctly stated the values in (a) were generally successful in part (b). Many candidates offered the whole term rather than the coefficient in part (b) and lost the final mark. Some candidates appeared to have misread the order of the variables, stating that \(p = 7\) (instead of \(r = 7\) ), \(q = 5\) (instead of \(p = 5\)) and \(r = 5\) or \(7\) (instead of \(q = 5\) or \(7\)). A large number of candidates did not make the connection between parts (a) and (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The first three terms of a geometric sequence are \({u_1} = 0.64,{\text{ }}{u_2} = 1.6\), and \({u_3} = 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(r\).</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">Find the value of \({S_6}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the least value of \(n\) such that \({S_n} > 75\,000\).</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 class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{{u_1}}}{{{u_2}}},{\text{ }}\frac{4}{{1.6}},{\text{ }}1.6 = r(0.64)\)</p>
<p class="p1">\(r = 2.5\;\;\;\left( { = \frac{5}{2}} \right)\) <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 class="p1">correct substitution into \({S_6}\) <strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\frac{{0.64({{2.5}^6} - 1)}}{{2.5 - 1}}\)</p>
<p class="p1">\({S_6} = 103.74\) (exact), \(104\) <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"><strong>METHOD 1 (analytic)</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>\(\;\;\;\frac{{0.64({{2.5}^n} - 1)}}{{2.5 - 1}} > 75\,000,{\text{ }}\frac{{0.64({{2.5}^n} - 1)}}{{2.5 - 1}} = 75\,000\)</p>
<p class="p1">correct inequality (accept equation) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;n > 13.1803,{\text{ }}n = 13.2\)</p>
<p class="p1">\(n = 14\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong>METHOD 2 (table of values)</strong></p>
<p class="p1"><strong>both </strong>crossover values <span class="Apple-converted-space"> </span><strong><em>A2</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{S_{13}} = 63577.8,{\text{ }}{S_{14}} = 158945\)</p>
<p class="p1">\(n = 14\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 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><span style="font-family: times new roman,times; font-size: medium;">The <em>n</em><sup>th</sup> term of an arithmetic sequence is given by \({u_n} = 5 + 2n\) .</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 common difference.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Given that the <em>n</em><sup>th</sup> term of this sequence is 115, find the value of <em>n</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) For this value of <em>n</em> , find the sum of the sequence.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</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;">\(d = 2\) <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;">(i) \(5 + 2n = 115\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n = 55\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({u_1} = 7\) (may be seen in above) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into formula for sum of arithmetic series <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({S_{55}} = \frac{{55}}{2}(7 + 115)\) , \({S_{55}} = \frac{{55}}{2}(2(7) + 54(2))\) , \(\sum\limits_{k = 1}^{55} {(5 + 2k)} \)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\({S_{55}} = 3355\) </span><span style="font-family: times new roman,times; font-size: medium;">(accept \(3360\)) <em><strong>A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</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;">The majority of candidates could either recognize the common difference in the formula for the <em>n</em><sup>th</sup> term or could find it by writing out the first few terms of the sequence. </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) demonstrated that candidates were not familiar with expression, "<em>n</em><sup>th</sup> term". Many stated that the first term was 5 and then decided to use their own version of the nth term formula leading to a great many errors in (b) (ii). </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">The third term in the expansion of \({(2x + p)^6}\) </span><span style="font-family: TimesNewRomanPSMT;">is \(60{x^4}\) </span><span style="font-family: TimesNewRomanPSMT;">. Find the possible values of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">p </span></em><span style="font-family: TimesNewRomanPSMT;">.</span></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 binomial <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({(2x)^6}{p^0} + \left( {\begin{array}{*{20}{c}}<br>6\\<br>1<br>\end{array}} \right){(2x)^5}{(p)^1} + \ldots \) , \(\left( {\begin{array}{*{20}{c}}<br>n\\<br>r<br>\end{array}} \right){(2x)^r}{(p)^{n - r}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">one correct calculation for term in \({x^4}\) in the expansion for power 6 <em><strong> (A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. 15 , \(16{x^4}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for term in \({x^4}\) <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>6\\<br>2<br>\end{array}} \right){(2x)^4}{(p)^2}\) , \({15.2^4}{p^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes:</strong> Accept sloppy notation e.g. omission of brackets around \(2x\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Accept absence of \(x\) in middle factor. </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 term <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(240{p^2}{x^4}\) (accept absence of \({x^4}\) ) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up equation with their coefficient equal to 60 <em><strong>M1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\left( {\begin{array}{*{20}{c}}<br>6\\<br>2<br>\end{array}} \right){(2)^4}{(p)^2} = 60\) , \(240{p^2}{x^4} = 60{x^4}\) , \({p^2} = \frac{{60}}{{240}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = \pm \frac{1}{2}(p = \pm 0.5)\) <em><strong>A1A1 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><span style="font-family: times new roman,times; font-size: medium;">This question proved challenging for many students. Most candidates recognized the need to expand a binomial but many executed this task incorrectly by selecting the wrong term, omitting brackets, or ignoring the binomial coefficient. Other candidates did not recognize that there were two values for <em>p</em> when solving their quadratic equation. </span></p>
</div>
<br><hr><br><div class="question">
<p class="p1">Ramiro walks to work each morning. During the first minute he walks \(80\) metres. In each subsequent minute he walks \(90\% \) of the distance walked during the previous minute.</p>
<p class="p1">The distance between his house and work is \(660\) metres. Ramiro leaves his house at 08:00 and has to be at work by 08:15.</p>
<p class="p1">Explain why he will not be at work on time.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>recognize that the distance walked each minute is a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;r = 0.9\), valid use of \(0.9\)</p>
<p>recognize that total distance walked is the sum of a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{S_n},{\text{ }}a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)\)</p>
<p>correct substitution into the sum of a geometric sequence <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;80\left( {\frac{{1 - {{0.9}^n}}}{{1 - 0.9}}} \right)\)</p>
<p>any correct equation with sum of a geometric sequence <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;80\left( {\frac{{{{0.9}^n} - 1}}{{0.9 - 1}}} \right) = 660,{\text{ }}1 - {0.9^n} = \frac{{66}}{{80}}\)</p>
<p>attempt to solve <strong>their </strong>equation involving the sum of a GP <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\)graph, algebraic approach</p>
<p>\(n = 16.54290788\) <strong><em>A1</em></strong></p>
<p>since \(n > 15\) <strong><em>R1</em></strong></p>
<p>he will be late <strong><em>AG N0</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Do not award the <strong><em>R </em></strong>mark without the preceding <strong><em>A </em></strong>mark.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>recognize that the distance walked each minute is a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;r = 0.9\), valid use of \(0.9\)</p>
<p>recognize that total distance walked is the sum of a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{S_n},{\text{ }}a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)\)</p>
<p>correct substitution into the sum of a geometric sequence <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;80\left( {\frac{{1 - {{0.9}^n}}}{{1 - 0.9}}} \right)\)</p>
<p>attempt to substitute \(n = 15\) into sum of a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{S_{15}}\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;80\left( {\frac{{{{0.9}^{15}} - 1}}{{0.9 - 1}}} \right)\)</p>
<p>\({S_{15}} = 635.287\) <strong><em>A1</em></strong></p>
<p>since \(S < 660\) <strong><em>R1</em></strong></p>
<p>he will not be there on time <strong><em>AG N0</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Do not award the <strong><em>R </em></strong>mark without the preceding <strong><em>A </em></strong>mark.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>recognize that the distance walked each minute is a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;r = 0.9\), valid use of \(0.9\)</p>
<p>recognize that total distance walked is the sum of a geometric sequence <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{S_n},{\text{ }}a\left( {\frac{{1 - {r^n}}}{{1 - r}}} \right)\)</p>
<p>listing at least 5 correct terms of the GP <strong><em>(M1)</em></strong></p>
<p>15 correct terms <strong><em>A1</em></strong></p>
<p>\(80,{\rm{ }}72,{\rm{ }}64.8,{\rm{ }}58.32,{\rm{ }}52.488,{\rm{ }}47.2392,{\rm{ }}42.5152,{\rm{ }}38.2637,{\rm{ }}34.4373,{\rm{ }}30.9936,{\rm{ }}27.8942,{\rm{ }}25.1048,{\rm{ }}22.59436,{\rm{ }}20.3349,{\rm{ }}18.3014\)</p>
<p>attempt to find the sum of the terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;{S_{15}},{\text{ }}80 + 72 + 64.8 + 58.32 + 52.488 + \ldots + 18.301433\)</p>
<p>\({S_{15}} = 635.287\) <strong><em>A1</em></strong></p>
<p>since \(S < 660\) <strong><em>R1</em></strong></p>
<p>he will not be there on time <strong><em>AG N0</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Do not award the <strong><em>R </em></strong>mark without the preceding <strong><em>A </em></strong>mark.</p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Many found this question accessible, although the most common approach was to calculate each term by brute force, which at times contained small errors or inaccuracies that affected the overall sum. Although this was a valid method, it meant an inefficient use of time that could have affected the performance on other questions.</p>
<p class="p1">Those who applied the formula for geometric series were typically more successful and far more efficient in answering the question.</p>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = {{\text{e}}^{2\,{\text{sin}}\left( {\frac{{\pi x}}{2}} \right)}}\), for <em>x</em> > 0.</p>
<p>The <em>k </em>th maximum point on the graph of <em>f</em> has <em>x</em>-coordinate <em>x<sub>k</sub></em> where \(k \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>x<sub>k</sub></em><sub> + 1</sub> = <em>x<sub>k</sub></em> + <em>a</em>, find <em>a</em>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <em>n</em> such that \(\sum\limits_{k = 1}^n {{x_k} = 861} \).</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>valid approach to find maxima <em><strong>(M1)</strong></em></p>
<p><em>eg</em> one correct value of <em>x<sub>k</sub></em>, sketch of <em>f</em></p>
<p>any two correct consecutive values of <em>x<sub>k</sub></em> <em><strong>(A1)(A1)</strong></em></p>
<p><em>eg x</em><sub>1</sub> = 1, <em>x</em><sub>2</sub> = 5</p>
<p><em>a</em> = 4 <em><strong>A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing the sequence <em>x</em><sub>1,<sup> </sup></sub> <em>x</em><sub>2</sub><sub>,<sup> </sup></sub> <em>x</em><sub>3, …,</sub> <em>x</em><sub>n</sub> is arithmetic <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <em>d</em> = 4</p>
<p>correct expression for sum<em> <strong>(A1)</strong><br></em></p>
<p><em>eg </em>\(\frac{n}{2}\left( {2\left( 1 \right) + 4\left( {n - 1} \right)} \right)\)</p>
<p>valid attempt to solve for <em>n</em> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> graph, 2<em>n</em><sup>2</sup> − <em>n</em> − 861 = 0</p>
<p><em>n</em> = 21 <em><strong>A1 N2</strong></em></p>
<p><em><strong>[4 marks]</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;">
[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">
<p>Consider the expansion of \({\left( {2x + \frac{k}{x}} \right)^9}\), where <em>k</em> > 0 . The coefficient of the term in <em>x</em><sup>3</sup> is equal to the coefficient of the term in <em>x</em><sup>5</sup>. Find <em>k</em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>valid approach to find one of the required terms (must have correct substitution for parameters but accept “<em>r</em>” or an incorrect value for <em>r</em>) <em><strong>(M1)</strong></em><br><em>eg</em> \(\left( \begin{gathered}<br> 9 \hfill \\<br> r \hfill \\ <br>\end{gathered} \right){\left( {2x} \right)^{9 - r}}{\left( {\frac{k}{x}} \right)^r},\,\,\left( \begin{gathered}<br> 9 \hfill \\<br> 6 \hfill \\ <br>\end{gathered} \right){\left( {2x} \right)^6}{\left( {\frac{k}{x}} \right)^3},\,\,\left( \begin{gathered}<br> 9 \hfill \\<br> 0 \hfill \\ <br>\end{gathered} \right){\left( {2x} \right)^0}{\left( {\frac{k}{x}} \right)^9} + \left( \begin{gathered}<br> 9 \hfill \\<br> 1 \hfill \\ <br>\end{gathered} \right){\left( {2x} \right)^1}{\left( {\frac{k}{x}} \right)^8} + \)…, Pascal’s triangle to 9th row</p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> if there is clear evidence of adding instead of multiplying.</p>
<p>identifying correct terms (must be clearly indicated if only seen in expansion) <em><strong>(A1)(A1)</strong></em></p>
<p><em>eg</em> for <em>x</em><sup>3</sup> term: <em>r</em> = 3, <em>r</em> = 6, 7th term, \(\left( \begin{gathered}<br> 9 \hfill \\<br> 6 \hfill \\ <br>\end{gathered} \right),\,\,\left( \begin{gathered}<br> 9 \hfill \\<br> 3 \hfill \\ <br>\end{gathered} \right),\,\,{\left( {2x} \right)^6}{\left( {\frac{k}{x}} \right)^3},\,\,5376{k^3}\)</p>
<p>for <em>x</em><sup>5</sup> term: <em>r</em> = 2, <em>r</em> = 7, 8th term, \(\left( \begin{gathered}<br> 9 \hfill \\<br> 7 \hfill \\ <br>\end{gathered} \right),\,\,\left( \begin{gathered}<br> 9 \hfill \\<br> 2 \hfill \\ <br>\end{gathered} \right),\,\,{\left( {2x} \right)^7}{\left( {\frac{k}{x}} \right)^2},\,\,4608{k^2}\)</p>
<p>correct equation (may include powers of<em> x</em>) <em><strong>A1</strong></em><br><em>eg</em> \(\left( \begin{gathered}<br> 9 \hfill \\<br> 3 \hfill \\ <br>\end{gathered} \right){\left( {2x} \right)^6}{\left( {\frac{k}{x}} \right)^3} = \left( \begin{gathered}<br> 9 \hfill \\<br> 2 \hfill \\ <br>\end{gathered} \right){\left( {2x} \right)^7}{\left( {\frac{k}{x}} \right)^2}\)</p>
<p>valid attempt to solve their equation in terms of <em>k</em> only <em><strong>(M1)</strong></em><br><em>eg</em> sketch, \(84 \times 64{k^3} - 36 \times 128{k^2} = 0,\,\,5376k - 4608 = 0,\,\,\left( \begin{gathered}<br> 9 \hfill \\<br> 3 \hfill \\ <br>\end{gathered} \right){2^6}{k^3} = \left( \begin{gathered}<br> 9 \hfill \\<br> 2 \hfill \\ <br>\end{gathered} \right){2^7}{k^2}\)</p>
<p>0.857142</p>
<p>\(k = \frac{{4608}}{{5376}}\left( { = \frac{6}{7}} \right)\) (exact), 0.857 <em><strong>A1N4</strong></em></p>
<p><em><strong>[6 marks]</strong></em></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 \(f(x) = {x^3} - 4x + 1\) .</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;">Expand \({(x + h)^3}\) .</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;">Use the formula \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}\) </span><span style="font-family: times new roman,times; font-size: medium;">to show that </span><span style="font-family: times new roman,times; font-size: medium;">the derivative of \(f(x)\) is \(3{x^2} - 4\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} - 2)\) is parallel to the tangent at </span><span style="font-family: times new roman,times; font-size: medium;">a point Q. Find the coordinates of Q.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is decreasing for \(p < x < q\) . Find the value of <em>p</em> and of <em>q</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the range of values for the gradient of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</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;">attempt to expand <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({(x + h)^3} = {x^3} + 3{x^2}h + 3x{h^2} + {h^3}\) <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><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(x + h\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{{{(x + h)}^3} - 4(x + h) + 1 - ({x^3} - 4x + 1)}}{h}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplifying <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{({x^3} + 3{x^2}h + 3x{h^2} + {h^3} - 4x - 4h + 1 - {x^3} + 4x - 1)}}{h}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">factoring out <em>h</em> <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{h(3{x^2} + 3xh + {h^2} - 4)}}{h}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f'(x) = 3{x^2} - 4\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>AG N0</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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(1) = - 1\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up an appropriate equation <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{x^2} - 4 = - 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">at Q, \(x = - 1,y = 4\) (Q is \(( - 1{\text{, }}4)\)) <em><strong>A1 A1</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>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that <em>f</em> is decreasing when \(f'(x) < 0\) <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct values for <em>p</em> and <em>q</em> (but do not accept \(p = 1.15{\text{, }}q = - 1.15\) ) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(p = - 1.15{\text{, }}q = 1.15\) ; \( \pm \frac{2}{{\sqrt 3 }}\) ; an interval such as \( - 1.15 \le x \le 1.15\)</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">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) \ge - 4\) , \(y \ge - 4\) , \(\left[ { - 4,\infty } \right[\) <em><strong>A2 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">e.</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;">In part (a), the basic expansion was not done well. Rather than use the binomial theorem, many candidates opted to expand by multiplication which resulted in algebraic 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), it was clear that many candidates had difficulty with differentiation from first principles. Those that successfully set the answer up, often got lost in the simplification. </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;">Part (c) was poorly done with many candidates assuming that the tangents were horizontal and then incorrectly estimating the maximum of <em>f</em> as the required point. Many candidates unnecessarily found the equation of the tangent and could not make any further progress. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d) many correct solutions were seen but only a very few earned the reasoning mark. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was often not attempted and if it was, candidates were not clear on what was expected. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Consider a geometric sequence where the first term is 768 and the second term is 576.</p>
<p>Find the least value of \(n\) such that the \(n\)th term of the sequence is less than 7.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to find \(r\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{{576}}{{768}},{\text{ }}\frac{{768}}{{576}},{\text{ }}0.75\)</p>
<p>correct expression for \({u_n}\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(768{(0.75)^{n - 1}}\)</p>
<p><strong>EITHER (solving inequality)</strong></p>
<p>valid approach (accept equation) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({u_n} < 7\)</p>
<p>valid approach to find \(n\) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(768{(0.75)^{n - 1}} = 7,{\text{ }}n - 1 > {\log _{0.75}}\left( {\frac{7}{{768}}} \right)\), sketch</p>
<p>correct value</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(n = 17.3301\) <strong><em>(A1)</em></strong></p>
<p>\(n = 18\) (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>OR (table of values)</strong></p>
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({u_n} > 7\), one correct crossover value</p>
<p>both crossover values, \({u_{17}} = 7.69735\) <strong>and</strong> \({u_{18}} = 5.77301\) <strong><em>A2</em></strong></p>
<p>\(n = 18\) (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong>OR (sketch of functions)</strong></p>
<p>valid approach <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)sketch of appropriate functions</p>
<p>valid approach <strong><em>(M1) </em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)finding intersections or roots (depending on function sketched)</p>
<p>correct value</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(n = 17.3301\) <strong><em>(A1)</em></strong></p>
<p>\(n = 18\) (must be an integer) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[6 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;">A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were 280 taxis in the city. After <em>n</em> years the number of taxis, <em>T</em>, in the city is given by\[T = 280 \times {1.12^n} .\]</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;">(i) Find the number of taxis in the city at the end of 2005. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the year in which the number of taxis is double the number of taxis there were at the end of 2000.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (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;">At the end of 2000 there were \(25600\) people in the city who used taxis. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">After <em>n</em> years the number of people, <em>P</em>, in the city who used taxis is given by\[P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1n}}}} .\](i) Find the value of <em>P</em> at the end of 2005, giving your answer to the nearest whole number. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) After seven complete years, will the value of <em>P</em> be double its value at the end of 2000? Justify your answer.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b(i) and (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;">Let <em>R</em> be the ratio of the number of people using taxis in the city to the number of taxis. The city will reduce the number of taxis if \(R < 70\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find the value of <em>R</em> at the end of 2000. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) After how many complete years will the city first reduce the number of taxis?</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</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;">(i) \(n = 5\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(T = 280 \times {1.12^5}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(T = 493\) <em><strong>A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of doubling <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. 560</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting up equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(280 \times {1.12^n} = 560\), \({1.12^n} = 2\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(n = 6.116 \ldots \) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">in the year 2007 <em><strong>A1 N3</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">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1(5)}}}}\) </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;">\(P = 39635.993 \ldots \) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 39636\) <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(P = \frac{{2560000}}{{10 + 90{{\rm{e}}^{ - 0.1(7)}}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(P = 46806.997 \ldots \) </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;">not doubled <em><strong>A1 N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reason for <strong>their</strong> answer <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(P < 51200\)</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(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) correct value <em><strong>A2 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{25600}}{{280}}\) , 91.4, \(640:7\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) setting up an inequality (accept an equation, or reversed inequality) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{P}{T} < 70\) , \(\frac{{2560000}}{{(10 + 90{{\rm{e}}^{ - 0.1n}})280 \times {{1.12}^n}}} < 70\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding the value \(9.31 \ldots \) <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">after 10 years <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</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;">A number of candidates found this question very accessible. In part (a), many correctly solved </span><span style="font-family: times new roman,times; font-size: medium;">for <em>n</em>, but often incorrectly answered with the year 2006, thus misinterpreting that 6.12 years </span><span style="font-family: times new roman,times; font-size: medium;">after the end of 2000 is in the year 2007.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many found correct values in part (b) and often justified their result by simply noting the value after seven years is less than 51200. A common alternative was to divide 46807 by 25600 and note that this ratio is less than two. There were still a good number of candidates who failed to provide any justification as instructed.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
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<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Part (c) proved more challenging to candidates. Many found the correct ratio for <em>R</em>, however </span><span style="font-family: times new roman,times; font-size: medium;">few candidates then created a proper equation or inequality by dividing the function for <em>P</em> by </span><span style="font-family: times new roman,times; font-size: medium;">the function for <em>T</em> and setting this equal (or less) than 70. Such a function, although unfamiliar, </span><span style="font-family: times new roman,times; font-size: medium;">can be solved using the graphing or solving features of the GDC. Many candidates chose a </span><span style="font-family: times new roman,times; font-size: medium;">tabular approach but often only wrote down one value of the table, such as \(n = 10\) , \(R = 68.3\) . </span><span style="font-family: times new roman,times; font-size: medium;">What is essential is to include the two values between which the correct answer falls. </span><span style="font-family: times new roman,times; font-size: medium;">Sufficient evidence would include \(n = 9\) , \(R = 70.8\) so that it is clear the value of \(R = 70\) has </span><span style="font-family: times new roman,times; font-size: medium;">been surpassed.</span></p>
<div class="question_part_label">c(i) and (ii).</div>
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