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<h2>SL Paper 1</h2><div class="specification">
<p>Mia baked a very large apple pie that she cuts into slices to share with her friends. The smallest slice is cut first. The volume of each successive slice of pie forms a geometric sequence.</p>
<p>The second smallest slice has a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>. The fifth smallest slice has a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>240</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio of the sequence.</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 volume of the smallest slice of pie.</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>The apple pie has a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>61</mn><mo> </mo><mn>425</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>.</p>
<p>Find the total number of slices Mia can cut from this pie.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. It appeared in a paper that permitted the use of a calculator, and so might not be suitable for all forms of practice.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mi>r</mi><mo>=</mo><mn>30</mn></math> <strong>and </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><msup><mi>r</mi><mn>4</mn></msup><mo>=</mo><mn>240</mn><mo>,</mo></math> <em><strong>(M1)</strong></em><em><strong><br></strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for both the given terms expressed in the formula for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub></math>.</p>
<p><strong>OR</strong><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><msup><mi>r</mi><mn>3</mn></msup><mo>=</mo><mn>240</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>r</mi><mn>3</mn></msup><mo>=</mo><mn>8</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for a correct equation seen.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>r</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>2</mn></math> <em><strong>(A1)</strong></em><em><strong> (C2)</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>×</mo><mn>2</mn><mo>=</mo><mn>30</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>×</mo><msup><mn>2</mn><mn>4</mn></msup><mo>=</mo><mn>240</mn></math> <em><strong>(M1)</strong></em><em><strong><br></strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution in geometric sequence formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo></mrow></mfenced><mo> </mo><mn>15</mn></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)<br></strong></em></p>
<p><em><strong><br></strong></em><strong>Note:</strong> Follow through from part (a).</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><mfenced><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>61425</mn></math> <em><strong>(M1)</strong></em><em><strong><br></strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correctly substituted geometric series formula equated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>61425</mn></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>12</mn></math> (slices) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)<br></strong></em></p>
<p><em><strong><br></strong></em><strong>Note:</strong> Follow through from parts (a) and (b).</p>
<p><em><strong><br>[2 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="question">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo> </mo><mi>ln</mi><mo> </mo><mn>9</mn><mo>+</mo><mn>4</mn></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi><msup><mi mathvariant="normal">e</mi><mi>q</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mo> </mo><mi>ln</mi><mo> </mo><mn>9</mn><mo>=</mo><mn>4</mn></math></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo> </mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>m</mi></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mo> </mo><mi>ln</mi><mo> </mo><mn>9</mn><mo>=</mo><mn>4</mn></math></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>a</mi><mo>-</mo><mo> </mo><mi>ln</mi><mo> </mo><mi>b</mi><mo>=</mo><mi>ln</mi><mo> </mo><mfrac><mi>a</mi><mi>b</mi></mfrac></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>9</mn></mfrac><mo>=</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>9</mn></mfrac><mo>=</mo><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup><mo>⇒</mo><mi>x</mi><mo>=</mo><msqrt><mn>9</mn><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><msup><mi mathvariant="normal">e</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mi>p</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>expresses <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi mathvariant="normal">e</mi></math> and uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>m</mi></msup><mo>=</mo><mi>m</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>3</mn><mo>+</mo><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi mathvariant="normal">e</mi><mo> </mo><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>3</mn><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi mathvariant="normal">e</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi mathvariant="normal">e</mi><mo>=</mo><mi>ln</mi><mo> </mo><msup><mi mathvariant="normal">e</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>a</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>b</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>a</mi><mi>b</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mfenced><mrow><mn>3</mn><msup><mi mathvariant="normal">e</mi><mn>2</mn></msup></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><msup><mi mathvariant="normal">e</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mi>p</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>expresses <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo> </mo><mi>ln</mi><mo> </mo><mi mathvariant="normal">e</mi></math> and uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>m</mi></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>ln</mi><mo> </mo><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup></math> <strong>A1</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>a</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>b</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>a</mi><mi>b</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>ln</mi><mo> </mo><mfenced><mrow><msup><mn>3</mn><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mn>3</mn><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup><mo>⇒</mo><mi>x</mi><mo>=</mo><msqrt><msup><mn>3</mn><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">e</mi><mn>4</mn></msup></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><msup><mi mathvariant="normal">e</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mi>p</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[5 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>p</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the case where the series is geometric.</p>
</div>
<div class="specification">
<p>Now consider the case where the series is arithmetic with common difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the series is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">attempt to use a ratio from consecutive terms <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><msup><mi>r</mi><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mfenced><mfrac><mn>1</mn><mrow><mn>3</mn><mi>p</mi></mrow></mfrac></mfenced></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Candidates may use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>…</mo></math> and consider the powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in geometric sequence</p>
<p style="text-align:left;">Award <em><strong>M1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>p</mi><mn>1</mn></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>p</mi></mfrac></math>.</p>
<p style="text-align:left;"><strong><br>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <em><strong>M0A0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> with no other working seen.</p>
<p style="text-align:left;"> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mstyle></mrow></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><mfrac><mn>3</mn><msqrt><mn>3</mn></msqrt></mfrac><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><mn>3</mn></msqrt></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;">attempt to find a difference from consecutive terms or from <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;">correct equation <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><strong><br>Note:</strong> Candidates may use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mi>p</mi></msup><mo>+</mo><mi>ln</mi><mo> </mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>3</mn></mfrac></msup><mo>+</mo><mo>…</mo></math> and consider the powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in arithmetic sequence.</p>
<p style="text-align:left;">Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mi>p</mi></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">attempt to use arithmetic mean <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>p</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 3</strong></p>
<p style="text-align:left;">attempt to find difference using <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>3</mn></msub></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>d</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>⇒</mo><mi>d</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>METHOD 1</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow></mfenced></math></p>
<p style="text-align:left;">attempt to substitute into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> and equate to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math></p>
<p style="text-align:left;">correct working with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>6</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfenced><mfrac><mrow><mn>4</mn><mo>-</mo><mi>n</mi></mrow><mn>3</mn></mfrac></mfenced><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math></p>
<p style="text-align:left;">correct equation without <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>6</mn></mfrac><mo>=</mo><mo>-</mo><mn>3</mn></math> or equivalent</p>
<p style="text-align:left;"><strong><br>Note:</strong> Award as above if the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mi>p</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>+</mo><mo>…</mo></math> is considered leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>7</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mi>n</mi><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mo>-</mo><mn>3</mn></math>.</p>
<p style="text-align:left;"><br>attempt to form a quadratic <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn><mi>n</mi><mo>-</mo><mn>18</mn><mo>=</mo><mn>0</mn></math></p>
<p style="text-align:left;">attempt to solve their quadratic <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>-</mo><mn>9</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>METHOD 2</strong></p>
<p style="text-align:left;">listing the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> terms of the sequence <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mn>0</mn><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></math></p>
<p style="text-align:left;">recognizing first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> terms sum to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math><sup>th</sup> term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math><sup>th</sup> term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">sum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math><sup>th</sup> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math><sup>th</sup> term <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>9</mn></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates were able to identify the key relationship between consecutive terms for both geometric and arithmetic sequences. Substitution into the infinity sum formula was good with solving involving the natural logarithm done quite well. The complexity of the equation formed using 𝑆𝑛 was a stumbling block for some candidates. Those who factored out and cancelled the ln𝑥 expression were typically successful in solving the resulting quadratic.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math><sup>th</sup> term of an arithmetic sequence is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mn>15</mn><mo>-</mo><mn>3</mn><mi>n</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the value of the first term, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math>.</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>Given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math><sup>th</sup> term of this sequence is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>33</mn></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</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 common difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>12</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo>-</mo><mn>3</mn><mi>n</mi><mo>=</mo><mo>-</mo><mn>33</mn><mo> </mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>16</mn></math> <em><strong>A1</strong></em></p>
<p> </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>valid approach to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>-</mo><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>9</mn><mo>-</mo><mn>12</mn></math> OR recognize gradient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>3</mn></math> OR attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>33</mn><mo>=</mo><mn>12</mn><mo>+</mo><mn>15</mn><mi>d</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 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;">
<p>A large majority of candidates earned full marks for this question. In part (a), a surprising number of candidates did not substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> into the given expression, erroneously stating <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>15</mn></math>. Many of these candidates were able to earn follow-through marks in later parts of the question. In part (b), algebraic errors led a few candidates to find inappropriate values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>-</mo><mn>6</mn></math>.</p>
<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>In an arithmetic sequence, the first term is 3 and the second term is 7.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the tenth term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the first ten terms of the sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to subtract terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = {u_2} - {u_1},{\text{ }}7 - 3">
<mi>d</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>7</mn>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 4">
<mi>d</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct approach <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{10}} = 3 + 9(4)">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mo>+</mo>
<mn>9</mn>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{10}} = 39">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>39</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into sum <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{10}} = 5(3 + 39),{\text{ }}{S_{10}} = \frac{{10}}{2}(2 \times 3 + 9 \times 4)">
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>5</mn>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>+</mo>
<mn>39</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>10</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>×</mo>
<mn>3</mn>
<mo>+</mo>
<mn>9</mn>
<mo>×</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{10}} = 210">
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>210</mn>
</math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {11} \\ a \end{array}} \right) = \frac{{11{\text{!}}}}{{a{\text{!}}\,9{\text{!}}}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>11</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mrow>
<mi>a</mi>
<mrow>
<mtext>!</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>9</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></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>Hence or otherwise find the coefficient of the term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^9}"> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> </math></span> in the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x + 3} \right)^{11}}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>11</mn> </mrow> </msup> </mrow> </math></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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11 - a = 9"> <mn>11</mn> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>9</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{11{\text{!}}}}{{9{\text{!}}\left( {11 - 9} \right){\text{!}}}}"> <mfrac> <mrow> <mn>11</mn> <mrow> <mtext>!</mtext> </mrow> </mrow> <mrow> <mn>9</mn> <mrow> <mtext>!</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>11</mn> <mo>−</mo> <mn>9</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>!</mtext> </mrow> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2"> <mi>a</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A1</strong></em><em><strong> 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>valid approach for expansion using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 11"> <mi>n</mi> <mo>=</mo> <mn>11</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {11} \\ r \end{array}} \right){x^{11 - r}}{3^r}"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>x</mi> <mrow> <mn>11</mn> <mo>−</mo> <mi>r</mi> </mrow> </msup> </mrow> <mrow> <msup> <mn>3</mn> <mi>r</mi> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^{11}}{b^0} + \left( {\begin{array}{*{20}{c}} {11} \\ 1 \end{array}} \right){a^{10}}{b^1} + \left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){a^9}{b^2} + \ldots "> <mrow> <msup> <mi>a</mi> <mrow> <mn>11</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mn>0</mn> </msup> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>a</mi> <mrow> <mn>10</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mn>1</mn> </msup> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>a</mi> <mn>9</mn> </msup> </mrow> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mo>…</mo> </math></span></p>
<p>evidence of choosing correct term <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){3^2}"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right){x^9}{3^2}"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {11} \\ 9 \end{array}} \right){3^2}"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>9</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> </math></span></p>
<p>correct working for binomial coefficient (seen anywhere, do not accept factorials) <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="55"> <mn>55</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {11} \\ 2 \end{array}} \right) = 55"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mn>11</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>55</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="55 \times {3^2}"> <mn>55</mn> <mo>×</mo> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {55 \times 9} \right){x^9}"> <mrow> <mo>(</mo> <mrow> <mn>55</mn> <mo>×</mo> <mn>9</mn> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{11 \times 10}}{2} \times 9"> <mfrac> <mrow> <mn>11</mn> <mo>×</mo> <mn>10</mn> </mrow> <mn>2</mn> </mfrac> <mo>×</mo> <mn>9</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="495"> <mn>495</mn> </math></span> <em><strong>A1</strong></em><em><strong> N2</strong></em></p>
<p><strong>Note:</strong> If there is clear evidence of adding instead of multiplying, award <em><strong>A1</strong> </em>for the correct working for binomial coefficient, but no other marks. For example, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="55{x^9} \times {3^2}"> <mn>55</mn> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> </math></span> would earn <em><strong>M0A0A1A0</strong></em>.</p>
<p>Do not award final <em><strong>A1</strong></em> for a final answer of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="495{x^9}"> <mn>495</mn> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> </math></span>, even if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="495"> <mn>495</mn> </math></span> is seen previously. If no working shown, award <em><strong>N1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="495{x^9}"> <mn>495</mn> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> </math></span>.</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="specification">
<p>Juan buys a bicycle in a sale. He gets a discount of 30% off the original price and pays 560 US dollars (USD).</p>
</div>
<div class="specification">
<p>To buy the bicycle, Juan takes a loan of 560 USD for 6 months at a nominal annual interest rate of 75%, <strong>compounded monthly</strong>. Juan believes that the total amount he will pay will be less than the original price of the bicycle.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the original price of the bicycle.</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>Calculate the difference between the original price of the bicycle and the total amount Juan will pay.</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="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{560}}{{70}} \times 100">
<mfrac>
<mrow>
<mn>560</mn>
</mrow>
<mrow>
<mn>70</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mn>100</mn>
</math></span> (or equivalent) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for dividing 560 by 0.7 or equivalent.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 800{\text{ (USD)}}">
<mo>=</mo>
<mn>800</mn>
<mrow>
<mtext> (USD)</mtext>
</mrow>
</math></span> <strong><em>(A1) (C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="560{\left( {1 + \frac{{75}}{{12 \times 100}}} \right)^{12 \times \frac{1}{2}}}">
<mn>560</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mn>75</mn>
</mrow>
<mrow>
<mn>12</mn>
<mo>×</mo>
<mn>100</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mn>12</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substitution into interest formula, <strong><em>(A1) </em></strong>for their correct substitution.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{N}} = \frac{1}{2}">
<mrow>
<mtext>N</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{I% }} = 75">
<mrow>
<mtext>I% </mtext>
</mrow>
<mo>=</mo>
<mn>75</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PV}} = ( \pm )560">
<mrow>
<mtext>PV</mtext>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<mn>560</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P/Y}} = 1">
<mrow>
<mtext>P/Y</mtext>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C/Y}} = 12">
<mrow>
<mtext>C/Y</mtext>
</mrow>
<mo>=</mo>
<mn>12</mn>
</math></span> <strong><em>(A1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C/Y}} = 12">
<mrow>
<mtext>C/Y</mtext>
</mrow>
<mo>=</mo>
<mn>12</mn>
</math></span> seen, <strong><em>(M1) </em></strong>for all other entries correct.</p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{N}} = 6">
<mrow>
<mtext>N</mtext>
</mrow>
<mo>=</mo>
<mn>6</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{I% }} = 75">
<mrow>
<mtext>I% </mtext>
</mrow>
<mo>=</mo>
<mn>75</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{PV}} = ( \pm )560">
<mrow>
<mtext>PV</mtext>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<mn>560</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P/Y}} = 12">
<mrow>
<mtext>P/Y</mtext>
</mrow>
<mo>=</mo>
<mn>12</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C/Y}} = 12">
<mrow>
<mtext>C/Y</mtext>
</mrow>
<mo>=</mo>
<mn>12</mn>
</math></span> <strong><em>(A1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C/Y}} = 12">
<mrow>
<mtext>C/Y</mtext>
</mrow>
<mo>=</mo>
<mn>12</mn>
</math></span> seen, <strong><em>(M1) </em></strong>for all other entries correct.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 805.678 \ldots {\text{ (USD)}}">
<mo>=</mo>
<mn>805.678</mn>
<mo>…</mo>
<mrow>
<mtext> (USD)</mtext>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A3) </em></strong>for 805.678… (806) seen without working.</p>
<p> </p>
<p>(Juan spends) 5.68 (USD) (5.67828… USD) (more than the original price) <strong><em>(A1)</em>(ft)<em> (C4)</em></strong></p>
<p><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;">
[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>Last year a South American candy factory sold 4.8 × 10<sup>8</sup> spherical sweets. Each sweet has a diameter of 2.5 cm.</p>
<p>The factory is producing an advertisement showing all of these sweets placed in a straight line.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The advertisement claims that the length of this line is <em>x</em> times the length of the Amazon River. The length of the Amazon River is 6400 km.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the length, in cm, of this line. Give your answer in the form <em>a</em> × 10<sup><em>k</em></sup> , where 1 ≤ <em>a</em> < 10 and k ∈ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{Z}">
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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>Write down the length of the Amazon River in cm.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>x</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>4.8 × 10<sup>8</sup> × 2.5 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for multiplying by 2.5.</p>
<p>1.2 × 10<sup>9</sup> (cm) (<em><strong>A1)</strong></em><strong>(ft)</strong><em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A0)(A0)</strong></em> for answers of the type 12 × 10<sup>8</sup>.</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>640 000 000 (cm) (6.4 × 10<sup>8</sup> (cm)) <em><strong>(A1)</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1.2 \times {{10}^9}}}{{6.4 \times {{10}^8}}}">
<mfrac>
<mrow>
<mn>1.2</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>9</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>6.4</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>8</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for division by 640 000 000.</p>
<p>= 1.88 (1.875) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a) and part (b)(i).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A hydraulic hammer drives a metal post vertically into the ground by striking the top of the post. The distance that the post is driven into the ground, by the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n{\text{th}}">
<mi>n</mi>
<mrow>
<mtext>th</mtext>
</mrow>
</math></span> strike of the hammer, is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_n}">
<mrow>
<msub>
<mi>d</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>.</p>
<p>The distances <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_1},{\text{ }}{d_2},{\text{ }}{d_3}{\text{ }} \ldots ,{\text{ }}{d_n}">
<mrow>
<msub>
<mi>d</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>d</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>d</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>d</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> form a geometric sequence.</p>
<p>The distance that the post is driven into the ground by the first strike of the hammer, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_1}">
<mrow>
<msub>
<mi>d</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>, is 64 cm.</p>
<p>The distance that the post is driven into the ground by the second strike of the hammer, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_2}">
<mrow>
<msub>
<mi>d</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, is 48 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the common ratio for this sequence.</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 distance that the post is driven into the ground by the eighth strike of the hammer.</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 <strong>total depth </strong>that the post has been driven into the ground after 10 strikes of the hammer.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="48 = 64r">
<mn>48</mn>
<mo>=</mo>
<mn>64</mn>
<mi>r</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into geometric sequence formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.75\left( {\frac{3}{4},{\text{ }}\frac{{48}}{{64}}} \right)">
<mo>=</mo>
<mn>0.75</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>3</mn>
<mn>4</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mrow>
<mn>48</mn>
</mrow>
<mrow>
<mn>64</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1) (C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="64 \times {(0.75)^7}">
<mn>64</mn>
<mo>×</mo>
<mrow>
<mo stretchy="false">(</mo>
<mn>0.75</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>7</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into geometric sequence formula or list of eight values using their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>. Follow through from part (a), only if answer is positive.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 8.54{\text{ }}({\text{cm}}){\text{ }}(8.54296 \ldots {\text{ cm}})">
<mo>=</mo>
<mn>8.54</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>8.54296</mn>
<mo>…</mo>
<mrow>
<mtext> cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{depth}} = \frac{{64\left( {1 - {{(0.75)}^{10}}} \right)}}{{1 - 0.75}}">
<mrow>
<mtext>depth</mtext>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>64</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>0.75</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>10</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>0.75</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into geometric series formula. Follow through from part (a), only if answer is positive.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 242({\text{cm}}){\text{ }}(241.583 \ldots )">
<mo>=</mo>
<mn>242</mn>
<mo stretchy="false">(</mo>
<mrow>
<mtext>cm</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>241.583</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>></mo><mn>1</mn></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> contains the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>4</mn></mrow></mfenced></math>.</p>
</div>
<div class="specification">
<p>Consider the arithmetic sequence <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo> </mo><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>8</mn></math>.</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>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><msqrt><mn>32</mn></msqrt></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are four consecutive terms in a geometric sequence.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mfrac><mn>2</mn><mn>3</mn></mfrac></mfenced><mo>=</mo><mn>4</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo>=</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msup><mn>4</mn><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msup><mfenced><msup><mn>2</mn><mn>2</mn></msup></mfenced><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>64</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mroot><mi>a</mi><mn>3</mn></mroot><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>8</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msub><mi>log</mi><mi>a</mi></msub><mo> </mo><mi>x</mi></math>.<br> Accept any equivalent expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mn>8</mn></mrow></mfrac></math>.</p>
<p> </p>
<p><em><strong>[1</strong></em><em><strong> mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><msqrt><mn>32</mn></msqrt></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>8</mn><mi>x</mi></msup><mo>=</mo><msup><mn>32</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math></p>
<p>correct working involving log/index law <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>32</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>2</mn></mfrac><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>2</mn><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><msup><mn>2</mn><mfrac><mn>5</mn><mn>2</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>8</mn><mo>=</mo><mn>3</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>ln</mi><mo> </mo><msup><mn>2</mn><mstyle displaystyle="true"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mi>ln</mi><mo> </mo><msup><mn>2</mn><mn>3</mn></msup></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mrow><mn>3</mn><mi>x</mi></mrow></msup><mo>=</mo><msup><mn>2</mn><mfrac><mn>5</mn><mn>2</mn></mfrac></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><msqrt><mn>32</mn></msqrt></mfenced><mo>=</mo><mfrac><mn>5</mn><mn>6</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>equating a pair of differences<strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>-</mo><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><msub><mi>u</mi><mn>4</mn></msub><mo>-</mo><msub><mi>u</mi><mn>3</mn></msub><mfenced><mrow><mo>=</mo><msub><mi>u</mi><mn>3</mn></msub><mo>-</mo><msub><mi>u</mi><mn>2</mn></msub></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo>-</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mi>p</mi><mn>27</mn></mfrac></mfenced><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>125</mn><mi>q</mi></mfrac></mfenced><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>125</mn><mi>q</mi></mfrac></mfenced><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mi>q</mi><mi>p</mi></mfrac></mfenced></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>p</mi><mn>27</mn></mfrac><mo>=</mo><mfrac><mn>125</mn><mi>q</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>125</mn><mi>q</mi></mfrac><mo>=</mo><mfrac><mi>q</mi><mi>p</mi></mfrac></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are in geometric sequence <strong><em>AG</em></strong></p>
<p><strong><br>Note:</strong> If candidate assumes the sequence is geometric, award no marks for part (i). If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math> has been found, this will be awarded marks in part (ii).</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>expressing a pair of consecutive terms, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><msup><mn>8</mn><mi>d</mi></msup><mo>×</mo><mn>27</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn><mo>=</mo><msup><mn>8</mn><mrow><mn>3</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></math></p>
<p>two correct pairs of consecutive terms, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mn>8</mn><mi>d</mi></msup><mo>×</mo><mn>27</mn></mrow><mn>27</mn></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></mstyle><mrow><msup><mn>8</mn><mi>d</mi></msup><mo>×</mo><mn>27</mn></mrow></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><msup><mn>8</mn><mrow><mn>3</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></mstyle><mrow><msup><mn>8</mn><mrow><mn>2</mn><mi>d</mi></mrow></msup><mo>×</mo><mn>27</mn></mrow></mfrac></math> (must include 3 ratios)<strong> <em>A1</em></strong></p>
<p>all simplify to <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>8</mn><mi>d</mi></msup></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are in geometric sequence <strong><em>AG</em></strong></p>
<p> </p>
<p><em><strong>[4</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (geometric, finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi></math>)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>4</mn></msub><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><msup><mi>r</mi><mn>3</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn><mo>=</mo><mn>27</mn><msup><mfenced><mi>r</mi></mfenced><mn>3</mn></msup></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>27</mn><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>125</mn><mi>q</mi></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>45</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>75</mn></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 2 (arithmetic)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>4</mn></msub><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><mi>d</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>+</mo><mn>3</mn><mi>d</mi></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>+</mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo>=</mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo>+</mo><mn>2</mn><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mfenced><mfrac><mn>5</mn><mn>3</mn></mfrac></mfenced></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>45</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>75</mn></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 3 (geometric using proportion)</strong></p>
<p>recognizing proportion<strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>q</mi><mo>=</mo><mn>125</mn><mo>×</mo><mn>27</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>125</mn><mi>p</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mn>27</mn><mi>q</mi></math></p>
<p>two correct proportion equations<strong> <em>A1</em></strong></p>
<p>attempt to eliminate either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>q</mi><mn>2</mn></msup><mo>=</mo><mn>125</mn><mo>×</mo><mfrac><mrow><mn>125</mn><mo>×</mo><mn>27</mn></mrow><mi>q</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><mn>27</mn><mo>×</mo><mfrac><mrow><mn>125</mn><mo>×</mo><mn>27</mn></mrow><mi>p</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>45</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mn>75</mn></math> <strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Iron in the asteroid <em>16 Psyche</em> is said to be valued at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8973</mn></math> quadrillion euros <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtext>EUR</mtext></mfenced></math>, where one quadrillion <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>10</mn><mn>15</mn></msup></math>.</p>
</div>
<div class="specification">
<p>James believes the asteroid is approximately spherical with radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>113</mn><mo> </mo><mtext>km</mtext></math>. He uses this information to estimate its volume.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of the iron in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>k</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn><mo> </mo><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</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>Calculate James’s estimate of its volume, in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>km</mtext><mn>3</mn></msup></math>.</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>The actual volume of the asteroid is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup><mo> </mo><msup><mtext>km</mtext><mn>3</mn></msup></math>.</p>
<p>Find the percentage error in James’s estimate of the volume.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. It appeared in a paper that permitted the use of a calculator, and so might not be suitable for all forms of practice.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>97</mn><mo>×</mo><msup><mn>10</mn><mn>18</mn></msup><mo> </mo><mo> </mo><mfenced><mtext>EUR</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>8</mn><mo>.</mo><mn>973</mn><mo>×</mo><msup><mn>10</mn><mn>18</mn></msup></mrow></mfenced></math> <em><strong>(A1)(A1) (C2)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>97</mn><mo> </mo><mo>(</mo><mn>8</mn><mo>.</mo><mn>973</mn><mo>)</mo></math>, <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><msup><mn>10</mn><mn>18</mn></msup></math>. Award <em><strong>(A1)(A0)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>97</mn><mtext>E</mtext><mn>18</mn></math>.<br>Award <em><strong>(A0)(A0)</strong></em> for answers of the type <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8973</mn><mo>×</mo><msup><mn>10</mn><mn>15</mn></msup></math>.</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><mo>×</mo><mi mathvariant="normal">π</mi><mo>×</mo><msup><mn>113</mn><mn>3</mn></msup></mrow><mn>3</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution in volume of sphere formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mn>040</mn><mo> </mo><mn>000</mn><mo> </mo><mfenced><msup><mtext>km</mtext><mn>3</mn></msup></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>04</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup><mo>,</mo><mo> </mo><mfrac><mrow><mn>5771588</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>6</mn><mo> </mo><mn>043</mn><mo> </mo><mn>992</mn><mo>.</mo><mn>82</mn></mrow></mfenced></math> <em><strong>(A1) (C2) </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfrac><mrow><mn>6</mn><mo> </mo><mn>043</mn><mo> </mo><mn>992</mn><mo>.</mo><mn>82</mn><mo>-</mo><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow><mrow><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfrac></mfenced><mo>×</mo><mn>100</mn></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)</strong></em> for their correct substitution into the percentage error formula (accept a consistent absence of “<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></math>” from all terms).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>494</mn><mo> </mo><mfenced><mo>%</mo></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>494026</mn><mo>…</mo><mfenced><mo>%</mo></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2) </strong></em></p>
<p><strong><em><br></em></strong><strong>Note:</strong> Follow through from their answer to part (b). If the final answer is negative, award at most <em><strong>(M1)(A0)</strong></em>.</p>
<p><em><strong><br>[2 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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {2n - 1} \right)^2} + {\left( {2n + 1} \right)^2} = 8{n^2} + 2">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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>Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.</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>attempting to expand the LHS <em><strong>(M1)</strong></em></p>
<p>LHS <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {4{n^2} - 4n + 1} \right) + \left( {4{n^2} + 4n + 1} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 8{n^2} + 2">
<mo>=</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
</math></span> (= RHS) <em><strong>AG</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><em><strong>METHOD 1</strong></em></p>
<p>recognition that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2n - 1}">
<mrow>
<mn>2</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2n + 1}">
<mrow>
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</math></span> represent two consecutive odd integers (for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>) <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8{n^2} + 2 = 2\left( {4{n^2} + 1} \right)">
<mn>8</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>valid reason <em>eg</em> divisible by 2 (2 is a factor) <em><strong>R1</strong></em></p>
<p>so the sum of the squares of any two consecutive odd integers is even <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>recognition, <em>eg</em> that <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_n">
<msub>
<mi></mi>
<mi>n</mi>
</msub>
</math></span></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{n + 2}">
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</math></span> represent two consecutive odd integers (for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>) <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{n^2} + {\left( {n + 2} \right)^2} = 2\left( {{n^2} + 2n + 2} \right)">
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>valid reason <em>eg</em> divisible by 2 (2 is a factor) <em><strong>R1</strong></em></p>
<p>so the sum of the squares of any two consecutive odd integers is even <em><strong>AG</strong></em></p>
<p><em><strong>[3 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="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand and simplify <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable substitution for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is a positive real constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;">attempt to use binomial expansion <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><mprescripts></mprescripts><mn>3</mn></mmultiscripts><mo>×</mo><mn>1</mn><mo>×</mo><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>3</mn></mmultiscripts><mo>×</mo><mn>1</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi></mrow></mfenced><mn>3</mn></msup></math></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>a</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>a</mi><mo>+</mo><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;">So, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;">attempt to substitute any double angle rule for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"><br><strong>Note:</strong> Allow working RHS to LHS.</p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">recognizing to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>×</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>EITHER</strong></p>
<p style="text-align:left;">applies integration by inspection <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn><mo>∫</mo><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>×</mo><msup><mfenced><mrow><mi>sin</mi><mo> </mo><mi>x</mi></mrow></mfenced><mn>6</mn></msup></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mi>m</mi></msubsup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>cos</mi><mo> </mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mn>32</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo> </mo><mfenced><mrow><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mn>32</mn><msup><mi>u</mi><mn>6</mn></msup><mo> </mo><mo>d</mo><mi>u</mi></math></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>u</mi><mn>7</mn></msup><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>0</mn><mi>m</mi></msubsup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>u</mi><mn>7</mn></msup></mrow></mfenced><mn>0</mn><mrow><mi>sin</mi><mo> </mo><mi>m</mi></mrow></msubsup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math> <em><strong>AG</strong></em></p>
<p style="text-align:left;"> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><strong>EITHER</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mfenced><mrow><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup></mrow></mfenced><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></mrow></mfenced><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>OR</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>32</mn><mn>7</mn></mfrac><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>+</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p style="text-align:left;"><br><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>128</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><msup><mn>2</mn><mn>7</mn></msup></mfrac></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>m</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates successfully expanded the binomial, with the most common error being to omit the negative sign with a. The connection between (a)(i) and (ii) was often noted but not fully utilised with candidates embarking on unnecessary complex algebraic expansions of expressions involving double angle rules. Candidates often struggled to apply inspection or substitution when integrating. As a 'show that' question, b(i) provided a useful result to be utilised in (ii). So even without successfully completing (i) candidates could apply it in part (ii). Not many managed to do so.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the binomial expansion <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>7</mn></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>7</mn></mrow></msup><mo>+</mo><mi>a</mi><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><mn>1</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>21</mn></math>.</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>The third term in the expansion is the mean of the second term and the fourth term in the expansion.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</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><strong>EITHER</strong></p>
<p>recognises the required term (or coefficient) in the expansion <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>7</mn></mmultiscripts><msup><mi>x</mi><mn>5</mn></msup><msup><mn>1</mn><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>7</mn></mmultiscripts></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><mprescripts></mprescripts><mn>7</mn></mmultiscripts></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mn>5</mn><mo>!</mo></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>7</mn><mo>!</mo></mrow><mrow><mn>2</mn><mo>!</mo><mfenced><mrow><mn>7</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac></mrow></mfenced></math></p>
<p>correct working <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>1</mn><mo>×</mo><mn>5</mn><mo>×</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mo>×</mo><mn>6</mn></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>42</mn><mn>2</mn></mfrac></math></p>
<p><br><strong>OR</strong></p>
<p>lists terms from row <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> of Pascal’s triangle <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>,</mo><mo> </mo><mn>21</mn><mo>,</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>21</mn></math> <em><strong>AG</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>7</mn></math> <em><strong>(A1)</strong></em></p>
<p>correct equation <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mfrac><mrow><mi>a</mi><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn><msup><mi>x</mi><mn>5</mn></msup><mo>=</mo><mfrac><mrow><mn>7</mn><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>2</mn></mfrac></math></p>
<p>correct quadratic equation <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>42</mn><mi>x</mi><mo>+</mo><mn>35</mn><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math> (or equivalent)</p>
<p>valid attempt to solve <strong>their</strong> quadratic <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mn>5</mn></mfenced></msqrt></mrow><mrow><mn>2</mn><mfenced><mn>1</mn></mfenced></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award final <em><strong>A0</strong> </em>for obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>5</mn></math>.</p>
<p> </p>
<p><em><strong>[5 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;">
<p>The majority of candidates answered part (a) correctly, either by using the <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mi>n</mi></mmultiscripts></math> formula or Pascal's Triangle. In part (b) of the question, most candidates were able to correctly find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>7</mn></math> and set up a correct equation showing the mean of the second and fourth terms. While some struggled to complete the required algebra to solve the equation, the majority of candidates who found a correct quadratic equation were able to solve it correctly. A few candidates included <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> in their final answer, thus not earning the final mark.</p>
<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 first three terms of a geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 486,{\text{ }}{u_2} = 162,{\text{ }}{u_3} = 54">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>486</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>162</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>54</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, the common ratio of the sequence.</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 value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = 2">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></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>Find the sum of the first 30 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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{162}}{{486}}">
<mfrac>
<mrow>
<mn>162</mn>
</mrow>
<mrow>
<mn>486</mn>
</mrow>
</mfrac>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{54}}{{162}}">
<mfrac>
<mrow>
<mn>54</mn>
</mrow>
<mrow>
<mn>162</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for dividing any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{n + 1}}">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}{\text{ }}(0.333,{\text{ }}0.333333 \ldots )">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.333</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.333333</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="486{\left( {\frac{1}{3}} \right)^{n - 1}} = 2">
<mn>486</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for their correct substitution into geometric sequence formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 6">
<mi>n</mi>
<mo>=</mo>
<mn>6</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p>Award <strong><em>(A1)(A0) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6} = 2">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</math></span> with or without working.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{30}} = \frac{{486\left( {1 - {{\frac{1}{3}}^{30}}} \right)}}{{1 - \frac{1}{3}}}">
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mn>30</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>486</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mrow>
<mn>30</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into geometric series formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 729">
<mo>=</mo>
<mn>729</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>1</mn></math>.</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>Hence or otherwise, solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>θ</mi><mo>≤</mo><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi>θ</mi><mo>≠</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>.</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><strong>METHOD 1</strong></p>
<p>attempt to write all LHS terms with a common denominator of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>-</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to use algebraic division on RHS <em><strong>(M1)</strong></em></p>
<p>correctly obtains quotient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></math> and remainder <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math> as required. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2</strong></em><em><strong> marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math></p>
<p><strong><br>EITHER</strong></p>
<p>attempt to factorise in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>+</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>+</mo><mi>b</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> Accept any variable in place of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><strong><br></strong><strong>OR</strong></p>
<p>attempt to substitute into quadratic formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><msqrt><mn>49</mn></msqrt></mrow><mn>4</mn></mfrac></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> only.</p>
<p><br>one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>11</mn><mi mathvariant="normal">π</mi></mrow><mn>6</mn></mfrac></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>330</mn></math>) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>12</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>11</mn><mtext>π</mtext></mrow><mn>12</mn></mfrac></math> (must be in radians) <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A0</strong></em> if additional answers given.</p>
<p> </p>
<p><em><strong>[5</strong></em><em><strong> 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="specification">
<p>The volume of a hemisphere, <em>V</em>, is given by the formula</p>
<p style="text-align: center;"><em>V</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{4\,{S^3}}}{{243\,\pi }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>S</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>243</mn>
<mspace width="thinmathspace"></mspace>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</msqrt>
</math></span>,</p>
<p>where <em>S</em> is the total surface area.</p>
<p>The total surface area of a given hemisphere is 350 cm<sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the volume of this hemisphere in cm<sup>3</sup>.</p>
<p>Give your answer correct to <strong>one decimal place</strong>.</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>Write down your answer to part (a) correct to the nearest integer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down your answer to <strong>part (b)</strong> in the form <em>a</em> × 10<sup><em>k</em></sup> , where 1 ≤ <em>a</em> < 10 and <em>k </em>∈<em> </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{Z}">
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{4\,{{\left( {350} \right)}^3}}}{{243\,\pi }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>350</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>243</mn>
<mspace width="thinmathspace"></mspace>
<mi>π</mi>
</mrow>
</mfrac>
</msqrt>
</math></span> <em><strong>OR </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{171500\,000}}{{763.407\, \ldots }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>171500</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</mrow>
<mrow>
<mn>763.407</mn>
<mspace width="thinmathspace"></mspace>
<mo>…</mo>
</mrow>
</mfrac>
</msqrt>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <strong>(M1)</strong> for substitution of 350 into volume formula.</p>
<p> </p>
<p>= 473.973… <em><strong>(A1)</strong></em> </p>
<p>= 474 (cm<sup>3</sup>) <em><strong>(A1)</strong></em><strong>(ft) <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>The final<strong> (A1)(ft) </strong>is awarded for rounding <strong>their</strong> answer to 1 decimal place provided the unrounded answer is seen.</p>
<p> </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>474 (cm<sup>3</sup>) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C1)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>4.74 × 10<sup>2</sup> (cm<sup>3</sup>) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Follow through from <strong>part (b) only</strong>.</p>
<p>Award<em><strong> (A0)(A0)</strong></em> for answers of the type 0.474 × 10<sup>3</sup>.</p>
<p> </p>
<p><em><strong>[2 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>In an arithmetic sequence, the first term is 8 and the second term is 5.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the tenth term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>subtracting terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5 - 8,{\text{ }}{u_2} - {u_1}">
<mn>5</mn>
<mo>−</mo>
<mn>8</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 3">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution into formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{10}} = 8 + (10 - 1)( - 3),{\text{ }}8 - 27,{\text{ }} - 3(10) + 11">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>10</mn>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>8</mn>
<mo>−</mo>
<mn>27</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>11</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{10}} = - 19">
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>19</mn>
</math></span> <strong><em>A1 N2</em></strong></p>
<p><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;">
[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>Sergei is training to be a weightlifter. Each day he trains at the local gym by lifting a metal bar that has heavy weights attached. He carries out successive lifts. After each lift, the same amount of weight is <strong>added</strong> to the bar to increase the weight to be lifted.</p>
<p>The weights of each of Sergei’s lifts form an arithmetic sequence.</p>
<p>Sergei’s friend, Yuri, records the weight of each lift. Unfortunately, last Monday, Yuri misplaced all but two of the recordings of Sergei’s lifts.</p>
<p>On that day, Sergei lifted 21 kg on the third lift and 46 kg on the eighth lift.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For that day find how much weight was added after each lift.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For that day find the weight of Sergei’s first lift.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.</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>5<em>d</em> = 46 − 21 <strong>OR</strong> <em>u</em><sub>1</sub> + 2<em>d</em> = 21 and <em>u</em><sub>1</sub> + 7<em>d</em> = 46 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for a correct equation in <em>d</em> or for two correct equations in <em>u</em><sub>1</sub> and <em>d</em>.</p>
<p>(<em>d =</em>) 5 (kg) <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>u</em><sub>1</sub> + 2 × 5 = 21 <em><strong>(M1)</strong></em></p>
<p><em><strong>OR</strong></em></p>
<p><em>u</em><sub>1</sub> + 7 × 5 = 46 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution of their <em>d</em> into either of the two equations.</p>
<p>(<em>u</em><sub>1 </sub>=) 11 (kg) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a)(i).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{12}}{2}\left( {2 \times 11 + \left( {12 - 1} \right) \times 5} \right)">
<mfrac>
<mrow>
<mn>12</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>11</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>12</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution into arithmetic series formula.</p>
<p>= 462 (kg) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from parts (a) and (b).</p>
<p><em><strong>[2 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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider any three consecutive integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that the sum of these three integers is always divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</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>Prove that the sum of the squares of these three integers is never divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mi>n</mi><mo>+</mo><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mi>n</mi></math> <em><strong>A1</strong></em></p>
<p>which is always divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> <em><strong>AG</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to expand either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> (do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math>) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>demonstrating recognition that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is not divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> seen after correct expression divided by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup></math> is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></math> is never divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></p>
<p>OR the first term is divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>, the second is not</p>
<p>OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><mn>3</mn></mfrac><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p>hence the sum of the squares is never divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> <em><strong>AG</strong></em></p>
<p> </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;">
<p>Most candidates were able to earn full marks in part (a), though some were not able to provide the required reasoning to earn full marks in part (b). In many cases, candidates did not seem to understand the nature of a general deductive proof, and instead substituted different consecutive integers (such as 1, 2,3 ), showing the desired result for these specific values, rather than an algebraic generalization for any three consecutive integers.</p>
<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>In the Canadian city of Ottawa:</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{l}} {{\text{97% of the population speak English,}}} \\ {{\text{38% of the population speak French,}}} \\ {{\text{36% of the population speak both English and French.}}} \end{array}">
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>97% of the population speak English,</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>38% of the population speak French,</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>36% of the population speak both English and French.</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
</div>
<div class="specification">
<p>The total population of Ottawa is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="985\,000">
<mn>985</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the percentage of the population of Ottawa that speak English but not French.</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>Calculate the number of people in Ottawa that speak both English and French.</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>Write down your answer to part (b) in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10">
<mn>1</mn>
<mo>⩽</mo>
<mi>a</mi>
<mo><</mo>
<mn>10</mn>
</math></span> and <em>k </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Z}">
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="97 - 36">
<mn>97</mn>
<mo>−</mo>
<mn>36</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for subtracting 36 from 97.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><img src="images/Schermafbeelding_2017-08-15_om_12.54.01.png" alt="M17/5/MATSD/SP1/ENG/TZ1/02.a/M"></p>
<p><strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for 61 <strong>and </strong>36 seen in the correct places in the Venn diagram.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 61{\text{ }}(\% )">
<mo>=</mo>
<mn>61</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mi mathvariant="normal">%</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept 61.0 (%).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{36}}{{100}} \times 985\,000">
<mfrac>
<mrow>
<mn>36</mn>
</mrow>
<mrow>
<mn>100</mn>
</mrow>
</mfrac>
<mo>×</mo>
<mn>985</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for multiplying 0.36 (or equivalent) by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="985\,000">
<mn>985</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 355\,000{\text{ }}(354\,600)">
<mo>=</mo>
<mn>355</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>354</mn>
<mspace width="thinmathspace"></mspace>
<mn>600</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3.55 \times {10^5}{\text{ }}(3.546 \times {10^5})">
<mn>3.55</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>5</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>3.546</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>5</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft)</strong><em> </em><strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1)(ft) </em></strong>for 3.55 (3.546) <strong>must </strong>match part (b), and <strong><em>(A1)(ft)</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times {10^5}">
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>5</mn>
</msup>
</mrow>
</math></span>.</p>
<p>Award <strong><em>(A0)(A0) </em></strong>for answers of the type: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="35.5 \times {10^4}">
<mn>35.5</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>4</mn>
</msup>
</mrow>
</math></span>. Follow through from part (b).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{\cos x + \sin y}}{{\sqrt {{w^2} - z} }}">
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>+</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>y</mi>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>w</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mi>z</mi>
</msqrt>
</mrow>
</mfrac>
</math></span>,</p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 36^\circ ,{\text{ }}y = 18^\circ ,{\text{ }}w = 29">
<mi>x</mi>
<mo>=</mo>
<msup>
<mn>36</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<msup>
<mn>18</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>w</mi>
<mo>=</mo>
<mn>29</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 21.8">
<mi>z</mi>
<mo>=</mo>
<mn>21.8</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>. Write down your full calculator display.</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>Write your answer to part (a)</p>
<p>(i) correct to two decimal places;</p>
<p>(ii) correct to three significant figures.</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>Write your answer to <strong>part (b)(ii) </strong>in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10,{\text{ }}k \in \mathbb{Z}">
<mn>1</mn>
<mo>⩽</mo>
<mi>a</mi>
<mo><</mo>
<mn>10</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\cos 36^\circ + \sin 18^\circ }}{{\sqrt {{{29}^2} - 21.8} }}">
<mfrac>
<mrow>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>36</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>18</mn>
<mo>∘</mo>
</msup>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mn>29</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>21.8</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into formula.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.0390625">
<mo>=</mo>
<mn>0.0390625</mn>
</math></span> <strong><em>(A1) (C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{{128}}">
<mfrac>
<mn>5</mn>
<mrow>
<mn>128</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) 0.04 <strong><em>(A1)</em>(ft) </strong></p>
<p>(ii) 0.0391 <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3.91 \times {10^{ - 2}}">
<mn>3.91</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note</strong><strong>: </strong>Answer should be consistent with their answer to part (b)(ii). Award <strong><em>(A1)</em>(ft) </strong>for 3.91, and <strong><em>(A1)</em>(ft) </strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{10^{ - 2}}">
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</math></span>. Follow through from part (b)(ii).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>4</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The following diagram shows the graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> intersect at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. The coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>In the following diagram, the shaded region is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The area of the shaded region can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>+</mo><mn>8</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>15</mn><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> <strong><em>(A1)</em></strong></p>
<p>valid attempt to solve <strong>their</strong> quadratic <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mo>±</mo><msqrt><msup><mn>8</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mn>15</mn></mfenced></msqrt></mrow><mrow><mn>2</mn><mfenced><mn>1</mn></mfenced></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mo>±</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>5</mn></mrow></mfenced></math> (may be seen in answer) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>2</mn></mrow></mfenced></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn></math>) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing two correct regions from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> <strong><em>(R1)</em></strong></p>
<p>triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><munderover><mo>∫</mo><mn>5</mn><mi>k</mi></munderover><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>3</mn><mn>5</mn></munderover><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><munderover><mo>∫</mo><mn>5</mn><mi>k</mi></munderover><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>3</mn><mn>5</mn></munderover><mfenced><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>+</mo><munderover><mo>∫</mo><mn>5</mn><mi>k</mi></munderover><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p>area of triangle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>·</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><msup><mn>5</mn><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mn>3</mn><mfenced><mn>5</mn></mfenced></mrow></mfenced><mo>-</mo><mfenced><mrow><mfrac><msup><mn>3</mn><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mn>3</mn><mfenced><mn>3</mn></mfenced></mrow></mfenced></math> <strong><em>(A1)</em></strong></p>
<p>correct integration <strong><em>(A1)</em></strong><strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>x</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.<br><strong>Note:</strong> The first three <em><strong>A</strong></em> marks may be awarded independently of the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p>substitution of <strong>their</strong> limits (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>) into <strong>their</strong> integrated function (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>) <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo>-</mo><mfenced><mrow><mi>ln</mi><mo> </mo><mn>1</mn><mo>+</mo><mn>5</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>x</mi></mrow></mfenced><mn>5</mn><mi>k</mi></msubsup><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p>adding <strong>their</strong> two areas (in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>) and equating to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>p</mi><mo>+</mo><mn>8</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>k</mi><mo>-</mo><mn>5</mn><mo>=</mo><mi>ln</mi><mo> </mo><mi>p</mi><mo>+</mo><mn>8</mn></math></p>
<p>equating <strong>their</strong> non-log terms to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> (equation must be in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>) <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>8</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>11</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>11</mn><mo>-</mo><mn>4</mn><mo>=</mo><mi>p</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[10 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;">
<p>Nearly all candidates knew to set up an equation with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> in order to find the intersection of the two graphs, and most were able to solve the resulting quadratic equation. Candidates were not as successful in part (b), however. While some candidates recognized that there were two regions to be added together, very few were able to determine the correct boundaries of these regions, with many candidates integrating one or both functions from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math>to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>. While a good number of candidates were able to correctly integrate the function(s), without the correct bounds the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> were unattainable.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mi>p</mi></msup></math>. Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</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>Hence, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>9</mn></msubsup><mfenced><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></mfenced><mo>d</mo><mi>x</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>=</mo><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>10</mn><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p>substituting limits into their integrated function and subtracting <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mn>9</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>9</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>10</mn><msup><mfenced><mn>1</mn></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>-</mo><mn>10</mn><mo>×</mo><mn>3</mn><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>10</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </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;">
<p>Many candidates could give the value of <em>p</em> correctly. However, many did struggle with the integration, including substituting limits into the integrand, without integrating at all. An incorrect value of <em>p</em> often resulted in arithmetic of greater complexity.</p>
<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><strong>In this question, give all answers to two decimal places.</strong></p>
<p>Karl invests 1000 US dollars (USD) in an account that pays a nominal annual interest of 3.5%, <strong>compounded quarterly</strong>. He leaves the money in the account for 5 years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the amount of money he has in the account after 5 years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the amount of <strong>interest</strong> he earned after 5 years.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Karl decides to donate this <strong>interest</strong> to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = <em>t</em> EUR.</p>
<p>Calculate the value of <em>t</em>.</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 class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1000{\left( {1 + \frac{{3.5}}{{4 \times 100}}} \right)^{4 \times 5}}"> <mn>1000</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mn>3.5</mn> </mrow> <mrow> <mn>4</mn> <mo>×</mo> <mn>100</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mn>4</mn> <mo>×</mo> <mn>5</mn> </mrow> </msup> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution in compound interest formula, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p><strong>OR </strong></p>
<p><em>N</em> = 5</p>
<p><em>I</em> = 3.5</p>
<p><em>PV</em> = 1000</p>
<p><em>P</em>/<em>Y</em> = 1</p>
<p><em>C</em>/<em>Y</em> = 4</p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <em>C</em>/<em>Y</em> = 4 seen, <em><strong>(M1)</strong></em> for other correct entries.</p>
<p><strong>OR</strong></p>
<p><em>N</em> = 5 × 4</p>
<p><em>I</em> = 3.5</p>
<p><em>PV</em> = 1000</p>
<p><em>P</em>/<em>Y</em> = 1</p>
<p><em>C</em>/<em>Y</em> = 4</p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <em>C</em>/<em>Y</em> = 4 seen, <em><strong>(M1)</strong></em> for other correct entries.</p>
<p>= 1190.34 (USD) <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substitution in compound interest formula, <em><strong>(A1)</strong></em> for correct substitution.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>190.34 (USD) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C4)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em><strong>(ft)</strong> for subtraction of 1000 from their part (a)(i). Follow through from (a)(i).</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{170}}{{190.34}}"> <mfrac> <mrow> <mn>170</mn> </mrow> <mrow> <mn>190.34</mn> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for division of 170 by their part (a)(ii).</p>
<p>= 0.89 <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from their part (a)(ii).</p>
<p><em><strong>[2 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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The first two terms of an infinite geometric sequence are <em>u</em><sub>1</sub> = 18 and <em>u</em><sub>2</sub> = 12sin<sup>2</sup> <em>θ</em> , where 0 < <em>θ</em> < 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span> , and <em>θ</em> ≠ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <em>r</em> in terms of <em>θ</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <em>θ</em> which give the greatest value of the sum.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{u_2}}}{{{u_1}}},\,\,\frac{{{u_1}}}{{{u_2}}}"> <mfrac> <mrow> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </mrow> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{{12\,{{\sin }^2}\,\theta }}{{18}}\left( { = \frac{{2\,{{\sin }^2}\,\theta }}{3}} \right)"> <mi>r</mi> <mo>=</mo> <mfrac> <mrow> <mn>12</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mrow> <mn>18</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><strong>METHOD 1 </strong>(using differentiation)</p>
<p>recognizing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}{S_\infty }}}{{{\text{d}}\theta }} = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">∞</mi> </msub> </mrow> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p>finding any correct expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}{S_\infty }}}{{{\text{d}}\theta }}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">∞</mi> </msub> </mrow> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>θ</mi> </mrow> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{0 - 54 \times \left( { - 2\,{\text{sin}}\,2\,\theta } \right)}}{{{{\left( {2 + {\text{cos}}\,2\,\theta } \right)}^2}}},\,\, - 54{\left( {2 + {\text{cos}}\,2\,\theta } \right)^{ - 2}}\,\left( { - 2\,{\text{sin}}\,2\,\theta } \right)"> <mfrac> <mrow> <mn>0</mn> <mo>−</mo> <mn>54</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mn>54</mn> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct working <em><strong> (A1)</strong></em></p>
<p><em>eg </em> sin 2<em>θ</em> = 0</p>
<p>any correct value for sin<sup>−1</sup>(0) (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em> 0, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>, … , sketch of sine curve with <em>x</em>-intercept(s) marked both correct values for 2<em>θ</em> (ignore additional values) <em><strong>(A1)</strong></em></p>
<p>2<em>θ </em>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> (accept values in degrees)</p>
<p>both correct answers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{2},\,\frac{{3\pi }}{2}"> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if either or both correct answers are given in degrees.<br>Award <em><strong>A0 </strong></em>if additional values are given.</p>
<p> </p>
<p><strong>METHOD 2 </strong>(using denominator)</p>
<p>recognizing when S<sub>∞</sub> is greatest <em><strong>(M1)</strong></em></p>
<p><em>eg</em> 2 + cos 2<em>θ </em>is a minimum, 1−<em>r</em> is smallest<br>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em>minimum value of 2 + cos 2<em>θ </em>is 1, minimum <em>r</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </math></span></p>
<p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2\,\theta = - 1,\,\,\frac{2}{3}\,{\text{si}}{{\text{n}}^2}\,\theta = \frac{2}{3},\,\,{\text{si}}{{\text{n}}^2}\theta = 1"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </math></span></p>
<p><strong>EITHER</strong> (using cos 2<em>θ</em>)</p>
<p>any correct value for cos<sup>−1</sup>(−1) (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>, … (accept values in degrees), sketch of cosine curve with <em>x</em>-intercept(s) marked</p>
<p>both correct values for 2<em>θ </em>(ignore additional values) <em><strong>(A1)</strong></em></p>
<p>2<em>θ </em>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> (accept values in degrees)</p>
<p><strong>OR</strong> (using sin<em>θ</em>)</p>
<p>sin<em>θ = </em>±1 (A1)</p>
<p>sin<sup>−1</sup>(1) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{2}"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> (accept values in degrees) (seen anywhere) <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p>both correct answers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{2},\,\frac{{3\pi }}{2}"> <mi>θ</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong>A1 N4</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> if either or both correct answers are given in degrees.<br>Award <em><strong>A0 </strong></em>if additional values are given.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The company Snakezen’s Ladders makes ladders of different lengths. All the ladders that the company makes have the same design such that:</p>
<p style="padding-left: 90px;">the first rung is 30 cm from the base of the ladder,</p>
<p style="padding-left: 90px;">the second rung is 57 cm from the base of the ladder,</p>
<p style="padding-left: 90px;">the distance between the first and second rung is equal to the distance between all adjacent rungs on the ladder.</p>
<p>The ladder in the diagram was made by this company and has eleven equally spaced rungs.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_10.59.54.png" alt="M17/5/MATSD/SP1/ENG/TZ1/05"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance from the base of this ladder to the top rung.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The company also makes a ladder that is 1050 cm long.</p>
<p>Find the maximum number of rungs in this 1050 cm long ladder.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="30 + (11 - 1) \times 27">
<mn>30</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>11</mn>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mn>27</mn>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substituted arithmetic sequence formula, <strong><em>(A1) </em></strong>for correct substitutions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 300{\text{ (cm)}}">
<mo>=</mo>
<mn>300</mn>
<mrow>
<mtext> (cm)</mtext>
</mrow>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Units are not required.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1050 \geqslant 30 + (n - 1) \times 27">
<mn>1050</mn>
<mo>⩾</mo>
<mn>30</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mn>27</mn>
</math></span> <strong><em>(M1)(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substituted arithmetic sequence formula <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \leqslant 1050">
<mo>⩽</mo>
<mn>1050</mn>
</math></span>, accept an equation, <strong><em>(A1) </em></strong>for correct substitutions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 38">
<mi>n</mi>
<mo>=</mo>
<mn>38</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong><em> </em><strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from their 27 in part (a). The answer must be an integer and rounded down.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>Consider the geometric sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 18,{\text{ }}{u_2} = 9,{\text{ }}{u_3} = 4.5,{\text{ }} \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>18</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>4.5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>…<!-- … --></mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the common ratio of the sequence.</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_5}">
<mrow>
<msub>
<mi>u</mi>
<mn>5</mn>
</msub>
</mrow>
</math></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>Find the smallest value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> is less than <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{10^{ - 3}}">
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{\text{ }}(0.5)">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>0.5</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C1)</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18 \times {\left( {\frac{1}{2}} \right)^4}">
<mn>18</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for their correct substitution into the geometric sequence formula. Accept a list of their five correct terms.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1.125{\text{ }}\left( {1.13,{\text{ }}\frac{9}{8}} \right)">
<mn>1.125</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.13</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>9</mn>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from their common ratio from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18 \times {\left( {\frac{1}{2}} \right)^{n - 1}} < {10^{ - 3}}">
<mn>18</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo><</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>(M1) </em></strong>for their correct substitution into the geometric sequence formula with a variable in the exponent, <strong><em>(M1) </em></strong>for comparing their expression with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{10^{ - 3}}{\text{ }}\left( {\frac{1}{{1000}}} \right)">
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Accept an equation.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 16">
<mi>n</mi>
<mo>=</mo>
<mn>16</mn>
</math></span> <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from their common ratio from part (a). “<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>” must be a positive integer for the <strong><em>(A1) </em></strong>to be awarded.</p>
<p> </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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {p^x} + q">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>p</mi>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mi>q</mi>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x{\text{, }}p{\text{, }}q \in \mathbb{R}{\text{, }}p > 1">
<mi>x</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>p</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>p</mi>
<mo>></mo>
<mn>1</mn>
</math></span>. The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( {0{\text{, }}a} \right)">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> lies on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span>.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {g^{ - 1}}\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>. The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> lies on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and is the reflection of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> in the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}"> <mrow> <mtext>B</mtext> </mrow> </math></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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> </math></span>, find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> <strong>in terms of</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> and has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left( {{\text{ln}}\,p} \right)x + q + 1"> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span>.</p>
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> passes through the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 2{\text{, }} - 2} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>The gradient of the normal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}"> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span>.</p>
<p> </p>
<p>Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {a{\text{, }}0} \right)"> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {q + 1{\text{, }}0} \right)"> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> <mrow> <mtext>, </mtext> </mrow> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>) <em><strong>A2</strong></em><em><strong> 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><strong>Note:</strong> There are many approaches to this part, and the steps may be done in any order. Please check working and award marks in line with the markscheme, noting that candidates may work with the equation of the line before finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span></strong></p>
<p>valid attempt to find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( 0 \right) = a{\text{, }}\,{p^0} + q = a"> <mi>g</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>p</mi> <mn>0</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>=</mo> <mi>a</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = q + 1"> <mi>a</mi> <mo>=</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>FINDING THE EQUATION OF</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span></p>
<p style="padding-left:30px;"><strong>EITHER</strong></p>
<p style="padding-left:30px;">attempt to substitute tangent gradient and coordinates into equation of straight line <em><strong>(M1)</strong></em></p>
<p style="padding-left:30px;"><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = f'\left( a \right)\left( {x - a} \right){\text{, }}\,y = f'\left( a \right)\left( {x - \left( {q + 1} \right)} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="padding-left:30px;">correct equation in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p style="padding-left:30px;"><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - 0 = \frac{1}{{{\text{ln}}\left( p \right)}}\left( {x - a} \right)"> <mi>y</mi> <mo>−</mo> <mn>0</mn> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="padding-left:30px;"><strong>OR</strong></p>
<p style="padding-left:30px;">attempt to substitute tangent gradient and coordinates to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span></p>
<p style="padding-left:30px;"><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 = \frac{1}{{{\text{ln}}\left( p \right)}}\left( a \right) + b"> <mn>0</mn> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> </math></span></p>
<p style="padding-left:30px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{ - a}}{{{\text{ln}}\left( p \right)}}"> <mi>b</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>THEN</strong> (must be in terms of <strong>both</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{{\text{ln}}\,p}}\left( {x - q - 1} \right){\text{, }}\,y = \frac{1}{{{\text{ln}}\,p}}x - \frac{{q + 1}}{{{\text{ln}}\,p}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong> N3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for final answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1} = \frac{1}{{{\text{ln}}\,p}}\left( {x - q - 1} \right)"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> There are many approaches to this part, and the steps may be done in any order. Please check working and award marks in line with the markscheme, noting that candidates may find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> before finding a value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span></strong></p>
<p>valid approach to find the gradient of the tangent <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_1}{m_2} = - 1{\text{, }}\,\, - \frac{1}{{\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}}}{\text{, }}\,\, - {\text{ln}}\left( {\frac{1}{3}} \right){\text{, }}\,\, - \frac{1}{{{\text{ln}}\,p}} = \frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}"><msub><mi>m</mi><mn>1</mn></msub><msub><mi>m</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mn>1</mn><mtext>, </mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo><mfrac><mn>1</mn><mfrac><mn>1</mn><mrow><mtext>ln</mtext><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac bevelled="true"><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mfrac></mfrac><mtext>, </mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo><mtext>ln</mtext><mrow><mo>(</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>)</mo></mrow><mtext>, </mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>−</mo><mfrac><mn>1</mn><mrow><mtext>ln</mtext><mspace width="thinmathspace"></mspace><mi>p</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mtext>ln</mtext><mrow><mo>(</mo><mstyle displaystyle="true"><mfrac bevelled="true"><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>)</mo></mrow></mrow></mfrac></math></span></p>
<p>correct application of log rule (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}{\left( {\frac{1}{3}} \right)^{ - 1}}{\text{, }}\,\, - \left( {{\text{ln}}\left( 1 \right) - {\text{ln}}\left( 3 \right)} \right)"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>correct equation (seen anywhere) <em><strong>A1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,p = {\text{ln}}\,3{\text{, }}\,\,p = 3"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>=</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>=</mo> <mn>3</mn> </math></span></p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span></strong></p>
<p>correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 2{\text{, }} - 2} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> equation <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 = \left( {{\text{ln}}\,p} \right)\left( { - 2} \right) + q + 1"> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 2\,{\text{ln}}\,p - 3{\text{, }}\,\,q = 2\,{\text{ln}}\,3 - 3"> <mi>q</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> <mo>−</mo> <mn>3</mn> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>q</mi> <mo>=</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>−</mo> <mn>3</mn> </math></span> (seen anywhere) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>FINDING <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span></strong></p>
<p>correct substitution of <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> into <strong>their</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{{\text{ln}}\,3}}\left( {x - \left( {2\,{\text{ln}}\,3 - 3} \right) - 1} \right)"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{{\text{ln}}\,3}}\left( {x - 2\,{\text{ln}}\,3 + 2} \right){\text{, }}\,\,y = \frac{1}{{{\text{ln}}\,3}}x - \frac{{2\,{\text{ln}}\,3 - 2}}{{{\text{ln}}\,3}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong> N2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for final answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1} = \frac{1}{{{\text{ln}}\,3}}\left( {x - 2\,{\text{ln}}\,3 + 2} \right)"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>3</mn> <mo>+</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p> </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>The speed of light is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{300}}\,{\text{000}}">
<mrow>
<mtext>300</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>000</mtext>
</mrow>
</math></span> kilometres per second. The average distance from the Sun to the Earth is 149.6 million km.</p>
</div>
<div class="specification">
<p>A light-year is the distance light travels in one year and is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{9}}\,{\text{467}}\,{\text{280}}">
<mrow>
<mtext>9</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>467</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>280</mtext>
</mrow>
</math></span> million km. Polaris is a bright star, visible from the Northern Hemisphere. The distance from the Earth to Polaris is 323 light-years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the time, <strong>in minutes</strong>, it takes for light from the Sun to reach the Earth.</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>Find the distance from the Earth to Polaris in millions of km. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span> with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10">
<mn>1</mn>
<mo>⩽</mo>
<mi>a</mi>
<mo><</mo>
<mn>10</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{149600000}}{{300000 \times 60}}">
<mfrac>
<mrow>
<mn>149600000</mn>
</mrow>
<mrow>
<mn>300000</mn>
<mo>×</mo>
<mn>60</mn>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for dividing the <strong>correct </strong>numerator (which can be presented in a different form such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="149.6 \times {10^6}">
<mn>149.6</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>6</mn>
</msup>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1.496 \times {10^8}">
<mn>1.496</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>8</mn>
</msup>
</mrow>
</math></span>) by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{300}}\,{\text{000}}">
<mrow>
<mtext>300</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>000</mtext>
</mrow>
</math></span> and <strong><em>(M1) </em></strong>for dividing by 60.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 8.31{\text{ }}({\text{minutes}}){\text{ }}(8.31111 \ldots {\text{, 8 minutes 19 seconds}})">
<mo>=</mo>
<mn>8.31</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>minutes</mtext>
</mrow>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mn>8.31111</mn>
<mo>…</mo>
<mrow>
<mtext>, 8 minutes 19 seconds</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1) (C3)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="323 \times 9467\,280">
<mn>323</mn>
<mo>×</mo>
<mn>9467</mn>
<mspace width="thinmathspace"></mspace>
<mn>280</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for multiplying 323 by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9\,467\,280">
<mn>9</mn>
<mspace width="thinmathspace"></mspace>
<mn>467</mn>
<mspace width="thinmathspace"></mspace>
<mn>280</mn>
</math></span>, seen with <strong>any </strong>power of 10; therefore only penalizing incorrect power of 10 once.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3.06 \times {10^9}{\text{ ( = }}3.05793 \ldots \times {10^9})">
<mo>=</mo>
<mn>3.06</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>9</mn>
</msup>
</mrow>
<mrow>
<mtext> ( = </mtext>
</mrow>
<mn>3.05793</mn>
<mo>…</mo>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>9</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)(A1) (C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for 3.06.</p>
<p>Award <strong><em>(A1) </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times {10^9}">
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>9</mn>
</msup>
</mrow>
</math></span></p>
<p>Award <strong><em>(A0)(A0) </em></strong>for answers of the type: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="30.6 \times {10^8}">
<mn>30.6</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>8</mn>
</msup>
</mrow>
</math></span></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>A sphere with diameter 3 474 000 metres can model the shape of the Moon.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this model to calculate the circumference of the Moon in <strong>kilometres</strong>. Give your full calculator display.</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>Give your answer to part (a) correct to three significant figures.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write your answer to<strong> part (b)</strong> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span>, where 1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> < 10 , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\,474\,000 \times \pi }}{{1000}}">
<mfrac>
<mrow>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mn>474</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mo>×</mo>
<mi>π</mi>
</mrow>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct numerator and <em><strong>(M1)</strong></em> for dividing by 1000 <strong>OR</strong> equivalent, such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3\,474\,000 \times 2 \times \pi }}{{2000}}">
<mfrac>
<mrow>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mn>474</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
<mo>×</mo>
<mn>2</mn>
<mo>×</mo>
<mi>π</mi>
</mrow>
<mrow>
<mn>2000</mn>
</mrow>
</mfrac>
</math></span> ie diameter.<br>Do not accept use of area formula ie <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi {r^2}">
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<p>10 913.89287… (km) <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>10 900 (km) <em><strong>(A1)</strong></em><strong>(ft) </strong><em><strong> (C1)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p><em><strong>[1 mark]</strong></em></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>1.09 × 10<sup>4</sup> <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em>(ft)<em> (C2)</em></strong></p>
<p><strong>Note:</strong> Follow through from part (b) only. Award <em><strong>(A1)</strong></em><strong>(ft)</strong> for 1.09, and <em><strong>(A1)</strong></em><strong>(ft)</strong> × 10<sup>4</sup>. Award <em><strong>(A0)(A0)</strong></em> for answers of the type: 10.9 × 10<sup>3</sup>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </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>The first three terms of an arithmetic sequence are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><mn>5</mn><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>8</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></math>.</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>Prove that the sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of this arithmetic sequence is a square number.</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="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><strong>EITHER</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>-</mo><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><msub><mi>u</mi><mn>3</mn></msub><mo>-</mo><msub><mi>u</mi><mn>2</mn></msub></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>8</mn></mrow></mfenced><mo>-</mo><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mfenced><mrow><mn>3</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>8</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>5</mn><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>8</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>24</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>2</mn></msub><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mn>3</mn></msub></mrow><mn>2</mn></mfrac></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>8</mn><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mn>3</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>8</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>12</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>8</mn></math> <strong>(A1)</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi></mrow></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>8</mn><mo>+</mo><mn>8</mn><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi></mrow></mfenced><mn>2</mn></msup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> The final <strong>A1</strong> can be awarded for clearly explaining that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup></math> is a square number.</p>
<p> </p>
<p>so sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms is a square number <strong>AG</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider an arithmetic sequence where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>8</mn></msub><mo>=</mo><msub><mi>S</mi><mn>8</mn></msub><mo>=</mo><mn>8</mn></math>. Find the value of the first term, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math>, and the value of the common difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 (finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">u</mi><mn mathvariant="bold">1</mn></msub></math> first, from <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold">S</mi><mn mathvariant="bold">8</mn></msub></math>)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>8</mn></mrow></mfenced><mo>=</mo><mn>8</mn></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mo>-</mo><mn>6</mn></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>7</mn><mi>d</mi><mo>=</mo><mn>8</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>7</mn><mi>d</mi></mrow></mfenced><mo>=</mo><mn>8</mn></math> (may be seen with their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math>) <em><strong> (A1)</strong></em></p>
<p>attempt to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>2</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2 (solving simultaneously)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>7</mn><mi>d</mi><mo>=</mo><mn>8</mn></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>8</mn></mrow></mfenced><mo>=</mo><mn>8</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>7</mn><mi>d</mi></mrow></mfenced><mo>=</mo><mn>8</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mo>-</mo><mn>3</mn><mi>d</mi></math> <em><strong> (A1)</strong></em></p>
<p>attempt to solve linear or simultaneous equations <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mo>-</mo><mn>6</mn><mo>,</mo><mo> </mo><mi>d</mi><mo>=</mo><mn>2</mn></math> <em><strong> A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 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>Consider the numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2.78 \times {10^{11}}">
<mi>p</mi>
<mo>=</mo>
<mn>2.78</mn>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mn>11</mn>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 3.12 \times {10^{ - 3}}">
<mi>q</mi>
<mo>=</mo>
<mn>3.12</mn>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{{\frac{p}{q}}}"> <mroot> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mn>3</mn> </mroot> </math></span>. Give your full calculator display.</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>Write down your answer to part (a) correct to two decimal places;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down your answer to part (a) correct to three significant figures.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write your answer to <strong>part (b)(ii) </strong>in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}"> <mi>a</mi> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mi>k</mi> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10,{\text{ }}k \in \mathbb{Z}"> <mn>1</mn> <mo>⩽</mo> <mi>a</mi> <mo><</mo> <mn>10</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{{\frac{{2.78 \times {{10}^{11}}}}{{3.12 \times {{10}^{ - 3}}}}}}"> <mroot> <mrow> <mfrac> <mrow> <mn>2.78</mn> <mo>×</mo> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>11</mn> </mrow> </msup> </mrow> </mrow> <mrow> <mn>3.12</mn> <mo>×</mo> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </mrow> </mfrac> </mrow> <mn>3</mn> </mroot> </math></span><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><strong>OR</strong><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{{8.91025 \ldots \times {{10}^{13}}}}"> <mroot> <mrow> <mn>8.91025</mn> <mo>…</mo> <mo>×</mo> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>13</mn> </mrow> </msup> </mrow> </mrow> <mn>3</mn> </mroot> </math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution into given expression.</p>
<p> </p>
<p>44664.59503 <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1) </em></strong>for a correct answer with at least 8 digits.</p>
<p>Accept 44664.5950301.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>44664.60 <strong><em>(A1)</em>(ft)</strong><em> </em><strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> For a follow through mark, the answer to part (a) must be to at least 3 decimal places.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>44700 <strong><em>(A1)</em>(ft)</strong><em> </em><strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Answer to part (a) must be to at least 4 significant figures.</p>
<p>Accept any equivalent notation which is correct to 3 significant figures.</p>
<p>For example <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="447 \times {10^2}"> <mn>447</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mn>2</mn> </msup> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="44.7 \times {10^3}"> <mn>44.7</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </math></span>.</p>
<p>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4.47 \times {10^4}"> <mn>4.47</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </math></span> <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>(A1)</em>(ft) </strong>for 4.47 and <strong><em>(A1)</em>(ft) </strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{10^4}"> <mrow> <msup> <mn>10</mn> <mn>4</mn> </msup> </mrow> </math></span>.</p>
<p>Award <strong><em>(A0)(A0) </em></strong>for answers such as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="44.7 \times {10^3}"> <mn>44.7</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </math></span>.</p>
<p>Follow through from part (b)(ii) <strong>only</strong>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>Give your answers in this question correct to the nearest whole number.</strong></p>
<p>Imon invested <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mn>000</mn></math> Singapore dollars (<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>SGD</mtext></math>) in a fixed deposit account with a nominal annual interest rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>6</mn><mo>%</mo></math>, compounded <strong>monthly</strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of Imon’s investment after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years.</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>At the end of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years, Imon withdrew <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mtext>SGD</mtext></math> from the fixed deposit account and reinvested this into a super-savings account with a nominal annual interest rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>7</mn><mo>%</mo></math>, compounded <strong>half-yearly</strong>.</p>
<p>The value of the super-savings account increased to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mn>000</mn><mo> </mo><mtext>SGD</mtext></math> after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math> months.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. It appeared in a paper that permitted the use of a calculator, and so might not be suitable for all forms of practice.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>F</mi><mi>V</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>25</mn><mo> </mo><mn>000</mn><mo>×</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>3</mn><mo>.</mo><mn>6</mn></mrow><mrow><mn>100</mn><mo>×</mo><mn>12</mn></mrow></mfrac></mrow></mfenced><mrow><mn>12</mn><mo>×</mo><mn>5</mn></mrow></msup></math> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for substituted compound interest formula, <em><strong>(A1)</strong></em> for correct substitutions.<br><br><strong>OR</strong><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>5</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>%</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>6</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mo>∓</mo><mn>25</mn><mo> </mo><mn>000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>12</mn></math> <em><strong>(A1)(M1)</strong></em><br><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>12</mn></math> seen, <em><strong>(M1)</strong></em> for <strong>all</strong> other correct entries.<br><br><strong>OR</strong><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>60</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>%</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>6</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>V</mi><mo>=</mo><mo>∓</mo><mn>25</mn><mo> </mo><mn>000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>12</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>12</mn></math> <em><strong>(A1)(M1)<br><br></strong></em><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>12</mn></math> seen, <em><strong>(M1)</strong></em> for <strong>all</strong> other correct entries.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>F</mi><mi>V</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>29</mn><mo> </mo><mn>922</mn><mo> </mo><mfenced><mtext>SGD</mtext></mfenced></math> <em><strong>(A1) (C3)</strong></em><br><br><strong>Note:</strong> Do not award the final <em><strong>(A1)</strong></em> if answer is not given correct to the nearest integer.<br><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mn>000</mn><mo>=</mo><mi>P</mi><mi>V</mi><mo>×</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><mo>.</mo><mn>7</mn></mrow><mrow><mn>100</mn><mo>×</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mrow><mn>2</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></msup></math> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for substituted compound interest equated to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mn>000</mn></math>. Award <em><strong>(A1)</strong></em> for correct substitutions.<br><br><strong>OR</strong><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>%</mo><mo>=</mo><mn>5</mn><mo>.</mo><mn>7</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mi>V</mi><mo>=</mo><mo>±</mo><mn>20</mn><mo> </mo><mn>000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>1</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math> <em><strong>(A1)(M1)</strong></em><br><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math> seen, <em><strong>(M1)</strong></em> for <strong>all</strong> other correct entries.<br><br><strong>OR</strong><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>3</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>%</mo><mo>=</mo><mn>5</mn><mo>.</mo><mn>7</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mi>V</mi><mo>=</mo><mo>±</mo><mn>20</mn><mo> </mo><mn>000</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math> <em><strong>(A1)(M1)<br><br></strong></em><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>/</mo><mi>Y</mi><mo>=</mo><mn>2</mn></math> seen, <em><strong>(M1)</strong></em> for <strong>all</strong> other correct entries.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>18</mn><mo> </mo><mn>383</mn><mo> </mo><mfenced><mtext>SGD</mtext></mfenced></math> <em><strong>(A1) (C3)</strong></em><br><br><strong>Note:</strong> Do not award the final <em><strong>(A1)</strong></em> if answer is not given correct to the nearest integer (unless already penalized in part(a)).<br><br><em><strong>[3 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="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = \sqrt x ">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mi>x</mi>
</msqrt>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0. The first time the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> intersect is at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The set of all non-zero values that satisfy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = g(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> can be described as an arithmetic sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = a + bn">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mi>n</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> ≥ 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <strong>two</strong> smallest non-zero values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = g(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>.</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>At point P, the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> intersect for the 21st time. Find the coordinates of P.</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>The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> <strong>reflected</strong> in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. It also shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the point P.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Find an expression for the area of the shaded region. Do not calculate the value of the expression.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>correct working <em><strong> (A1)</strong></em></p>
<p><em>eg </em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\left( {\frac{\pi }{4}x} \right) = 1"> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt x \left( {1 - {\text{sin}}\left( {\frac{\pi }{4}x} \right)} \right) = 0"> <msqrt> <mi>x</mi> </msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\left( {\frac{\pi }{2}} \right) = 1"> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p>correct working (ignore additional values) <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{4}x = \frac{\pi }{2}"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\pi }{4}x = \frac{\pi }{2} + 2\pi "> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>+</mo> <mn>2</mn> <mi>π</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 2, 10 <em><strong>A1A1 N1N1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em>first intersection at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 20"> <mi>n</mi> <mo>=</mo> <mn>20</mn> </math></span></p>
<p>correct working <em><strong>A1</strong></em></p>
<p><em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 + 8 \times 20"> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mn>8</mn> <mo>×</mo> <mn>20</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + \left( {20 - 1} \right) \times 8"> <mn>2</mn> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>20</mn> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mn>8</mn> </math></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{20}} = 154"> <mrow> <msub> <mi>u</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>154</mn> </math></span></span></p>
<p>P(154, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {154} "> <msqrt> <mn>154</mn> </msqrt> </math></span>) (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 154"> <mi>x</mi> <mo>=</mo> <mn>154</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \sqrt {154} "> <mi>y</mi> <mo>=</mo> <msqrt> <mn>154</mn> </msqrt> </math></span>) <em><strong>A1A1 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find upper boundary <em><strong> (M1)</strong></em></p>
<p><em>eg</em> half way between <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{20}}"> <mrow> <msub> <mi>u</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> </math></span></span> and <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{21}}"> <mrow> <msub> <mi>u</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </math></span></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_{20}} + \frac{d}{2}"> <mrow> <msub> <mi>u</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> <mo>+</mo> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> </math></span></span>, 154 + 4, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 + 8n"> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>8</mn> <mi>n</mi> </math></span>, at least two values of new sequence {6, 14, ...}</p>
<p>upper boundary at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 158"> <mi>x</mi> <mo>=</mo> <mn>158</mn> </math></span> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p>correct integral expression (accept missing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{d}}x}"> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span>) <em><strong>A1A1 N4</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{158} {\left( {\sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right) + \sqrt x } \right)} {\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>158</mn> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mi>x</mi> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{158} {\left( {g + f} \right)} \left. {{\text{d}}x} \right)"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>158</mn> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mo>+</mo> <mi>f</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{158} {\sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right)} {\text{d}}x - \int_0^{158} { - \sqrt x \,{\text{d}}x} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>158</mn> </mrow> </msubsup> <mrow> <msqrt> <mi>x</mi> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>158</mn> </mrow> </msubsup> <mrow> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for two correct limits and <em><strong>A1</strong></em> for correct integrand. The <em><strong>A1</strong> </em>for correct integrand may be awarded independently of all the other marks.</p>
<p><em><strong>[4 marks]</strong></em></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">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>In an arithmetic sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2} = 5">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>5</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_3} = 11">
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>11</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20"> <mn>20</mn> </math></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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11 - 5"> <mn>11</mn> <mo>−</mo> <mn>5</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="11 = 5 + d"> <mn>11</mn> <mo>=</mo> <mn>5</mn> <mo>+</mo> <mi>d</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 6"> <mi>d</mi> <mo>=</mo> <mn>6</mn> </math></span> <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>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2} - d"> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mo>−</mo> <mi>d</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5 - 6"> <mn>5</mn> <mo>−</mo> <mn>6</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} + \left( {3 - 1} \right)\left( 6 \right) = 11"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>11</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = - 1"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </math></span> <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>correct substitution into sum formula</p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{20}}{2}\left( {2\left( { - 1} \right) + 19\left( 6 \right)} \right)"><mfrac><mn>20</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>19</mn><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow></mfenced></math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{20}}{2}\left( { - 1 + 113} \right)"> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>113</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{20}} = 1120"> <mrow> <msub> <mi>S</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> <mo>=</mo> <mn>1120</mn> </math></span> <em><strong>A1 N2</strong></em></p>
<p><em><strong>[2 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>In an arithmetic sequence, <em>u</em><sub>1</sub> = −5 and <em>d</em> = 3.</p>
</div>
<div class="question">
<p>Find <em>u</em><sub>8</sub>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>correct working <em><strong>(A1)</strong></em></p>
<p><em>eg </em>−5 + (8 − 1)(3)</p>
<p><em>u</em><sub>8</sub> = 16 <em><strong>A1 N2</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why any integer can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k">
<mn>4</mn>
<mi>k</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 1">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 2">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 3">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></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>Hence prove that the square of any integer can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4t">
<mn>4</mn>
<mi>t</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4t + 1">
<mn>4</mn>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \in {\mathbb{Z}^ + }">
<mi>t</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></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>Upon division by 4 <em><strong>M1</strong></em></p>
<p>any integer leaves a remainder of 0, 1, 2 or 3. <em><strong>R1</strong></em></p>
<p>Hence, any integer can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k">
<mn>4</mn>
<mi>k</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 1">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 2">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 3">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span> <em><strong>AG</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {4k} \right)^2} = 16{k^2} = 4t">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>16</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mi>t</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {4k + 1} \right)^2} = 16{k^2} + 8k + 1 = 4t + 1">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>16</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mn>4</mn>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {4k + 2} \right)^2} = 16{k^2} + 16k + 4 = 4t">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>16</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>16</mn>
<mi>k</mi>
<mo>+</mo>
<mn>4</mn>
<mo>=</mo>
<mn>4</mn>
<mi>t</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {4k + 3} \right)^2} = 16{k^2} + 24k + 9 = 4t + 1">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>16</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>24</mn>
<mi>k</mi>
<mo>+</mo>
<mn>9</mn>
<mo>=</mo>
<mn>4</mn>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>Hence, the square of any integer can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4t">
<mn>4</mn>
<mi>t</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4t + 1">
<mn>4</mn>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \in {\mathbb{Z}^ + }">
<mi>t</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>. <em><strong>AG</strong></em></p>
<p><em><strong>[6 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>The following diagram shows [CD], with length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b{\text{ cm}}"> <mi>b</mi> <mrow> <mtext> cm</mtext> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b > 1"> <mi>b</mi> <mo>></mo> <mn>1</mn> </math></span>. Squares with side lengths <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k{\text{ cm}},{\text{ }}{k^2}{\text{ cm}},{\text{ }}{k^3}{\text{ cm}},{\text{ }} \ldots "> <mi>k</mi> <mrow> <mtext> cm</mtext> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mrow> <mtext> cm</mtext> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>k</mi> <mn>3</mn> </msup> </mrow> <mrow> <mtext> cm</mtext> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>…</mo> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < k < 1"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>1</mn> </math></span>, are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.</p>
<p><img src="images/Schermafbeelding_2018-02-12_om_09.48.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/10.b"></p>
<p>The <strong>total</strong> sum of the areas of all the squares is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{9}{{16}}"> <mfrac> <mn>9</mn> <mrow> <mn>16</mn> </mrow> </mfrac> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>recognizing infinite geometric series with squares <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} + {k^4} + {k^6} + \ldots ,{\text{ }}\frac{{{k^2}}}{{1 - {k^2}}}"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>6</mn> </msup> </mrow> <mo>+</mo> <mo>…</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span></p>
<p>correct substitution into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = \frac{9}{{16}}"> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">∞</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mn>9</mn> <mrow> <mn>16</mn> </mrow> </mfrac> </math></span> (must substitute into formula) <strong><em>(A2)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{k^2}}}{{1 - {k^2}}} = \frac{9}{{16}}"> <mfrac> <mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>9</mn> <mrow> <mn>16</mn> </mrow> </mfrac> </math></span></p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="16{k^2} = 9 - 9{k^2},{\text{ }}25{k^2} = 9,{\text{ }}{k^2} = \frac{9}{{25}}"> <mn>16</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>9</mn> <mo>−</mo> <mn>9</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>25</mn> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>9</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mn>9</mn> <mrow> <mn>25</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = \frac{3}{5}"> <mi>k</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </math></span> (seen anywhere) <strong><em>A1</em></strong></p>
<p>valid approach with segments and CD (may be seen earlier) <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = k,{\text{ }}{S_\infty } = b"> <mi>r</mi> <mo>=</mo> <mi>k</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">∞</mi> </msub> </mrow> <mo>=</mo> <mi>b</mi> </math></span></p>
<p>correct expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> (may be seen earlier) <strong><em>(A1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{k}{{1 - k}},{\text{ }}b = \sum\limits_{n = 1}^\infty {{k^n},{\text{ }}b = k + {k^2} + {k^3} + \ldots } "> <mi>b</mi> <mo>=</mo> <mfrac> <mi>k</mi> <mrow> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>b</mi> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi mathvariant="normal">∞</mi> </munderover> <mrow> <mrow> <msup> <mi>k</mi> <mi>n</mi> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>k</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mo>…</mo> </mrow> </math></span></p>
<p>substituting <strong>their</strong> value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> into <strong>their</strong> formula for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\frac{3}{5}}}{{1 - \frac{3}{5}}},{\text{ }}\frac{{\left( {\frac{3}{5}} \right)}}{{\left( {\frac{2}{5}} \right)}}"> <mfrac> <mrow> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{3}{2}"> <mi>b</mi> <mo>=</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></span> <strong><em>A1 N3</em></strong></p>
<p><strong><em>[9 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>A comet orbits the Sun and is seen from Earth every 37 years. The comet was first seen from Earth in the year 1064.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the year in which the comet was seen from Earth for the fifth time.</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>Determine how many times the comet has been seen from Earth up to the year 2014.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1064 + (5 - 1) \times 37">
<mn>1064</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>5</mn>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mn>37</mn>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substituted arithmetic sequence formula, <strong><em>(A1) </em></strong>for correct substitution.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1212">
<mo>=</mo>
<mn>1212</mn>
</math></span> <strong><em>(A1) (C3)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2014 > 1064 + (n - 1) \times 37">
<mn>2014</mn>
<mo>></mo>
<mn>1064</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mn>37</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for a correct substitution into arithmetic sequence formula.</p>
<p>Accept an equation.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(n < ){\text{ }}26.6756 \ldots ">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo><</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>26.6756</mn>
<mo>…</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p>26 (times) <strong><em>(A1) (C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the final <strong><em>(A1) </em></strong>for the correct rounding <strong>down </strong>of their unrounded answer.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2014 > 1064 + 37t">
<mn>2014</mn>
<mo>></mo>
<mn>1064</mn>
<mo>+</mo>
<mn>37</mn>
<mi>t</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for a correct substitution into a linear model (where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = n - 1">
<mi>t</mi>
<mo>=</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</math></span>).</p>
<p>Accept an equation or weak inequality.</p>
<p>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2014 - 1064}}{{37}}">
<mfrac>
<mrow>
<mn>2014</mn>
<mo>−</mo>
<mn>1064</mn>
</mrow>
<mrow>
<mn>37</mn>
</mrow>
</mfrac>
</math></span> for <strong><em>(M1)</em></strong>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(t < ){\text{ }}25.6756 \ldots ">
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo><</mo>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>25.6756</mn>
<mo>…</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p>26 (times) <strong><em>(A1) (C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award the final <strong><em>(A1) </em></strong>for adding 1 to the correct rounding down of their unrounded answer.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>In the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>+</mo><mi>k</mi><msup><mo>)</mo><mn>7</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, the coefficient of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>EITHER</strong></p>
<p>attempt to use the binomial expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mn>7</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>0</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>x</mi><mn>7</mn></msup><msup><mi>k</mi><mn>0</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>x</mi><mn>6</mn></msup><msup><mi>k</mi><mn>1</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>x</mi><mn>5</mn></msup><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>0</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>k</mi><mn>7</mn></msup><msup><mi>x</mi><mn>0</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>1</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>k</mi><mn>5</mn></msup><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>k</mi><mn>5</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></math>)</p>
<p>identifying the correct term <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>x</mi><mn>5</mn></msup><msup><mi>k</mi><mn>2</mn></msup></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>k</mi><mn>2</mn></msup><msup><mi>x</mi><mn>5</mn></msup></math>) <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to use the general term <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>x</mi><mi>r</mi></msup><msup><mi>k</mi><mrow><mn>7</mn><mo>-</mo><mi>r</mi></mrow></msup></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mi>r</mi><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><msup><mi>k</mi><mi>r</mi></msup><msup><mi>x</mi><mrow><mn>7</mn><mo>-</mo><mi>r</mi></mrow></msup></math>) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>5</mn></math>) <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mo>=</mo><mn>21</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mo>=</mo><mn>21</mn></math> (seen anywhere) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>21</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><mn>63</mn><msup><mi>x</mi><mn>5</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>21</mn><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><mn>63</mn><mo> </mo><mo>,</mo><mo> </mo><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><mn>3</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>±</mo><msqrt><mn>3</mn></msqrt></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> If working shown, award <em><strong>M1A1A1A1A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><msqrt><mn>3</mn></msqrt></math>.</p>
<p> </p>
<p><em><strong>[5 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>The first three terms of a geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln {x^{16}}">
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mn>16</mn>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln {x^8}">
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>8</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln {x^4}">
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question">
<p>Find the common ratio.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>correct use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\log {x^n} = n\log x"> <mi>log</mi> <mo></mo> <mrow> <msup> <mi>x</mi> <mi>n</mi> </msup> </mrow> <mo>=</mo> <mi>n</mi> <mi>log</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>A1</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="16\ln x"> <mn>16</mn> <mi>ln</mi> <mo></mo> <mi>x</mi> </math></span></p>
<p>valid approach to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r"> <mi>r</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><em>eg</em><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,"> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </math><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{u_{n + 1}}}}{{{u_n}}},{\text{ }}\frac{{\ln {x^8}}}{{\ln {x^{16}}}},{\text{ }}\frac{{4\ln x}}{{8\ln x}},{\text{ }}\ln {x^4} = \ln {x^{16}} \times {r^2}"> <mfrac> <mrow> <mrow> <msub> <mi>u</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mi>u</mi> <mi>n</mi> </msub> </mrow> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mi>ln</mi> <mo></mo> <mrow> <msup> <mi>x</mi> <mn>8</mn> </msup> </mrow> </mrow> <mrow> <mi>ln</mi> <mo></mo> <mrow> <msup> <mi>x</mi> <mrow> <mn>16</mn> </mrow> </msup> </mrow> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mfrac> <mrow> <mn>4</mn> <mi>ln</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mn>8</mn> <mi>ln</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>ln</mi> <mo></mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>=</mo> <mi>ln</mi> <mo></mo> <mrow> <msup> <mi>x</mi> <mrow> <mn>16</mn> </mrow> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{1}{2}"> <mi>r</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></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>Hence or otherwise solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></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>attempting to use the change of base rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = \frac{{{\text{lo}}{{\text{g}}_3}\left( {{\text{cos}}\,2x + 2} \right)}}{{{\text{lo}}{{\text{g}}_3}9}}">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mn>9</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}{\text{lo}}{{\text{g}}_3}\left( {{\text{cos}}\,2x + 2} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>AG</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{2}}\,{\text{sin}}\,x = \sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{4}}\,{\text{si}}{{\text{n}}^2}\,x = {\text{cos}}\,2x + 2">
<mrow>
<mtext>4</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,2x = 1 - 2\,{\text{si}}{{\text{n}}^2}\,x">
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6\,{\text{si}}{{\text{n}}^2}\,x = 3">
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\,x = \left( \pm \right)\frac{1}{{\sqrt 2 }}">
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mo>±</mo>
<mo>)</mo>
</mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: Award <em><strong>A0</strong></em> if solutions other than <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> are included.</p>
<p><em><strong>[5 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 curve with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>The tangent to the curve at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is parallel to the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup><mi>x</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>evidence of using product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfenced><mrow><mi>k</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>+</mo><mn>2</mn><mo>×</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup><mfenced><mrow><mn>2</mn><mi>k</mi><mi>x</mi><mo>-</mo><mi>k</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>correct working for one of (seen anywhere) <em><strong>A1</strong></em></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>⇒</mo><mi>k</mi><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><mn>2</mn><msup><mtext>e</mtext><mi>k</mi></msup></math></p>
<p style="padding-left:30px;"><br><strong>OR</strong></p>
<p style="padding-left:30px;">slope of tangent is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup></math></p>
<p><br>their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> equals the <em>slope</em> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup></mrow></mfenced></math> (seen anywhere) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msup><mtext>e</mtext><mi>k</mi></msup><mo>+</mo><mn>2</mn><msup><mtext>e</mtext><mi>k</mi></msup><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>The product rule was well recognised and used with 𝑥=1 properly substituted into this expression. Although the majority of the candidates tried equating the derivative to the slope of the tangent line, the slope of the tangent line was not correctly identified; many candidates incorrectly substituted 𝑥=1 into the tangent equation, thus finding the <em>y</em>-coordinate instead of the slope.</p>
</div>
<br><hr><br><div class="specification">
<p>The diameter of a spherical planet is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo> </mo><mtext>km</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the radius of the planet.</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>The volume of the planet can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>k</mi></msup></mrow></mfenced><mo> </mo><msup><mtext>km</mtext><mn>3</mn></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30000</mn><mo> </mo><mfenced><mtext>km</mtext></mfenced></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>∙</mo><msup><mn>10</mn><mn>4</mn></msup></math>) <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><msup><mfenced><mrow><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></mrow></mfenced><mn>3</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><msup><mfenced><mn>30000</mn></mfenced><mn>3</mn></msup></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><mo>×</mo><mn>27</mn><mo>×</mo><msup><mn>10</mn><mn>12</mn></msup><mo> </mo><mfenced><mrow><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>36</mn><mo>×</mo><msup><mn>10</mn><mn>12</mn></msup></mrow></mfenced></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi mathvariant="normal">π</mi><mo>×</mo><mn>27000000000000</mn></math> <em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mn>36</mn><mo>×</mo><msup><mn>10</mn><mn>13</mn></msup></mrow></mfenced><mo> </mo><mfenced><msup><mtext>km</mtext><mn>3</mn></msup></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>6</mn><mo>,</mo><mo> </mo><mi>k</mi><mo>=</mo><mn>13</mn></math> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[3 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 two consecutive positive integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math>.</p>
<p>Show that the difference of their squares is equal to the sum of the two integers.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to subtract squares of integers <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mi>n</mi><mn>2</mn></msup></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>correct order of subtraction and correct expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math>, seen anywhere <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>−</mo><msup><mi>n</mi><mn>2</mn></msup><mo> </mo><mo>(</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>correct order of subtraction and correct factorization of difference of squares <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>n</mi><mo>)</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>)</mo><mo>(</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>n</mi><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>=</mo><mtext>RHS</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award final <em><strong>A1</strong></em> unless all previous working is correct.</p>
<p> </p>
<p>which is the sum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> If expansion and order of subtraction are correct, award full marks for candidates who find the sum of the integers as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> and then show that the difference of the squares (subtracted in the correct order) is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math>.</p>
<p> </p>
<p><em><strong>[4 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>Tomás is playing with sticks and he forms the first three diagrams of a pattern. These diagrams are shown below.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_17.25.44.png" alt="M17/5/MATSD/SP1/ENG/TZ2/05"></p>
<p>Tomás continues forming diagrams following this pattern.</p>
</div>
<div class="specification">
<p>Tomás forms a total of 24 diagrams.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Diagram <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> is formed with 52 sticks. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></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>Find the total number of sticks used by Tomás for all 24 diagrams.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 + 3(n - 1) = 52">
<mn>4</mn>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>52</mn>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substitution into the formula of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>th term of an arithmetic sequence, <strong><em>(A1) </em></strong>for correct substitution.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 17">
<mi>n</mi>
<mo>=</mo>
<mn>17</mn>
</math></span> <strong><em>(A1)</em></strong> <strong><em>(C3)</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{24}}{2}(2 \times 4 + 23 \times 3)">
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>×</mo>
<mn>4</mn>
<mo>+</mo>
<mn>23</mn>
<mo>×</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{24}}{2}(4 + 73)">
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>+</mo>
<mn>73</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Notes:</strong> Award <strong><em>(M1) </em></strong>for substitution into the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms of an arithmetic sequence formula, <strong><em>(A1)</em>(ft) </strong>for their correct substitution, consistent with part (a).</p>
<p> </p>
<p>924 <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>Jashanti is saving money to buy a car. The price of the car, in US Dollars (USD), can be modelled by the equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="P = 8500{\text{ }}{(0.95)^t}.">
<mi>P</mi>
<mo>=</mo>
<mn>8500</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>0.95</mn>
<msup>
<mo stretchy="false">)</mo>
<mi>t</mi>
</msup>
</mrow>
<mo>.</mo>
</math></span></p>
<p>Jashanti’s savings, in USD, can be modelled by the equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="S = 400t + 2000.">
<mi>S</mi>
<mo>=</mo>
<mn>400</mn>
<mi>t</mi>
<mo>+</mo>
<mn>2000.</mn>
</math></span></p>
<p>In both equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is the time in months since Jashanti started saving for the car.</p>
</div>
<div class="specification">
<p>Jashanti does not want to wait too long and wants to buy the car two months after she started saving. She decides to ask her parents for the extra money that she needs.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the amount of money Jashanti saves per month.</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>Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car.</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>Calculate how much extra money Jashanti needs.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>400 (USD) <strong><em>(A1)</em></strong> <strong><em>(C1)</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8500{\text{ }}{(0.95)^t} = 400 \times t + 2000"> <mn>8500</mn> <mrow> <mtext> </mtext> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>0.95</mn> <msup> <mo stretchy="false">)</mo> <mi>t</mi> </msup> </mrow> <mo>=</mo> <mn>400</mn> <mo>×</mo> <mi>t</mi> <mo>+</mo> <mn>2000</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for equating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8500{(0.95)^t}"> <mn>8500</mn> <mrow> <mo stretchy="false">(</mo> <mn>0.95</mn> <msup> <mo stretchy="false">)</mo> <mi>t</mi> </msup> </mrow> </math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="400 \times t + 2000"> <mn>400</mn> <mo>×</mo> <mi>t</mi> <mo>+</mo> <mn>2000</mn> </math></span> or for comparing the difference between the two expressions to zero or for showing a sketch of both functions.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(t = ){\text{ }}8.64{\text{ (months) }}\left( {8.6414 \ldots {\text{ (months)}}} \right)"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>=</mo> <mo stretchy="false">)</mo> <mrow> <mtext> </mtext> </mrow> <mn>8.64</mn> <mrow> <mtext> (months) </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>8.6414</mn> <mo>…</mo> <mrow> <mtext> (months)</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept 9 months.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8500{(0.95)^2} - (400 \times 2 + 2000)"> <mn>8500</mn> <mrow> <mo stretchy="false">(</mo> <mn>0.95</mn> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mo stretchy="false">(</mo> <mn>400</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mn>2000</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correct substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 2"> <mi>t</mi> <mo>=</mo> <mn>2</mn> </math></span> into equation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span>, <strong><em>(M1) </em></strong>for finding the difference between a value/expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> and a value/expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S"> <mi>S</mi> </math></span>. The first <strong><em>(M1) </em></strong>is implied if 7671.25 seen.</p>
<p> </p>
<p>4870 (USD) (4871.25) <strong><em>(A1)</em></strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept 4871.3.</p>
<p> </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>