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<h2>HL Paper 2</h2><div class="specification">
<p>Consider the identity <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>+</mo><mn>7</mn><mi>x</mi></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
</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>A</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</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, expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo>+</mo><mn>7</mn><mi>x</mi></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give a reason why the series expansion found in part (b) is not valid for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[1]</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 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 style="color:#999;font-size:90%;font-style:italic;"> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>7</mn><mi>x</mi><mo>≡</mo><mi>A</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mi>B</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> and attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> and substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> and attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>=</mo><mn>3</mn><mi>B</mi><mo>⇒</mo><mi>B</mi><mo>=</mo><mn>3</mn><mo> </mo><mo>;</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi>A</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>⇒</mo><mi>A</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>forms <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>A</mi><mo>+</mo><mn>2</mn><mi>B</mi><mo>=</mo><mn>7</mn></math> and attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>3</mn></math> <strong>A1</strong><strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>uses the binomial expansion on either <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p>so the expansion is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>5</mn><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></math> (in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>) <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></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"><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> (is convergent) requires <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>x</mi></mfenced><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> is outside this so the expansion is not valid <strong>R1</strong></p>
<p> </p>
<p><strong>[1 mark]</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">
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>d</mtext><mi>n</mi></msup><mrow><mtext>d</mtext><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>=</mo><msup><mi>p</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>n</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo> </mo><mo>∈</mo><mo> </mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>p</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℚ</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>:</mo><mo> </mo><mtext>LHS</mtext><mo>=</mo><mfrac><mrow><mtext>d</mtext><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mi>p</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mi mathvariant="normal">=</mi><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mo mathvariant="italic">,</mo><mo mathvariant="italic"> </mo><mtext>RHS</mtext><mo mathvariant="italic">=</mo><msup><mi>p</mi><mn>0</mn></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo mathvariant="italic">+</mo><mn mathvariant="italic">1</mn></mrow></mfenced><msup><mtext mathvariant="italic">e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>LHS</mtext><mo>=</mo><mtext>RHS</mtext></math> so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>:</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math> is proved.</p>
<p><br>assume proposition true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>d</mtext><mi>k</mi></msup><mrow><mtext>d</mtext><msup><mi>x</mi><mi>k</mi></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>=</mo><msup><mi>p</mi><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><br><strong>Notes:</strong> Do not award <em><strong>M1</strong></em> if using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.<br>Assumption of truth must be present.<br>Subsequent marks are not dependent on this <em><strong>M1</strong></em> mark.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>d</mtext><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mrow><mtext>d</mtext><msup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced><mo>=</mo><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mfenced><mrow><mfrac><msup><mtext>d</mtext><mi>k</mi></msup><mrow><mtext>d</mtext><msup><mi>x</mi><mi>k</mi></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mfenced><mrow><msup><mi>p</mi><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>p</mi><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mi>p</mi><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mfenced><msup><mi>p</mi><mi>k</mi></msup></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>p</mi><mi>k</mi></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>k</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup><mfenced><msup><mi>p</mi><mi>k</mi></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct derivative.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>p</mi><mi>k</mi></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>p</mi><mfenced><mrow><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></msup><mfenced><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>p</mi><mi>x</mi></mrow></msup></math></p>
<p><br><strong>Note:</strong> The final <em><strong>A1</strong></em> can be awarded for either of the two lines above.</p>
<p><br>hence true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> true <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> true <em><strong>R1</strong></em></p>
<p>therefore true for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo> </mo><mo>∈</mo><mo> </mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math></p>
<p><br><strong>Note:</strong> Only award the final <em><strong>R1</strong></em> if the three method marks have been awarded.</p>
<p><em><strong><br>[7 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>At a gathering of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> teachers, seven are male and five are female. A group of five of these teachers go out for a meal together. Determine the possible number of groups in each of the following situations:</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>There are more males than females in the group.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Two of the teachers, Gary and Gerwyn, refuse to go out for a meal together.</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>identifying two or three possible cases <em><strong>(M1)</strong></em></p>
<p>total number of possible groups is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for any two correct cases, <em><strong>A1</strong></em> for the other one.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>21</mn><mo>+</mo><mfenced><mrow><mn>35</mn><mo>×</mo><mn>5</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>35</mn><mo>×</mo><mn>10</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>546</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>identifying at least two of the three possible cases- Gary goes, Gerwyn goes or neither goes <em><strong>(M1)</strong></em></p>
<p>total number of possible groups is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>252</mn><mo>+</mo><mn>210</mn><mo>+</mo><mn>210</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>672</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>identifying the overall number of groups and no. of cases where both Gary and Gerwyn go. <em><strong>(M1)</strong></em></p>
<p>total number of possible groups is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>792</mn><mo>-</mo><mn>120</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>672</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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 the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>2</mn><mfenced><mrow><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mn>5</mn></mfrac><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>5</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mn>8</mn><mfenced><mrow><mi>cos</mi><mfrac><mrow><mn>2</mn><mi>k</mi><mi mathvariant="normal">π</mi></mrow><mn>5</mn></mfrac><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi>k</mi><mi mathvariant="normal">π</mi></mrow><mn>5</mn></mfrac></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi><mo> </mo><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</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>Find the argument of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</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 minimum 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">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> found in part (i), find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced open="|" close="|"><mrow><mi>z</mi><mi>w</mi></mrow></mfenced><mo>=</mo></mrow></mfenced><mn>16</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>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arg</mtext><mfenced><mi>z</mi></mfenced><mo>+</mo><mtext>arg</mtext><mfenced><mi>w</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arg</mtext><mfenced><mrow><mi>z</mi><mi>w</mi></mrow></mfenced><mo>=</mo><mtext>arg</mtext><mfenced><mi>z</mi></mfenced><mo>+</mo><mtext>arg</mtext><mfenced><mi>w</mi></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>5</mn></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mi>k</mi><mi mathvariant="normal">π</mi></mrow><mn>5</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi>k</mi></mrow></mfenced><mi mathvariant="normal">π</mi></mrow><mn>5</mn></mfrac></mrow></mfenced></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi><mo> </mo><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>⇒</mo><mtext>arg</mtext><mfenced><mrow><mi>z</mi><mi>w</mi></mrow></mfenced></math> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi>k</mi></math> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> <em><strong>(M1)</strong></em></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>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mi>w</mi><mo> </mo><mo>=</mo><mn>16</mn><mfenced><mrow><mi>cos</mi><mfenced><mrow><mo>-</mo><mi mathvariant="normal">π</mi></mrow></mfenced><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mfenced><mrow><mo>-</mo><mi mathvariant="normal">π</mi></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>16</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} > 2x + 1">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</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>Use mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{n + 1}} > {n^2}">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> 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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 3">
<mi>n</mi>
<mo>⩾</mo>
<mn>3</mn>
</math></span>.</p>
<div class="marks">[7]</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="x < - 0.414,\,\,x > 2.41">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>0.414</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>></mo>
<mn>2.41</mn>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x < 1 - \sqrt 2 ,\,\,x > 1 + \sqrt 2 } \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
<mo>−</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>></mo>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for −0.414, 2.41 and <em><strong>A1</strong></em> for correct inequalities.</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>check for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 3">
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</math></span>,</p>
<p>16 > 9 so true when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 3">
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>assume true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{k + 1}} > {k^2}">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M0</strong></em> for statements such as “let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span>”.</p>
<p><strong>Note:</strong> Subsequent marks after this <em><strong>M1</strong></em> are independent of this mark and can be awarded.</p>
<p>prove true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k + 1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^{k + 2}} = 2 \times {2^{k + 1}}">
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" > 2{k^2}">
<mo>></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {k^2} + {k^2}">
<mo>=</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" > {k^2} + 2k + 1">
<mo>></mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> (from part (a)) <em><strong>A1</strong></em></p>
<p> which is true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> ≥ 3 <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Only award the <em><strong>A1</strong></em> or the <em><strong>R1</strong></em> if it is clear why. Alternate methods are possible.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {K + 1} \right)^2}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>K</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>hence if true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span> true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k +1">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, true for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 3">
<mi>n</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> so true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> ≥ 3 <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Only award the final <em><strong>R1</strong></em> provided at least three of the previous marks are awarded.</p>
<p><em><strong>[7 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 the set of six-digit positive integers that can be formed from the digits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>,</mo><mo> </mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math>.</p>
<p>Find the total number of six-digit positive integers that can be formed such that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the digits are distinct.</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 digits are distinct and are in increasing order.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>9</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>5</mn><mprescripts></mprescripts><mn>9</mn></mmultiscripts></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>136080</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>9</mn><mo>×</mo><mfrac><mrow><mn>9</mn><mo>!</mo></mrow><mrow><mn>4</mn><mo>!</mo></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1A0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>×</mo><mn>9</mn><mo>×</mo><mn>8</mn><mo>×</mo><mn>7</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>5</mn><mo> </mo><mfenced><mrow><mo>=</mo><mmultiscripts><mi>P</mi><mn>6</mn><mprescripts></mprescripts><mn>10</mn></mmultiscripts><mo>=</mo><mn>151200</mn><mo>=</mo><mfrac><mrow><mn>10</mn><mo>!</mo></mrow><mrow><mn>4</mn><mo>!</mo></mrow></mfrac></mrow></mfenced></math>.</p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>P</mi><mn>6</mn><mprescripts></mprescripts><mn>9</mn></mmultiscripts><mo>=</mo><mn>60480</mn></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><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p>every unordered subset of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> digits from the set of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> non-zero digits can be arranged in exactly one way into a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>-digit number with the digits in increasing order. <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>6</mn><none></none><mprescripts></mprescripts><none></none><mn>9</mn></mmultiscripts><mfenced><mrow><mo>×</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>84</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>removes <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> digits from the set of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> non-zero digits and these <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> remaining digits can be arranged in exactly one way into a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>-digit number with the digits in increasing order. <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>9</mn></mmultiscripts><mfenced><mrow><mo>×</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>84</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>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) A number of candidates got the correct answer here with the valid approach and recognising that zero could not occupy the first position. Some lost a mark by including zero. Many candidates used an incorrect method with combinations or simplified permutations.</p>
<p>Part (b) Only very few candidates got the correct answer. Many left it blank or provided unreasonably enormous numbers as their answers.</p>
<p>Some candidates had the answer to part (b) showing in part (a).</p>
<p>A small number of candidates tried to list all possibilities but mostly unsuccessfully.</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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</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><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses the</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</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>Write down the equation of the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></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>The oblique asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</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>b</mi></math>.</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>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>, clearly indicating the points of intersection with each axis and any asymptotes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac></math> in partial fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mo>d</mo><mi>x</mi></math>, expressing your answer as a single logarithm.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In part (a), penalise once only, if correct values are given instead of correct coordinates.</p>
<p><br>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </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><strong>Note:</strong> In part (a), penalise once only, if correct values are given instead of correct coordinates.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.<br> Award <em><strong>A1</strong></em> in part (b), if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math> is seen on their graph in part (d).<br><br></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><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mo>≡</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></math></p>
<p>attempts to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>15</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>2</mn><mi>b</mi><mi>x</mi><mo>-</mo><mn>15</mn><mi>b</mi><mo>≡</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>equates coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>15</mn><mn>2</mn></mfrac><mo>+</mo><mn>2</mn><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts division on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>≡</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn><mo>≡</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mi>x</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mi>b</mi><mo>+</mo><mi>c</mi></math></p>
<p>equates coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> : <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>15</mn><mn>2</mn></mfrac><mo>+</mo><mn>2</mn><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>attempts division on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>13</mn><mi>x</mi></mrow><mn>2</mn></mfrac></mstyle><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>13</mn><mi>x</mi></mrow><mn>2</mn></mfrac></mstyle><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </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> <img 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"></p>
<p>two branches with approximately correct shape (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>) <em><strong>A1</strong></em></p>
<p>their vertical and oblique asymptotes in approximately correct positions with both branches showing correct asymptotic behaviour to these asymptotes <em><strong>A1</strong></em></p>
<p>their axes intercepts in approximately the correct positions <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Points of intersection with the axes and the equations of asymptotes are not required to be labelled.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to split into partial fractions: <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn><mo>≡</mo><mi>A</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>B</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfenced><mrow><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p>attempts to integrate and obtains two terms involving ‘ln’ <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mn>3</mn><mo> </mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></mrow></mfenced><mn>0</mn><mn>3</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>6</mn><mo>-</mo><mi>ln</mi><mo> </mo><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>3</mn><mo>+</mo><mi>ln</mi><mo> </mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>+</mo><mi>ln</mi><mo> </mo><mn>4</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><mn>8</mn><mo>+</mo><mi>ln</mi><mo> </mo><mn>4</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>ln</mi><mo> </mo><mn>32</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The final <em><strong>A1</strong></em> is dependent on the previous two <em><strong>A</strong></em> marks.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _{10}}\left( {\frac{1}{{2\sqrt 2 }}\left( {p + 2q} \right)} \right) = \frac{1}{2}\left( {{{\log }_{10}}p + {{\log }_{10}}q} \right),{\text{ }}p > 0,{\text{ }}q > 0">
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mrow>
<mi>log</mi>
</mrow>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>p</mi>
<mo>+</mo>
<mrow>
<msub>
<mrow>
<mi>log</mi>
</mrow>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>q</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>p</mi>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>q</mi>
<mo>></mo>
<mn>0</mn>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</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>.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _{10}}\frac{1}{{2\sqrt 2 }}(p + 2q) = \frac{1}{2}({\log _{10}}p + {\log _{10}}q)">
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>p</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>q</mi>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _{10}}\frac{1}{{2\sqrt 2 }}(p + 2q) = \frac{1}{2}{\log _{10}}pq">
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mi>p</mi>
<mi>q</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _{10}}\frac{1}{{2\sqrt 2 }}(p + 2q) = {\log _{10}}{(pq)^{\frac{1}{2}}}">
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msub>
<mi>log</mi>
<mrow>
<mn>10</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{2\sqrt 2 }}(p + 2q) = {(pq)^{\frac{1}{2}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mi>q</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(p + 2q)^2} = 8pq">
<mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mi>p</mi>
<mi>q</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p^2} + 4pq + 4{q^2} = 8pq">
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>p</mi>
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>q</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mi>p</mi>
<mi>q</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p^2} - 4pq + 4{q^2} = 0">
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>p</mi>
<mi>q</mi>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>q</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(p - 2q)^2} = 0">
<mrow>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo>−</mo>
<mn>2</mn>
<mi>q</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2q">
<mi>p</mi>
<mo>=</mo>
<mn>2</mn>
<mi>q</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = a + b{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{b}} \in {\mathbb{R}^ + }">
<mrow>
<mtext>b</mtext>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\,z = \theta ">
<mrow>
<mtext>arg</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>=</mo>
<mi>θ<!-- θ --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show the points represented by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z - 2a">
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> on the following Argand diagram.</p>
<p><img src="data:image/png;base64,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"></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 an expression in terms of <em>θ</em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\left( {z - 2a} \right)">
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</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 an expression in terms of <em>θ</em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\left( {\frac{z}{{z - 2a}}} \right)">
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find the value of <em>θ</em> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0">
<mrow>
<mtext>Re</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</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><img src="data:image/png;base64,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"> A1</p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in first quadrant and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z - 2a">
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> its reflection in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi - \theta ">
<mi>π</mi>
<mo>−</mo>
<mi>θ</mi>
</math></span> (or any equivalent) <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="{\text{arg}}\left( {\frac{z}{{z - 2a}}} \right) = {\text{arg}}\left( z \right) - {\text{arg}}\left( {z - 2a} \right)">
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>arg</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\theta - \pi ">
<mo>=</mo>
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mi>π</mi>
</math></span> (or any equivalent) <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0">
<mrow>
<mtext>Re</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\theta - \pi = \frac{{n\pi }}{2}">
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mi>π</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span>, (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> odd) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \pi < 2\theta - \pi < 0 \Rightarrow n = - 1">
<mo>−</mo>
<mi>π</mi>
<mo><</mo>
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mi>π</mi>
<mo><</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\theta - \pi = - \frac{\pi }{2}">
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mi>π</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{4}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{a + b{\text{i}}}}{{ - a + b{\text{i}}}} = \frac{{{b^2} - {a^2} - 2ab{\text{i}}}}{{{a^2} + {b^2}}}">
<mfrac>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mrow>
<mo>−</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Re}}\left( {\frac{z}{{z - 2a}}} \right) = 0 \Rightarrow {b^2} - {a^2} = 0">
<mrow>
<mtext>Re</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = a">
<mi>b</mi>
<mo>=</mo>
<mi>a</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{4}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept any equivalent, <em>eg </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = - \frac{{7\pi }}{4}">
<mi>θ</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mn>7</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<p> </p>
<p> </p>
<p><em><strong>[3 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.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>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^2} = 4{{\text{e}}^{\frac{\pi }{2}{\text{i}}}}">
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, giving your answers in the form</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r{{\text{e}}^{{\text{i}}\theta }}">
<mi>r</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</msup>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta \in \mathbb{R}">
<mi>θ</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r > 0">
<mi>r</mi>
<mo>></mo>
<mn>0</mn>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{i}}b">
<mi>a</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>b</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{R}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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="z = 2{{\text{e}}^{\frac{\pi }{4}{\text{i}}}}\,\left( { = 2{{\text{e}}^{0.785{\text{i}}}}} \right)">
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>0.785</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept all answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{{\text{e}}^{\left( {\frac{\pi }{4} + 2\pi n} \right){\text{i}}}}">
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>n</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2{{\text{e}}^{\frac{{5\pi }}{4}{\text{i}}}}\,\left( { = 2{{\text{e}}^{3.93{\text{i}}}}} \right)">
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3.93</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2{{\text{e}}^{ - \frac{{3\pi }}{4}{\text{i}}}}\,\left( { = 2{{\text{e}}^{ - 2.36{\text{i}}}}} \right)">
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2.36</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Accept all answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{{\text{e}}^{\left( { - \frac{{3\pi }}{4} + 2\pi n} \right){\text{i}}}}">
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>n</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for correct answers in the incorrect form, <em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2{{\text{e}}^{\frac{\pi }{4}{\text{i}}}}">
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>.</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="z = 1.41 + 1.41{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mn>1.41</mn>
<mo>+</mo>
<mn>1.41</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = - 1.41 - 1.41{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1.41</mn>
<mo>−</mo>
<mn>1.41</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {z = \sqrt 2 + \sqrt 2 {\text{i}},\,\,z = - \sqrt 2 - \sqrt 2 {\text{i}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>z</mi>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mo>+</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mo>−</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></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.</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 differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> for <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>y</mi><mo>></mo><mn>2</mn><mi>x</mi></math>. It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method, with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math>, to find an approximate value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use Euler’s method <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>;</mo><mo> </mo><mo> </mo><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>, where <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><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math></p>
<p>correct intermediate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-values <em><strong>(A1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>63140</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>5</mn><mo>.</mo><mn>92098</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>.</mo><mn>79542</mn><mo>…</mo></math></p>
<p> </p>
<p><strong>Note:</strong> <em><strong>A1</strong> </em>for any two correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-values seen</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>6958</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> For the final <em><strong>A1</strong></em>, the value <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>7</mn></math> must be the last value in a table or a list, or be given as a final answer, not just embedded in a table which has further lines.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p>replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>x</mi></math> and <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> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to separate variables <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mfenced><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math> <em><strong>(A1)</strong></em></p>
<p>attempt to express in partial fraction form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>v</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>ln</mi><mfenced open="|" close="|"><mi>x</mi></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p style="text-align:left;"><strong>Note:</strong> Condone absence of modulus signs throughout.</p>
<p style="text-align:left;"><strong><br>EITHER</strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mn>3</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>ln</mi><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p>expressing both sides as a single logarithm <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mfrac><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>ln</mi><mfenced><mfrac><msup><mfenced open="|" close="|"><mi>x</mi></mfenced><mn>3</mn></msup><mn>4</mn></mfrac></mfenced></math></p>
<p><strong><br>OR</strong></p>
<p>expressing both sides as a single logarithm <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mfrac><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>A</mi><msup><mfenced open="|" close="|"><mi>x</mi></mfenced><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mn>3</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfrac><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>)</p>
<p>substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mi>y</mi><mi>x</mi></mfrac></math> (seen anywhere) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>y</mi><mi>x</mi></mfrac></mstyle><mo>-</mo><mn>2</mn></mrow><mrow><mstyle displaystyle="true"><mfrac><mi>y</mi><mi>x</mi></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2</mn><mi>x</mi></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to make <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> the subject <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mi>y</mi></mrow><mn>4</mn></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>actual value at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>27</mn><mo>.</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient changes rapidly (during the interval considered) OR</p>
<p>the curve has a vertical asymptote at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mroot><mn>4</mn><mn>3</mn></mroot><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>5874</mn><mo>…</mo></mrow></mfenced></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Most candidates showed evidence of an attempt to use Euler's method in part a), although very few explicitly wrote down the formulae, they used in order to calculate successive <em>y</em>-value. In addition, many seemed to take a step-by-step approach rather than using the recursive capabilities of the graphical display calculator.</p>
<p>There were many good attempts at part b), but not all candidates recognised that this would help them to solve part c).</p>
<p>Part c) was done very well by many candidates, although there were a significant number who failed to recognise the need for partial fractions and could not make further progress. A common error was to integrate without a constant of integration, which meant that the initial condition could not be used. The reasoning given for the estimate being poor was often too vague and did not address the specific nature of the function given clearly enough.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove the identity <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></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 equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> has two real roots, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>n</mi><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math> and which has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>α</mi><mn>3</mn></msup></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>β</mi><mn>3</mn></msup></mfrac></math>.<br>Without solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and <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.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math></p>
<p>attempts to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>+</mo><mn>3</mn><mi>p</mi><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>+</mo><mn>3</mn><mi>p</mi><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>+</mo><mn>3</mn><mi>p</mi><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>-</mo><mn>3</mn><mi>p</mi><msup><mi>q</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note: </strong>Condone the use of equals signs throughout.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math></p>
<p>attempts to factorise <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mfenced><mrow><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>≡</mo><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mi>p</mi><mi>q</mi><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>-</mo><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>+</mo><mi>p</mi><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>p</mi><mn>2</mn></msup><mi>q</mi><mo>-</mo><mi>p</mi><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math> <em><strong>AG</strong></em></p>
<p><em><br></em><strong>Note: </strong>Condone the use of equals signs throughout.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup><mo>≡</mo><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced></math></p>
<p>attempts to factorise <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>3</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mi>p</mi><mi>q</mi><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mfenced><mrow><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><msup><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>p</mi><mi>q</mi><mfenced><mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note: </strong>Condone the use of equals signs throughout.</p>
<p><br><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> Award a maximum of <em><strong>A1M0A0A1M0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>95</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>8</mn></math> found by using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>±</mo><msqrt><mn>17</mn></msqrt></mrow><mn>4</mn></mfrac><mo> </mo><mfenced><mrow><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>219</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>28</mn><mo>…</mo></mrow></mfenced></math>.<br>Condone, as appropriate, solutions that state but clearly do not use the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math>.<br>Special case: Award a maximum of <em><strong>A1M1A0A1M0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>95</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>8</mn></math> obtained by solving simultaneously for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> from product of roots and sum of roots equations.</p>
<p><br>product of roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> (seen anywhere) <em><strong>A1</strong></em></p>
<p>considers <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mn>1</mn><msup><mi>α</mi><mn>3</mn></msup></mfrac></mfenced><mfenced><mfrac><mn>1</mn><msup><mi>β</mi><mn>3</mn></msup></mfrac></mfenced></math> by stating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mn>3</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mi>n</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempting to substitute their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mn>3</mn></msup></mfrac></math>.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mn>3</mn></msup></mfrac><mo>=</mo><mfrac><mn>1</mn><msup><mfenced><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mfenced><mn>3</mn></msup></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p>sum of roots of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>β</mi><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></math> (seen anywhere) <em><strong>A1</strong></em></p>
<p>considers <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>α</mi><mn>3</mn></msup></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>β</mi><mn>3</mn></msup></mfrac></math> by stating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi><mi>β</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></mrow><msup><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mn>3</mn></msup></mfrac><mo> </mo><mfenced><mrow><msup><mfenced><mfrac><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow><mrow><mi>α</mi><mi>β</mi></mrow></mfrac></mfenced><mn>3</mn></msup><mo>-</mo><mfrac><mrow><mn>3</mn><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></mrow><msup><mfenced><mrow><mi>α</mi><mi>β</mi></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced><mfenced><mrow><mo>=</mo><mo>-</mo><mi>m</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempting to substitute their values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>+</mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>β</mi></math> into their expression. Award <em><strong>M0</strong></em> for use of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>α</mi><mi>β</mi><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced></math> only.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mstyle displaystyle="true"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle></mfenced><mn>3</mn></msup><mo>-</mo><mfenced><mstyle displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mfenced><mfenced><mstyle displaystyle="true"><mfrac><mn>5</mn><mn>2</mn></mfrac></mstyle></mfenced></mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>8</mn></mfrac></mstyle></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>125</mn><mo>-</mo><mn>30</mn><mo>=</mo><mn>95</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>95</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>95</mn><mi>x</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><br><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="specification">
<p>Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio of this sequence.</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 sum to infinity of this sequence.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_4} = {u_1}{r^3} \Rightarrow - 2.916 = 4{r^3}">
<mrow>
<msub>
<mi>u</mi>
<mn>4</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">⇒</mo>
<mo>−</mo>
<mn>2.916</mn>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>solving, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = - 0.9">
<mi>r</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.9</mn>
</math></span> <em><strong> (M1)A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_\infty } = \frac{4}{{1 - \left( { - 9} \right)}}"> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">∞</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mn>4</mn> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>9</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{40}}{{19}}\,\left( { = 2.11} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>40</mn> </mrow> <mrow> <mn>19</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2.11</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> A1</strong></em></p>
<p> </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.</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 particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves in a straight line such that after time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>−</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mo> </mo><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> has displacement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>; at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>At successive times when the acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo> </mo></math>, the velocities of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> form a geometric sequence. The acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is zero at times <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo><</mo><msub><mi>t</mi><mn>3</mn></msub></math> and the respective velocities are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>3</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> comes to instantaneous rest.</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 an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in metres, from its initial position.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> seconds of its motion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, at these times, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>524</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>05</mn></mrow></mfenced></math> <em><strong>A1</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>attempt to use integration by parts <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mo>∫</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math></p>
<p><strong><br>EITHER</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mo>∫</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mo>∫</mo><mo>-</mo><mn>4</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>+</mo><mn>4</mn><mi>s</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>s</mi><mo>=</mo><mfrac><mrow><mo>-3</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>9</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mo>∫</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac><mo>+</mo><mo>∫</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>s</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>s</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced></mrow><mn>15</mn></mfrac><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>s</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mn>0</mn><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>+</mo><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced></mrow><mn>15</mn></mfrac></math></p>
<p><em><strong><br>[7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>s</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mi mathvariant="normal">π</mi></mrow></mfenced></mrow><mn>15</mn></mfrac></mrow></mfenced></math></p>
<p><br><strong>OR</strong><br><br></p>
<p>using GDC to find maximum value <em><strong>(M1)</strong></em><br><br></p>
<p><strong>OR</strong></p>
<p>evaluating <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></msubsup><mi>v</mi><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>161</mn><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle></mrow></msup></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p><em><strong><br>[2 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><strong><br>EITHER</strong></p>
<p>distance required <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></munderover><mfenced open="|" close="|"><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>distance required <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi><mo>+</mo><mfenced open="|" close="|"><mrow><munderover><mo>∫</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced><mo>+</mo><munderover><mo>∫</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>033479</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>006806</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>201</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><br>using successive minimum and maximum values on the displacement graph <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>12757</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>13453</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>12757</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>201</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac></math> using product rule and set <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> in exact form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong></em> is for any two consecutive correct, or showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub></math>.</p>
<p><br>showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>6</mn><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>-</mo><mi>sin</mi><mo> </mo><mn>6</mn><msub><mi>t</mi><mi>n</mi></msub></math></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn><mo>⇒</mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mo>±</mo><mfrac><mn>2</mn><msqrt><mn>5</mn></msqrt></mfrac></math> <em><strong>M1A1</strong></em></p>
<p>showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><msub><mi>t</mi><mrow><mi>n</mi><mo>+1</mo></mrow></msub></mrow></msup><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><msub><mi>t</mi><mi>n</mi></msub></mrow></msup></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>M1</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>÷</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>k</mi></mrow></msup><mo>=</mo><msup><mtext>e</mtext><mrow><mi>-</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math></p>
<p><br><strong>Note:</strong> Award the <em><strong>A1</strong></em> for any two consecutive terms.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">e.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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Find the term independent of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup></mfrac><msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mi>x</mi><mn>2</mn></mfrac></mrow></mfenced><mn>9</mn></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 question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>use of Binomial expansion to find a term in either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mi>x</mi><mn>2</mn></mfrac></mrow></mfenced><mn>9</mn></msup><mo>,</mo><mo> </mo><msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mstyle displaystyle="true"><mfrac bevelled="true"><mn>7</mn><mn>3</mn></mfrac></mstyle></msup></mrow></mfrac><mo>-</mo><mfrac><msup><mi>x</mi><mstyle displaystyle="true"><mfrac bevelled="true"><mn>2</mn><mn>3</mn></mfrac></mstyle></msup><mn>2</mn></mfrac></mrow></mfenced><mn>9</mn></msup><mo>,</mo><mo> </mo><msup><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>2</mn></mfrac></mrow></mfenced><mn>9</mn></msup><mo>,</mo><mo> </mo><msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>9</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>2</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mn>9</mn></msup></math> <em><strong>(</strong><strong>M1)(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for a product of three terms including a binomial coefficient and powers of the two terms, and <em><strong>A1</strong></em> for a correct expression of a term in the expansion.</p>
<p><br>finding the powers required to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> <em><strong>(</strong><strong>M1)(A1)</strong></em></p>
<p>constant term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>C</mi><mn>2</mn><mprescripts></mprescripts><mn>9</mn></mmultiscripts><mo>×</mo><msup><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>7</mn></msup></math> <em><strong>(</strong><strong>M1)</strong></em></p>
<p><br><strong>Note:</strong> Ignore all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>’s in student’s expression.<br><br>therefore term independent of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>32</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>03125</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express the binomial coefficient <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> as a polynomial in <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>Hence find the least 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="\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right) > {10^6}">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>></mo>
<mrow>
<msup>
<mn>10</mn>
<mn>6</mn>
</msup>
</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="\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right) = \frac{{\left( {3n + 1} \right){\text{!}}}}{{\left( {3n - 2} \right){\text{!}}3{\text{!}}}}">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>!</mtext>
</mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\left( {3n + 1} \right)3n\left( {3n - 1} \right)}}{{3{\text{!}}}}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <strong>A1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{9}{2}{n^3} - \frac{1}{2}n">
<mo>=</mo>
<mfrac>
<mn>9</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>n</mi>
</math></span> or equivalent <em><strong>A1</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>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{9}{2}{n^3} - \frac{1}{2}n > {10^6}">
<mo>=</mo>
<mfrac>
<mn>9</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>n</mi>
<mo>></mo>
<mrow>
<msup>
<mn>10</mn>
<mn>6</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 60.57 \ldots ">
<mi>n</mi>
<mo>></mo>
<mn>60.57</mn>
<mo>…</mo>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> Allow equality.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow n = 61">
<mo stretchy="false">⇒</mo>
<mi>n</mi>
<mo>=</mo>
<mn>61</mn>
</math></span> <em><strong>A1</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>The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mi>r</mi><mi>x</mi><mo>)</mo></math> . The graph has a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> and a local minimum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the area of the shaded region.</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>the principal axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>+</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>the amplitude is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mn>3</mn></mrow></mfenced></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p>one period is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>-</mo><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></mrow></mfenced></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>r</mi></mfrac><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi></math></p>
<p><strong><br>OR</strong></p>
<p>Substituting a point eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo>+</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn><mo>⇒</mo><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi><mo>=</mo><mo>…</mo><mo>-</mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo>…</mo></math></p>
<p>Choice of correct solution <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><mi>y</mi><mo>=</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mn>3</mn></mfrac></mfenced></mrow></mfenced></math></p>
<p><strong><br>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> can be both given as negatives for full marks</p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>09459</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>617797</mn><mo>…</mo></math><em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>3</mn><mo>.</mo><mn>617797</mn><mo>…</mo></mrow><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>09459</mn><mo>…</mo></mrow></msubsup><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mn>3</mn></mfrac></mfenced></mrow></mfenced><mtext>d</mtext><mi>x</mi></math><em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>66</mn><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>66179</mn><mo>…</mo></mrow></mfenced></math><em><strong> (A1)</strong></em></p>
<p>so area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>66</mn><mo> </mo><mo> </mo><mfenced><msup><mtext>units</mtext><mn>2</mn></msup></mfenced></math><em><strong> A1</strong></em></p>
<p><em><strong><br>[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>Mary, three female friends, and her brother, Peter, attend the theatre. In the theatre there is a row of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> empty seats. For the first half of the show, they decide to sit next to each other in this row.</p>
</div>
<div class="specification">
<p>For the second half of the show, they return to the same row of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> empty seats. The four girls decide to sit at least one seat apart from Peter. The four girls do not have to sit next to each other.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of ways these five people can be seated in this row.</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 number of ways these five people can now be seated in this row.</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"><mn>6</mn><mo>×</mo><mn>5</mn><mo>!</mo></math> <em><strong>(A1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>720</mn></math> (accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>!</mo></math>) <em><strong>A1</strong></em></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><strong>METHOD 1</strong></p>
<p>(Peter apart from girls, in an end seat) <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>P</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>1680</mn></mrow></mfenced></math> OR</p>
<p>(Peter apart from girls, not in end seat) <math xmlns="http://www.w3.org/1998/Math/MathML"><mmultiscripts><mi>P</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>840</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>case 1: Peter at either end </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>3360</mn></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mmultiscripts><mi>C</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts><mo>×</mo><mn>4</mn><mo>!</mo><mfenced><mrow><mo>=</mo><mn>3360</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>case 2: Peter not at the end</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>6720</mn></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><mmultiscripts><mi>C</mi><mn>4</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mo>×</mo><mn>4</mn><mo>!</mo><mfenced><mrow><mo>=</mo><mn>6720</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>Total number of ways <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3360</mn><mo>+</mo><mn>6720</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10080</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>(Peter next to girl, in an end seat) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>1344</mn></mrow></mfenced></math> OR</p>
<p>(Peter next to one girl, not in end seat) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>4</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>1680</mn></mrow></mfenced></math> OR</p>
<p>(Peter next to two girls, not in end seat) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>504</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>case 1: Peter at either end</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>4</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>8</mn></mmultiscripts><mfenced><mrow><mo>=</mo><mn>2688</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>case 2: Peter not at the end</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mfenced><mrow><mn>2</mn><mo>×</mo><mn>4</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>3</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts><mo>+</mo><mn>4</mn><mo>×</mo><mn>3</mn><mo>×</mo><mmultiscripts><mi>P</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>7</mn></mmultiscripts></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>17472</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>Total number of ways <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mmultiscripts><mi>P</mi><mn>5</mn><none></none><mprescripts></mprescripts><none></none><mn>10</mn></mmultiscripts><mo>-</mo><mfenced><mrow><mn>2688</mn><mo>+</mo><mn>17472</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10080</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;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down and simplify the first three terms, in ascending powers of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, in the Extended Binomial expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - x} \right)^{\frac{1}{3}}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</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>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{9}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>9</mn>
</mfrac>
</math></span> find a rational approximation to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{9}">
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - x} \right)^{\frac{1}{3}}} = 1 + \frac{1}{3}\left( { - x} \right) + \frac{1}{3}\left( {\frac{{ - 2}}{3}} \right)\frac{{{{\left( { - x} \right)}^2}}}{{2{\text{!}}}} \ldots = 1 - \frac{x}{3} - \frac{{{x^2}}}{9} \ldots ">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
<mo>…</mo>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mi>x</mi>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
<mo>…</mo>
</math></span> <em><strong>M1A1A1</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="{\left( {\frac{8}{9}} \right)^{\frac{1}{3}}} \simeq 1 - \frac{1}{{27}} - \frac{1}{{729}} \Rightarrow \frac{2}{{\sqrt[3]{9}}} \simeq \frac{{701}}{{729}} \Rightarrow \sqrt[3]{9} \simeq \frac{{1458}}{{701}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>8</mn>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>≃</mo>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>27</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>729</mn>
</mrow>
</mfrac>
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</mrow>
</mfrac>
<mo>≃</mo>
<mfrac>
<mrow>
<mn>701</mn>
</mrow>
<mrow>
<mn>729</mn>
</mrow>
</mfrac>
<mo stretchy="false">⇒</mo>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
<mo>≃</mo>
<mfrac>
<mrow>
<mn>1458</mn>
</mrow>
<mrow>
<mn>701</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1A1A1</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 the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{R}">
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The equation has three distinct real roots which can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,b">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,c">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span>.</p>
<p>The equation also has two imaginary roots, one of which is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d{\text{i}}">
<mi>d</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The values <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> are consecutive terms in a geometric sequence.</p>
</div>
<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="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </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>Show that one of the real roots is equal to 1.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 8{d^2}"> <mi>q</mi> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span>, find the other two real roots.</p>
<div class="marks">[9]</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>recognition of the other root <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - d{\text{i}}"> <mo>=</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </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}}_2}\,a + {\text{lo}}{{\text{g}}_2}\,b + {\text{lo}}{{\text{g}}_2}\,c + d{\text{i}} - d{\text{i}} = 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>M1A</strong></em><em><strong>1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for sum of the roots, <em><strong>A1</strong> </em>for 3. Award <em><strong>A0M1A0</strong></em> for just <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a + {\text{lo}}{{\text{g}}_2}\,b + {\text{lo}}{{\text{g}}_2}\,c = 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>=</mo> <mn>3</mn> </math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,abc = 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow abc = {2^3}"> <mo stretchy="false">⇒</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> </mrow> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></span> <em><strong>AG</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><strong>METHOD 1</strong></p>
<p>let the geometric series be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}r"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mi>r</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{r^2}"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <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="{\left( {{u_1}r} \right)^3} = 8"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>8</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}r = 2"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mi>r</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>hence one of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}2 = 1"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{b}{a} = \frac{c}{b}"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> <mo>=</mo> <mfrac> <mi>c</mi> <mi>b</mi> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{b^2} = ac \Rightarrow {b^3} = abc = 8"> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>b</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 2"> <mi>b</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>hence one of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}2 = 1"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mn>2</mn> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>product of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} \times {r_2} \times 1 \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>×</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>×</mo> <mn>1</mn> <mo>×</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>×</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} \times {r_2} = - 8"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>×</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>sum of the roots is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} + {r_2} + 1 + d{\text{i}} + - d{\text{i}} = 3"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>+</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mn>3</mn> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} + {r_2} = 2"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p>solving simultaneously <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_1} = - 2"> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r_2} = 4"> <mrow> <msub> <mi>r</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>product of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>×</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>×</mo> <mo>−</mo> <mi>d</mi> <mrow> <mtext>i</mtext> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c = - 8"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{r}"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2"> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2r"> <mn>2</mn> <mi>r</mi> </math></span> <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{lo}}{{\text{g}}_2}\frac{2}{r}} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2r} \right) = - 8"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span></p>
<p>attempt to solve <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {\text{lo}}{{\text{g}}_2}\,r} \right)\left( {1 + {\text{lo}}{{\text{g}}_2}\,r} \right) = - 8"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,r = \pm 3"> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>r</mi> <mo>=</mo> <mo>±</mo> <mn>3</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{1}{8}{\text{,}}\,\,8"> <mi>r</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>8</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2"> <mn>2</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{4}{a}"> <mfrac> <mn>4</mn> <mi>a</mi> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{lo}}{{\text{g}}_2}\,a} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,\frac{4}{a}} \right) = - 8"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>lo</mtext> </mrow> <mrow> <msub> <mrow> <mtext>g</mtext> </mrow> <mn>2</mn> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mn>4</mn> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>8</mn> </math></span></p>
<p>attempt to solve <em><strong>M</strong></em><em><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{1}{4}{\text{,}}\,\,16"> <mi>a</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>16</mn> </math></span> <em><strong>A</strong></em><em><strong>1</strong></em><em><strong>A</strong></em><em><strong>1</strong></em></p>
<p><strong>THEN</strong></p>
<p><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="c"> <mi>c</mi> </math></span> are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4}{\text{,}}\,\,16"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>,</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>16</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p>roots are −2, 4 <em><strong>A</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><em><strong>[9 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 following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} = {\text{si}}{{\text{n}}^3}\,y">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant y \leqslant \pi ">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>y</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The shaded region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \pi ">
<mi>y</mi>
<mo>=</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using implicit differentiation, find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to the curve at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</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>Find the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span> is now rotated about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis, through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians, to form a solid.</p>
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{si}}{{\text{n}}^3}\,y}"> <mrow> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </math></span>, show that the volume of the solid formed is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span>.</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 attempt to differentiate implicitly <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x = 3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mn>4</mn> <mi>x</mi> <mo>=</mo> <mn>3</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>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4x}}{{3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>3</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>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right){\text{, }}\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4x}}{{3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y}} = \frac{1}{{3{{\left( {\frac{1}{2}} \right)}^2}\left( { - \frac{{\sqrt 3 }}{2}} \right)}}"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>3</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>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{8}{{3\sqrt 3 }}\left( { = - 1.54} \right)"> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>8</mn> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>hence equation of tangent is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - \frac{{5\pi }}{6} = - 1.54\left( {x - \frac{1}{4}} \right)"> <mi>y</mi> <mo>−</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1.54x + 3.00"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> <mi>x</mi> <mo>+</mo> <mn>3.00</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1.54x + 3"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> </math></span>. </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt {\frac{1}{2}{\text{si}}{{\text{n}}^3}\,y} "> <mi>x</mi> <mo>=</mo> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </msqrt> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {\sqrt {\frac{1}{2}{\text{si}}{{\text{n}}^3}\,y\,{\text{d}}y} } "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </msqrt> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1.24"> <mo>=</mo> <mn>1.24</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int {\pi {x^2}} \,{\text{d}}y"> <mo>=</mo> <mo>∫</mo> <mrow> <mi>π</mi> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int_0^\pi {\frac{1}{2}} \pi \,{\text{si}}{{\text{n}}^3}\,y\,{\text{d}}y"> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </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}\pi \int_0^\pi {\left( {{\text{sin}}\,y - {\text{sin}}\,y\,{\text{co}}{{\text{s}}^2}\,y} \right){\text{d}}y} "> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> </math></span></p>
<p><strong>Note:</strong> Condone absence of limits up to this point.</p>
<p>reasonable attempt to integrate <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\pi \left[ { - {\text{cos}}\,y + \frac{1}{3}{\text{co}}{{\text{s}}^3}\,y} \right]_0^\pi "> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mi>π</mi> </msubsup> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct limits (not to be awarded if previous <em><strong>M1</strong></em> has not been awarded) and <em><strong>A1</strong></em> for correct integrand.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\pi \left( {1 - \frac{1}{3}} \right) - \frac{1}{2}\pi \left( { - 1 + \frac{1}{3}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math> <em><strong>A1</strong></em></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2\pi }}{3}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Do not accept decimal answer equivalent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span>.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Phil takes out a bank loan of $150 000 to buy a house, at an annual interest rate of 3.5%. The interest is calculated at the end of each year and added to the amount outstanding.</p>
</div>
<div class="specification">
<p>To pay off the loan, Phil makes annual deposits of $<em>P </em>at the end of every year in a savings account, paying an annual interest rate of 2% . He makes his first deposit at the end of the first year after taking out the loan.</p>
</div>
<div class="specification">
<p>David visits a different bank and makes a single deposit of $<em>Q </em>, the annual interest rate being 2.8%.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.</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>Show that the total value of Phil’s savings after 20 years is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>20</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1.02</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that Phil’s aim is to own the house after 20 years, find the value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> to the nearest dollar.</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>David wishes to withdraw $5000 at the end of each year for a period of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> years. Show that an expression for the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="150000 \times {1.035^{20}}"> <mn>150000</mn> <mo>×</mo> <mrow> <msup> <mn>1.035</mn> <mrow> <mn>20</mn> </mrow> </msup> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \$ 298468"> <mo>=</mo> <mi mathvariant="normal">$</mi> <mn>298468</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only accept answers to the nearest dollar. Accept $298469.</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>attempt to look for a pattern by considering 1 year, 2 years <em>etc </em><strong><em>(M1)</em></strong></p>
<p>recognising a geometric series with first term <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> and common ratio 1.02 <strong><em>(M1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P + 1.02P + \ldots + {1.02^{19}}P{\text{ }}\left( { = P(1 + 1.02 + \ldots + {{1.02}^{19}})} \right)"> <mi>P</mi> <mo>+</mo> <mn>1.02</mn> <mi>P</mi> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow> <msup> <mn>1.02</mn> <mrow> <mn>19</mn> </mrow> </msup> </mrow> <mi>P</mi> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1.02</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>19</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>explicitly identify <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = P,{\text{ }}r = 1.02"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mi>P</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>r</mi> <mo>=</mo> <mn>1.02</mn> </math></span> and <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> (may be seen as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_{20}}"> <mrow> <msub> <mi>S</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> </math></span>). <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_{20}} = \frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}"> <mrow> <msub> <mi>s</mi> <mrow> <mn>20</mn> </mrow> </msub> </mrow> <mo>=</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>20</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1.02</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[3 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="24.297 \ldots P = 298468"> <mn>24.297</mn> <mo>…</mo> <mi>P</mi> <mo>=</mo> <mn>298468</mn> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = 12284"> <mi>P</mi> <mo>=</mo> <mn>12284</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers which round to 12284.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q({1.028^n}) = 5000(1 + 1.028 + {1.028^2} + {1.028^3} + \ldots + {1.028^{n - 1}})"> <mi>Q</mi> <mo stretchy="false">(</mo> <mrow> <msup> <mn>1.028</mn> <mi>n</mi> </msup> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5000</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1.028</mn> <mo>+</mo> <mrow> <msup> <mn>1.028</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>1.028</mn> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mrow> <msup> <mn>1.028</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q = \frac{{5000\left( {1 + 1.028 + {{1.028}^2} + {{1.028}^3} + ... + {{1.028}^{n - 1}}} \right)}}{{{{1.028}^n}}}"> <mi>Q</mi> <mo>=</mo> <mfrac> <mrow> <mn>5000</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>1.028</mn> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mo>=</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>the initial value of the first withdrawal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>the initial value of the second withdrawal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{{{1.028}^2}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>R1</em></strong></p>
<p>the investment required for these two withdrawals is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q = \frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mi>Q</mi> <mo>=</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[3 Marks]</em></strong></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sum to infinity is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\frac{{5000}}{{1.028}}}}{{1 - \frac{1}{{1.028}}}}"> <mfrac> <mrow> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mn>1.028</mn> </mrow> </mfrac> </mrow> </mfrac> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 178571.428 \ldots "> <mo>=</mo> <mn>178571.428</mn> <mo>…</mo> </math></span></p>
<p>so minimum amount is $178572 <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers which round to $178571 or $178572.</p>
<p> </p>
<p><strong><em>[3 Marks]</em></strong></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>Eight boys and two girls sit on a bench. Determine the number of possible arrangements, given that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the girls do not sit together.</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 girls do not sit on either end.</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 girls do not sit on either end and do not sit together.</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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10{\text{!}} - 2 \times 9{\text{!}}\,\left( { = 2903040} \right)"> <mn>10</mn> <mrow> <mtext>!</mtext> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>9</mn> <mrow> <mtext>!</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2903040</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> (A1)</strong></em><em><strong>(A1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10{\text{!}}"> <mn>10</mn> <mrow> <mtext>!</mtext> </mrow> </math></span>, <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times 9{\text{!}}"> <mn>2</mn> <mo>×</mo> <mn>9</mn> <mrow> <mtext>!</mtext> </mrow> </math></span>, <em><strong>A1</strong></em> for final answer.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 \times 8 \times 8{\text{!}}"> <mn>9</mn> <mo>×</mo> <mn>8</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> <em><strong> (A1)</strong></em><em><strong>(A1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 \times 8"> <mn>9</mn> <mo>×</mo> <mn>8</mn> </math></span> or equivalent, <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8{\text{!}}"> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> and <em><strong>A1</strong></em> for answer.</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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 \times 7 \times 8{\text{!}}\,\left( { = 2257920} \right)"> <mn>8</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2257920</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> (A1)</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(</strong><strong>A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 \times 7"> <mn>8</mn> <mo>×</mo> <mn>7</mn> </math></span>, <em><strong>A1</strong></em> for final answer.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10{\text{!}} - 2 \times 8{\text{!}} - 2 \times 2 \times 7 \times 8{\text{!}}"> <mn>10</mn> <mrow> <mtext>!</mtext> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> </math></span></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10{\text{!}}"> <mn>10</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> minus EITHER subtracted terms and <em><strong>A1</strong></em> for final correct answer.</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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 \times 7 \times \left( {8{\text{!}} - 2 \times 7{\text{!}}} \right)\,\left( { = 1693440} \right)"> <mn>8</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mrow> <mtext>!</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>1693440</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> (A1)(A1)</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(</strong><strong>A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8 \times 7"> <mn>8</mn> <mo>×</mo> <mn>7</mn> </math></span>, <em><strong>(</strong><strong>A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2 \times 7{\text{!}}}"> <mrow> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mrow> <mtext>!</mtext> </mrow> </mrow> </math></span>, <em><strong>A1</strong></em> for final answer. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {8{\text{!}} - 2 \times 7{\text{!}}} \right)"> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mrow> <mtext>!</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> can be replaced by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6 \times 7{\text{!}}"> <mn>6</mn> <mo>×</mo> <mn>7</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{}^7{P_2} \times 6{\text{!}}"> <msup> <mrow> </mrow> <mn>7</mn> </msup> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> </mrow> <mo>×</mo> <mn>6</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> which may be awarded the second <em><strong>A1</strong></em>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>their answer to (a) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \times 8{\text{!}} - 2 \times 2 \times 7 \times 8{\text{!}}"> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> <em><strong> (A1)(A1)</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for subtracting each of the terms and <em><strong>A1</strong></em> for final answer.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>their answer to (b) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \times 7 \times 8{\text{!}}"> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>8</mn> <mrow> <mtext>!</mtext> </mrow> </math></span> or equivalent <em><strong> (A1)</strong></em><em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for the subtraction and <em><strong>A2</strong> </em>for final answer.</p>
<p> </p>
<p><em><strong>[3 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>The complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> satisfy the equations</p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{w}{z} = 2{\text{i}}">
<mfrac>
<mi>w</mi>
<mi>z</mi>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p style="padding-left:150px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{z}}^ * } - 3w = 5 + 5{\text{i}}">
<mrow>
<msup>
<mrow>
<mtext>z</mtext>
</mrow>
<mo>∗</mo>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>w</mi>
<mo>=</mo>
<mn>5</mn>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b{\text{i}}">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{b}} \in \mathbb{Z}">
<mrow>
<mtext>b</mtext>
</mrow>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 2{\text{i}}z">
<mi>w</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>z</mi>
</math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{z}}^ * } - 3w = 5 + 5{\text{i}}">
<mrow>
<msup>
<mrow>
<mtext>z</mtext>
</mrow>
<mo>∗</mo>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>w</mi>
<mo>=</mo>
<mn>5</mn>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{z}}^ * } - 6{\text{i}}z = 5 + 5{\text{i}}">
<mrow>
<msup>
<mrow>
<mtext>z</mtext>
</mrow>
<mo>∗</mo>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = x + y{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p>comparing real and imaginary parts of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - y{\text{i}}} \right) - 6{\text{i}}\left( {x + y{\text{i}}} \right) = 5 + 5{\text{i}}">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>5</mn>
<mo>+</mo>
<mn>5</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>to obtain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 6y = 5">
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6x - y = 5">
<mo>−</mo>
<mn>6</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = - 1 + {\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = - 2 - 2{\text{i}}">
<mi>w</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[7 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">
<p>Boxes of mixed fruit are on sale at a local supermarket.</p>
<p>Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.</p>
<p>Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.</p>
<p>Box C contains 5 bananas and 4 kiwifruit and costs $3.00.</p>
<p>Find the cost of each type of fruit.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> be the cost of one banana, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> the cost of one kiwifruit, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> the cost of one melon</p>
<p>attempt to set up three linear equations <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2b + 3k + 4m = 658">
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mn>3</mn>
<mi>k</mi>
<mo>+</mo>
<mn>4</mn>
<mi>m</mi>
<mo>=</mo>
<mn>658</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5b + 2k + 8m = 1232">
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>+</mo>
<mn>8</mn>
<mi>m</mi>
<mo>=</mo>
<mn>1232</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5b + 4k = 300">
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mn>4</mn>
<mi>k</mi>
<mo>=</mo>
<mn>300</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve three simultaneous equations <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 36,{\text{ }}k = 30,{\text{ }}m = 124">
<mi>b</mi>
<mo>=</mo>
<mn>36</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
<mo>=</mo>
<mn>30</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>m</mi>
<mo>=</mo>
<mn>124</mn>
</math></span></p>
<p>banana costs ($)0.36, kiwifruit costs ($)0.30, melon costs ($)1.24 <strong><em>A1</em></strong></p>
<p><strong><em>[4 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 random variable <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X">
<mi>X</mi>
</math></span> has probability density function</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {3a}&{\text{,}}&{0 \leqslant x < 2}&{} \\ {a\left( {x - 5} \right)\left( {1 - x} \right)}&{\text{,}}&{2 \leqslant x \leqslant b}&{a{\text{, }}b \in {\mathbb{R}^ + }{\text{, }}3 < b \leqslant 5.} \\ 0&{\text{,}}&{{\text{otherwise}}}&{} \end{array}} \right.">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>a</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>b</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mn>3</mn>
<mo><</mo>
<mi>b</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>5.</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mtext>,</mtext>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<mtext>otherwise</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
<p> </p>
</div>
<div class="specification">
<p>Consider the case where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 5">
<mi>b</mi>
<mo>=</mo>
<mn>5</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Find the value of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>, the probability that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> lies between 1 and 3.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>. State the coordinates of the end points and any local maximum or minimum points, giving your answers in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right)"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the median of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</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="\left( {{\text{P}}\left( {1 < X < 3} \right) = } \right)\int_1^2 {3a\,{\text{d}}x + a} \int_2^3 { - {x^2} + 6x - 5} \,{\text{d}}x"> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo><</mo> <mi>X</mi> <mo><</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> </mrow> <mo>)</mo> </mrow> <msubsup> <mo>∫</mo> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mn>3</mn> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>+</mo> <mi>a</mi> </mrow> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>3</mn> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>(M1)(A1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3a + \frac{{11}}{3}a"> <mo>=</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mfrac> <mrow> <mn>11</mn> </mrow> <mn>3</mn> </mfrac> <mi>a</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{20}}{3}a\,\left( { = 6.67a} \right)"> <mo>=</mo> <mfrac> <mrow> <mn>20</mn> </mrow> <mn>3</mn> </mfrac> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>6.67</mn> <mi>a</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A4</strong></em></p>
<p>award <em><strong>A1</strong></em> for (0, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>), <em><strong>A1</strong></em> for continuity at (2, 3<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>), <em><strong>A1</strong></em> for maximum at (3, 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>), <em><strong>A1</strong></em> for (5, 0)</p>
<p><strong>Note:</strong> Award <em><strong>A3</strong></em> if correct four points are not joined by a straight line and a quadratic curve.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {0 \leqslant X \leqslant 5} \right) = 6a + a\int_2^5 { - {x^2} + 6x - 5} \,{\text{d}}x"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>X</mi> <mo>⩽</mo> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>6</mn> <mi>a</mi> <mo>+</mo> <mi>a</mi> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>5</mn> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <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=" = 15a"> <mo>=</mo> <mn>15</mn> <mi>a</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="15a = 1"> <mn>15</mn> <mi>a</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = \frac{1}{{15}}\left( { = 0.0667} \right)"> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>0.0667</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{E}}\left( X \right) = \frac{1}{5}\int_0^2 {x\,{\text{d}}x} + \frac{1}{{15}}\int_2^5 { - {x^3} + 6{x^2} - 5} x\,{\text{d}}x"> <mrow> <mtext>E</mtext> </mrow> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </msubsup> <mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> </mrow> </mfrac> <msubsup> <mo>∫</mo> <mn>2</mn> <mn>5</mn> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>5</mn> </mrow> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>= 2.35 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^m {f\left( x \right)} {\text{d}}x = 0.5"> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>m</mi> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.5</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.4 + a\int_2^m { - {x^2} + 6x - 5} \,{\text{d}}x = 0.5"> <mn>0.4</mn> <mo>+</mo> <mi>a</mi> <msubsup> <mo>∫</mo> <mn>2</mn> <mi>m</mi> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.5</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\int_2^m { - {x^2} + 6x - 5} \,{\text{d}}x = 0.1"> <mi>a</mi> <msubsup> <mo>∫</mo> <mn>2</mn> <mi>m</mi> </msubsup> <mrow> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.1</mn> </math></span></p>
<p>attempt to solve integral using GDC and/or analytically <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{15}}\left[ { - \frac{1}{3}{x^3} + 3{x^2} - 5x} \right]_2^m = 0.1"> <mfrac> <mn>1</mn> <mrow> <mn>15</mn> </mrow> </mfrac> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>5</mn> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mn>2</mn> <mi>m</mi> </msubsup> <mo>=</mo> <mn>0.1</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 2.44"> <mi>m</mi> <mo>=</mo> <mn>2.44</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.iii.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Eight runners compete in a race where there are no tied finishes. Andrea and Jack are two of the eight competitors in this race.</p>
<p>Find the total number of possible ways in which the eight runners can finish if Jack finishes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>in the position immediately after Andrea.</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>in any position after Andrea.</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>Jack and Andrea finish in that order (as a unit) so we are considering the arrangement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> objects <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>!</mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5040</mn></mrow></mfenced></math> ways <em><strong>A1</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><strong>METHOD 1</strong></p>
<p>the number of ways that Andrea finishes in front of Jack is equal to the number of ways that Jack finishes in front of Andrea <em><strong>(M1)</strong></em></p>
<p>total number of ways is 8! <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>8</mn><mo>!</mo></mrow><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>20160</mn></mrow></mfenced></math> ways <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>the other six runners can finish in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>!</mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>720</mn></mrow></mfenced></math> ways <em><strong> (A1)</strong></em></p>
<p>when Andrea finishes first, Jack can finish in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> different positions</p>
<p>when Andrea finishes second, Jack can finish in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> different positions etc</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><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><mo> </mo><mo>(</mo><mo>=</mo><mn>28</mn><mo>)</mo></math> ways <em><strong>(A1)</strong></em></p>
<p>hence there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><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><mo>)</mo><mo>×</mo><mn>6</mn><mo>!</mo></math> ways</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn><mo>×</mo><mn>6</mn><mo>!</mo><mo> </mo><mo>(</mo><mo>=</mo><mn>20160</mn><mo>)</mo></math> ways <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>Prove by contradiction that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>5</mn></math> is an irrational number.</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>assume there exist <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>≥</mo><mn>1</mn></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>5</mn><mo>=</mo><mfrac><mi>p</mi><mi>q</mi></mfrac></math> <strong>M1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to write the negation of the statement as an assumption. Award <strong>A1</strong> for a correctly stated assumption.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>5</mn><mo>=</mo><mfrac><mi>p</mi><mi>q</mi></mfrac><mo>⇒</mo><mn>5</mn><mo>=</mo><msup><mn>2</mn><mfrac><mi>p</mi><mi>q</mi></mfrac></msup></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mi>q</mi></msup><mo>=</mo><msup><mn>2</mn><mi>p</mi></msup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> is a factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mi>q</mi></msup></math> but not a factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>p</mi></msup></math> <strong>R1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> is a factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>p</mi></msup></math> but not a factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mi>q</mi></msup></math> <strong>R1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mi>q</mi></msup></math> is odd and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mi>p</mi></msup></math> is even <strong>R1</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>no <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>≥</mo><mn>1</mn></math>) satisfy the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mi>q</mi></msup><mo>=</mo><msup><mn>2</mn><mi>p</mi></msup></math> and this is a contradiction <strong>R1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mn>5</mn></math> is an irrational number <strong>AG</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>A geometric sequence has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_4} = - 70"> <mrow> <msub> <mi>u</mi> <mn>4</mn> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mn>70</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_7} = 8.75"> <mrow> <msub> <mi>u</mi> <mn>7</mn> </msub> </mrow> <mo>=</mo> <mn>8.75</mn> </math></span>. Find the second term of the sequence.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{r^3} = - 70"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <mi>r</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mo>−</mo> <mn>70</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{r^6} = 8.75"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <mi>r</mi> <mn>6</mn> </msup> </mrow> <mo>=</mo> <mn>8.75</mn> </math></span><em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^3} = \frac{{8.75}}{{ - 70}} = - 0.125"> <mrow> <msup> <mi>r</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>8.75</mn> </mrow> <mrow> <mo>−</mo> <mn>70</mn> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>0.125</mn> </math></span><em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow r = - 0.5"> <mo stretchy="false">⇒</mo> <mi>r</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </math></span><em><strong> (A1)</strong></em></p>
<p>valid attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2}"> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> </math></span><em><strong> (M1)</strong></em></p>
<p>for example: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = \frac{{ - 70}}{{ - 0.125}} = 560"> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>70</mn> </mrow> <mrow> <mo>−</mo> <mn>0.125</mn> </mrow> </mfrac> <mo>=</mo> <mn>560</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2} = 560 \times - 0.5"> <mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> </mrow> <mo>=</mo> <mn>560</mn> <mo>×</mo> <mo>−</mo> <mn>0.5</mn> </math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 280"> <mo>=</mo> <mo>−</mo> <mn>280</mn> </math></span> <em><strong>A1</strong></em></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 complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{2 + 7{\text{i}}}}{{6 + 2{\text{i}}}}">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>7</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mrow>
<mn>6</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{i}}b">
<mi>a</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>b</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,\,b \in \mathbb{Q}">
<mi>a</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</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>Find the exact value of the modulus of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</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 argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span>, giving your answer to 4 decimal places.</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="z = \frac{{\left( {2 + 7{\text{i}}} \right)}}{{\left( {6 + 2{\text{i}}} \right)}} \times \frac{{\left( {6 - 2{\text{i}}} \right)}}{{\left( {6 - 2{\text{i}}} \right)}}">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>7</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{26 + 38{\text{i}}}}{{40}} = \left( {\frac{{13 + 19{\text{i}}}}{{20}} = 0.65 + 0.95{\text{i}}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>26</mn>
<mo>+</mo>
<mn>38</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mrow>
<mn>40</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>13</mn>
<mo>+</mo>
<mn>19</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mrow>
<mn>20</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.65</mn>
<mo>+</mo>
<mn>0.95</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</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>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {{a^2} + {b^2}} ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {\frac{{53}}{{40}}} \left( { = \frac{{\sqrt {530} }}{{20}}} \right)">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mn>53</mn>
</mrow>
<mrow>
<mn>40</mn>
</mrow>
</mfrac>
</msqrt>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>530</mn>
</msqrt>
</mrow>
<mrow>
<mn>20</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or equivalent <em><strong>A1</strong></em></p>
<p><strong>Note:<em> A1</em></strong> is only awarded for the correct exact value.</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><strong>EITHER</strong></p>
<p>arg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = arg(2 + 7i) − arg(6 + 2i) <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p>arg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = arctan<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{19}}{{13}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>19</mn>
</mrow>
<mrow>
<mn>13</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>arg <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = 0.9707 (radians) (= 55.6197 degrees) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Only award the last <em><strong>A1</strong> </em>if 4 decimal places are given.</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>It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.</p>
<p>At the start of 2018, there are 2500 fish in the lake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there will be approximately 2645 fish in the lake at the start of 2020.</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 approximate number of fish in the lake at the start of 2042.</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 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><strong>EITHER</strong></p>
<p>2019: 2500 × 0.93 + 250 = 2575 <em><strong>(M1)A1</strong></em></p>
<p>2020: 2575 × 0.93 + 250 <em><strong> M1</strong></em></p>
<p><strong>OR</strong></p>
<p>2020: 2500 × 0.93<sup>2</sup> + 250(0.93 + 1) <em><strong>M1M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for starting with 2500, <em><strong>M1</strong></em> for multiplying by 0.93 and adding 250 twice. <em><strong>A1</strong></em> for correct expression. Can be shown in recursive form.</p>
<p><strong>THEN</strong></p>
<p>(= 2644.75) = 2645 <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>2020: 2500 × 0.93<sup>2</sup> + 250(0.93 + 1)<br>2042: 2500 × 0.93<sup>24</sup> + 250(0.93<sup>23</sup> + 0.93<sup>22</sup> + … + 1) <em> <strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2500 \times {0.93^{24}} + 250\frac{{\left( {{{0.93}^{24}} - 1} \right)}}{{\left( {0.93 - 1} \right)}}">
<mo>=</mo>
<mn>2500</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>0.93</mn>
<mrow>
<mn>24</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>250</mn>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mn>0.93</mn>
</mrow>
<mrow>
<mn>24</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.93</mn>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>=3384 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> If recursive formula used, award <em><strong>M1</strong> </em>for <em>u<sub>n</sub></em> = 0.93 <em>u<sub>n</sub></em><sub>−1</sub> <strong>and</strong> <em>u</em><sub>0</sub> or <em>u</em><sub>1</sub> seen (can be awarded if seen in part (a)). Then award <em><strong>M1A1</strong> </em>for attempt to find <em>u</em><sub>24</sub> or <em>u</em><sub>25</sub> respectively (different term if other than 2500 used) (<em><strong>M1A0</strong></em> if incorrect term is being found) and <em><strong>A2</strong> </em>for correct answer.</p>
<p><strong>Note:</strong> Accept all answers that round to 3380.</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>The coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span> in the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{1}{x} + 5x} \right)^8}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> <mo>+</mo> <mn>5</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>8</mn> </msup> </mrow> </math></span> is equal to the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4}"> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </math></span> in the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a + 5x} \right)^7},{\text{ }}a \in \mathbb{R}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mn>5</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>7</mn> </msup> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>a</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>. 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>
<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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="^8{C_r}{\left( {\frac{1}{x}} \right)^{8 - r}}{(5x)^r} = {}^8Cr{\left( 5 \right)^r}{x^{2r - 8}}"> <msup> <mi></mi> <mn>8</mn> </msup> <mrow> <msub> <mi>C</mi> <mi>r</mi> </msub> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mn>8</mn> <mo>−</mo> <mi>r</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mi>r</mi> </msup> </mrow> <mo>=</mo> <msup> <mrow> </mrow> <mn>8</mn> </msup> <mi>C</mi> <mi>r</mi> <mrow> <msup> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> <mi>r</mi> </msup> </mrow> <mrow> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mi>r</mi> <mo>−</mo> <mn>8</mn> </mrow> </msup> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 5"> <mi>r</mi> <mo>=</mo> <mn>5</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="^8{C_5} \times {5^5}{ = ^7}{C_4}{a^3} \times {5^4}"> <msup> <mi></mi> <mn>8</mn> </msup> <mrow> <msub> <mi>C</mi> <mn>5</mn> </msub> </mrow> <mo>×</mo> <mrow> <msup> <mn>5</mn> <mn>5</mn> </msup> </mrow> <mrow> <msup> <mo>=</mo> <mn>7</mn> </msup> </mrow> <mrow> <msub> <mi>C</mi> <mn>4</mn> </msub> </mrow> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>5</mn> <mn>4</mn> </msup> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="56 \times 5 = 35{a^3}"> <mn>56</mn> <mo>×</mo> <mn>5</mn> <mo>=</mo> <mn>35</mn> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^3} = 8"> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>8</mn> </math></span> <strong><em>(A1)</em></strong></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> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to expand both binomials <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="175000{x^2}"> <mn>175000</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="21875{a^3}{x^4}"> <mn>21875</mn> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="175000 = 21875{a^3}"> <mn>175000</mn> <mo>=</mo> <mn>21875</mn> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^3} = 8"> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>8</mn> </math></span> <strong><em>(A1)</em></strong></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> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first three terms of the binomial expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><msup><mo>)</mo><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</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>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and the result from part (a), show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi></math> up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>x</mi></math> and the result from part (b), find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mtext> arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mi>t</mi><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mn>2</mn></msup></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"><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac></mstyle><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfenced><mrow><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfenced></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>24</mn></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>so the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi></math> up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Condone the absence of ‘…’ </p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mfrac><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mtext> arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup><mn>3</mn></mfrac></mstyle><mo>+</mo><mo>…</mo></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></mstyle></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>8</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>3</mn></mfrac></mstyle><mo>+</mo><mo>…</mo></mrow><mrow><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></mstyle></mrow></mfrac></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></mstyle></mrow></mfenced></mrow><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>12</mn></mfrac></mrow></mfenced></mstyle></mfrac></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><strong>Note:</strong> Condone missing ‘lim’ and errors in higher derivatives.<br>Do not award <em><strong>M1</strong></em> unless <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext></math>.</p>
<p> </p>
<p><em><strong>[3 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>A biased coin is weighted such that the probability, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, of obtaining a tail is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>6</mn></math>. The coin is tossed repeatedly and independently until a tail is obtained.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> be the event “obtaining the first tail on an even numbered toss”.</p>
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced></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><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>n</mi></msub></math> is the event “the first tail occurs on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>nd, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>th, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>th, …, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math>th toss”</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mtext>P</mtext><mfenced><msub><mi>E</mi><mi>n</mi></msub></mfenced></math> <strong>(A1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for deducing that either <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> head before a tail or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> heads before a tail or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> heads before a tail etc. is required. In other words, deduces <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math> heads before a tail.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>3</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>5</mn></msup><mo>×</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>+</mo><mo>…</mo></math> <strong>M1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to form an infinite geometric series.</p>
<p><strong>Note:</strong> Award <strong>A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>6</mn></mrow></mfenced></math>.</p>
<p> </p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>6</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></math> <strong>(M1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for using <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>6</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow><mrow><mn>1</mn><mo>-</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>286</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>1</mn></msub></math> be the event “tail occurs on the first toss”</p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub></menclose></mrow></mfenced><mtext>P</mtext><mfenced><msub><mi>T</mi><mn>1</mn></msub></mfenced><mo>+</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced><mtext>P</mtext><mfenced><mrow><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></mrow></mfenced></math> <strong>M1</strong></p>
<p>concludes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub></menclose></mrow></mfenced><mo>=</mo><mn>0</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced><mtext>P</mtext><mfenced><mrow><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></mrow></mfenced></math> <strong>R1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced><mo>=</mo><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo>'</mo></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mi>E</mi></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>A1</strong> for concluding: given that a tail is not obtained on the first toss, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo> </mo><menclose notation="left"><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></menclose></mrow></mfenced></math> is the probability that the first tail is obtained after a further odd number of tosses, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>E</mi><mo>'</mo></mrow></mfenced></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><msub><mi>T</mi><mn>1</mn></msub><mspace width="-0.2em"></mspace><mo>'</mo></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.4</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced><mo>=</mo><mn>0</mn><mo>.4</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mtext>P</mtext><mfenced><mi>E</mi></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p>attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mi>E</mi></mfenced></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>286</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>7</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>≠</mo><mn>1</mn></math>.</p>
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mi>z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow></mfrac></mfenced><mo>=</mo><mn>0</mn></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>1</mn><mo>+</mo><mi>z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfrac></math></p>
<p>attempt to use the complex conjugate of their denominator <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mi>z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi></mrow><mrow><mn>2</mn><mo>-</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>θ</mi></mrow></mfrac></mrow></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1</strong></em> for expanding the numerator and <em><strong>A1</strong></em> for a correct numerator. Condone either an incorrect denominator or the absence of a denominator.</p>
<p><br>using <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn></math> to simplify the numerator <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mfrac><mrow><mn>1</mn><mo>+</mo><mi>z</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow></mfrac></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>AG</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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3} = 8{\text{i}}">
<mrow>
<msup>
<mi>w</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w \in \mathbb{C}">
<mi>w</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>. Give your answers in Cartesian form.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One of the roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_1}">
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> satisfies the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Re}}\left( {{w_1}} \right) = 0">
<mrow>
<mtext>Re</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_1} = \frac{z}{{z - {\text{i}}}}">
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>, express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b{\text{i}}">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{Q}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3} = 8{\text{i}}">
<mrow>
<msup>
<mi>w</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p>writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8{\text{i}} = 8\left( {{\text{cos}}\left( {\frac{\pi }{2} + 2\pi k} \right) + {\text{i}}\,{\text{sin}}\left( {\frac{\pi }{2} + 2\pi k} \right)} \right)">
<mn>8</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for an attempt to find cube roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span> using modulus-argument form.</p>
<p>cube roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 2\left( {{\text{cos}}\left( {\frac{{\frac{\pi }{2} + 2\pi k}}{3}} \right) + {\text{i}}\,{\text{sin}}\left( {\frac{{\frac{\pi }{2} + 2\pi k}}{3}} \right)} \right)">
<mi>w</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>π</mi>
<mi>k</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>i.e. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \sqrt 3 + {\text{i,}}\,\, - \sqrt 3 + {\text{i,}}\,\, - 2{\text{i}}">
<mi>w</mi>
<mo>=</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong> </em>for all 3 correct, <em><strong>A1</strong></em> for 2 correct.</p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 1.73 + {\text{i}}">
<mi>w</mi>
<mo>=</mo>
<mn>1.73</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = - 1.73 + {\text{i}}">
<mi>w</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1.73</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3} + {\left( {2{\text{i}}} \right)^3} = 0">
<mrow>
<msup>
<mi>w</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {w + 2{\text{i}}} \right)\left( {{w^2} - 2w{\text{i}} - 4} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>w</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>w</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>w</mi>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \frac{{2{\text{i}} \pm \sqrt {12} }}{2}">
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mo>±</mo>
<msqrt>
<mn>12</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \sqrt 3 + {\text{i,}}\,\, - \sqrt 3 + {\text{i,}}\,\, - 2{\text{i}}">
<mi>w</mi>
<mo>=</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>+</mo>
<mrow>
<mtext>i,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong> </em>for all 3 correct, <em><strong>A1</strong></em> for 2 correct.</p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 1.73 + {\text{i}}">
<mi>w</mi>
<mo>=</mo>
<mn>1.73</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = - 1.73 + {\text{i}}">
<mi>w</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1.73</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w_1} = - 2{\text{i}}">
<mrow>
<msub>
<mi>w</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{z}{{z - {\text{i}}}} = - 2{\text{i}}">
<mfrac>
<mi>z</mi>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = - 2{\text{i}}\left( {z - {\text{i}}} \right)">
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>z</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z\left( {1 + 2{\text{i}}} \right) = - 2">
<mi>z</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{{ - 2}}{{1 + 2{\text{i}}}}">
<mi>z</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = - \frac{2}{5} + \frac{4}{5}{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>4</mn>
<mn>5</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - \frac{2}{5}{\text{,}}\,\,b = \frac{4}{5}">
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>5</mn>
</mfrac>
</math></span>.</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>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a derivative given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></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><mi>o</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>k</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is a positive constant.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, the population of a colony of ants, which has an initial value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn></math>.</p>
<p>The rate of change of the population can be modelled by the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow><mrow><mn>5</mn><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in days, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is the upper bound for the population.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math> the population of the colony has doubled in size from its initial value.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> in terms 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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</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>, giving your answer correct to four significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when the rate of change of the population is at its maximum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mi>b</mi><mi>x</mi><mo>=</mo><mn>1</mn></math> <em><strong> (A1)</strong></em></p>
<p>attempt to compare coefficients OR substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and solve <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></math></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>attempt to integrate their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mfrac><mn>1</mn><mi>k</mi></mfrac><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mfenced><mrow><mi>ln</mi><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mi>ln</mi><mfenced open="|" close="|"><mfrac><mi>x</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct term. Award <em><strong>A1A0</strong></em> for a correct answer without modulus signs. Condone the absence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to separate variables and integrate both sides <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>k</mi><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mn>1</mn><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> There are variations on this which should be accepted, such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>k</mi></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mn>5</mn><mi>k</mi></mrow></mfrac><mi>t</mi><mo>+</mo><mi>c</mi></math>. Subsequent marks for these variations should be awarded as appropriate.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>1200</mn></math> into an equation involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mn>1200</mn><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>5</mn><mo> </mo><mi>ln</mi><mfenced><mfrac><mn>1200</mn><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mn>1200</mn><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow><mrow><mn>1200</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac></mfenced><mo>=</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow><mrow><mn>1200</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mi>P</mi><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mi>t</mi><mo>+</mo><mi>c</mi></mrow><mn>5</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>P</mi><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfrac><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> <em><strong>A1</strong></em></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>1200</mn></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1200</mn><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfrac><mo>=</mo><mi>A</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>P</mi><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1200</mn><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></msup></mrow><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>attempt to rearrange and isolate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>k</mi><mo>-</mo><mn>1200</mn><mi>P</mi><mo>=</mo><mn>1200</mn><mi>k</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup><mo>-</mo><mn>1200</mn><mi>P</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>k</mi><msup><mtext>e</mtext><mrow><mi>-</mi><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo>-</mo><mn>1200</mn><mi>P</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo> </mo><mi mathvariant="normal">=</mi><mn>1200</mn><mi>k</mi><mo>-</mo><mn>1200</mn><mi>P</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><mi>P</mi></mfrac><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow><mrow><mn>1200</mn><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn><mo>+</mo><mn>1200</mn><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></mrow></mfenced><mo>=</mo><mn>1200</mn><mi>k</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>k</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo>-</mo><mn>1200</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfenced><mo>=</mo><mn>1200</mn><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>2400</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2400</mn><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2845</mn><mo>.</mo><mn>34</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2845</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)A0</strong></em> for any other value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> which rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2850</mn></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the maximum of the first derivative graph OR zero of the second derivative graph OR that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>k</mi><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1422</mn><mo>.</mo><mn>67</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>57814</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>58</mn></math> (days) <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any value which rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>6</mn></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = a{z^3} - 37{z^2} + 66z - 10"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>37</mn> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>66</mn> <mi>z</mi> <mo>−</mo> <mn>10</mn> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z \in \mathbb{C}"> <mi>z</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{Z}"> <mi>a</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<p>One of the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = 0"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + {\text{i}}"> <mn>3</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span>. 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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>one other root is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - {\text{i}}"> <mn>3</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>let third root be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p>considering sum or product of roots <em><strong>(M1)</strong></em></p>
<p>sum of roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6 + \alpha = \frac{{37}}{a}"> <mo>=</mo> <mn>6</mn> <mo>+</mo> <mi>α</mi> <mo>=</mo> <mfrac> <mrow> <mn>37</mn> </mrow> <mi>a</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>product of roots <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 10\alpha = \frac{{10}}{a}"> <mo>=</mo> <mn>10</mn> <mi>α</mi> <mo>=</mo> <mfrac> <mrow> <mn>10</mn> </mrow> <mi>a</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 6"> <mi>a</mi> <mo>=</mo> <mn>6</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>one other root is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - {\text{i}}"> <mn>3</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>quadratic factor will be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^2} - 6z + 10"> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>z</mi> <mo>+</mo> <mn>10</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = a{z^3} - 37{z^2} + 66z - 10 = \left( {{z^2} - 6z + 10} \right)\left( {az - 1} \right)"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <msup> <mi>z</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>37</mn> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>66</mn> <mi>z</mi> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>z</mi> <mo>+</mo> <mn>10</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mi>z</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p>comparing coefficients <em><strong>(M1)</strong></em></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 6"> <mi>a</mi> <mo>=</mo> <mn>6</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + {\text{i}}"> <mn>3</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right)"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {18 + 26{\text{i}}} \right) - 37\left( {8 + 6{\text{i}}} \right) + 66\left( {3 + {\text{i}}} \right) - 10 = 0"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mn>18</mn> <mo>+</mo> <mn>26</mn> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>37</mn> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mo>+</mo> <mn>6</mn> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>66</mn> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>equating real or imaginary parts or dividing <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18a - 296 + 198 - 10 = 0"> <mn>18</mn> <mi>a</mi> <mo>−</mo> <mn>296</mn> <mo>+</mo> <mn>198</mn> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="26a - 222 + 66 = 0"> <mn>26</mn> <mi>a</mi> <mo>−</mo> <mn>222</mn> <mo>+</mo> <mn>66</mn> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{10 - 66\left( {3 + {\text{i}}} \right) + 37\left( {8 + 6{\text{i}}} \right)}}{{18 + 26{\text{i}}}}"> <mfrac> <mrow> <mn>10</mn> <mo>−</mo> <mn>66</mn> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>37</mn> <mrow> <mo>(</mo> <mrow> <mn>8</mn> <mo>+</mo> <mn>6</mn> <mrow> <mtext>i</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>18</mn> <mo>+</mo> <mn>26</mn> <mrow> <mtext>i</mtext> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 6"> <mi>a</mi> <mo>=</mo> <mn>6</mn> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The population, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in years and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mo> </mo><mi>N</mi></math> are positive constants.</p>
<p>The constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> represents the maximum population of this species of marsupial that the island can sustain indefinitely.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> be the initial population of marsupials.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the population model, interpret the meaning of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the population of marsupials will increase at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>. Justify your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> in terms 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>N</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the logistic differential equation, show that its solution can be expressed in the form</p>
<p style="padding-left:150px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years, the population of marsupials is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math>. It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> for this population model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>rate of growth (change) of the (marsupial) population (with respect to time) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark] </strong></em></p>
<p><strong><br>Note:</strong> Do not accept growth (change) in the (marsupials) population per year.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts implicit differentiation on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mo>-</mo><mfrac><mrow><mi>k</mi><msup><mi>P</mi><mn>2</mn></msup></mrow><mi>N</mi></mfrac></math> be expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>k</mi><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mi>k</mi><mi>P</mi></mrow><mi>N</mi></mfrac><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>k</mi><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts implicit differentiation (product rule) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>k</mi><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>+</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mo>-</mo><mfenced><mfrac><mn>1</mn><mi>N</mi></mfrac></mfenced><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> into their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>+</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mo>-</mo><mfenced><mfrac><mn>1</mn><mi>N</mi></mfrac></mfenced><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mfrac><mi>P</mi><mi>N</mi></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></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>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>0</mn><mo>,</mo><mfrac><mi>N</mi><mn>2</mn></mfrac><mo>,</mo><mi>N</mi></math> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> only.</p>
<p>uses the second derivative to show that concavity changes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> or the first derivative to show a local maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> <em><strong>M1</strong></em><br><br><strong>EITHER</strong></p>
<p>a clearly labelled correct sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> corresponding to a local maximum point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>R1</strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p><br><strong>OR</strong></p>
<p>a correct and clearly labelled sign diagram (table) showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> corresponding to a local maximum point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mi>N</mi></mrow><mn>32</mn></mfrac><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>4</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mi>N</mi></mrow><mn>32</mn></mfrac><mfenced><mrow><mo><</mo><mn>0</mn></mrow></mfenced></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>N</mi></mrow><mn>4</mn></mfrac></math> showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> corresponds to a local maximum point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>R1</strong></em></p>
<p>so the population is increasing at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks] </strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mfrac><mi>N</mi><mn>2</mn></mfrac></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mstyle displaystyle="true"><mfrac><mi>N</mi><mn>2</mn></mfrac></mstyle><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>k</mi><mi>N</mi></mrow><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to separate variables <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p>attempts to write <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac></math> in partial fractions form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mi>P</mi></mfrac><mo>+</mo><mfrac><mi>B</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mfrac><mo>⇒</mo><mi>N</mi><mo>≡</mo><mi>A</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced><mo>+</mo><mi>B</mi><mi>P</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mfrac></mrow></mfenced><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math>. Absolute value signs are not required.</p>
<p> </p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>ln</mi><mo> </mo><msub><mi>P</mi><mn>0</mn></msub><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>-</mo><mi>ln</mi><mfenced><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mi>o</mi></msub></mrow></mfrac></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mstyle></mfrac></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to separate variables <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p>attempts to write <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac></math> in partial fractions form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mi>P</mi></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfrac><mo>⇒</mo><mn>1</mn><mo>≡</mo><mi>A</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>+</mo><mi>B</mi><mi>P</mi></math> </p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>N</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac></math></strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>N</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></strong></em> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math>. Absolute value signs are not required.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mi>P</mi><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfrac></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><mi>P</mi></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mo>+</mo><mi>C</mi></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><mi>P</mi></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>-</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mfrac><mstyle displaystyle="true"><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>lets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mfrac><mn>1</mn><mi>P</mi></mfrac></math> and forms <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><msup><mi>P</mi><mn>2</mn></msup></mfrac><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p>multiplies both sides of the differential equation by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><msup><mi>P</mi><mn>2</mn></msup></mfrac></math> and makes the above substitutions <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><msup><mi>P</mi><mn>2</mn></msup></mfrac><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>P</mi></mfrac></mrow></mfenced><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>-</mo><mi>u</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>k</mi><mi>u</mi><mo>=</mo><mfrac><mi>k</mi><mi>N</mi></mfrac></math> (linear first-order DE)<em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>IF</mtext><mo>=</mo><msup><mtext>e</mtext><mrow><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></mrow></msup><mo>=</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mo>⇒</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>k</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mi>u</mi><mo>=</mo><mfrac><mi>k</mi><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup></math><em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mfenced><mrow><mi>u</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><mfrac><mi>k</mi><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mi>P</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>P</mi><mn>0</mn></msub></mfrac></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>P</mi><mn>0</mn></msub></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow><mrow><mi>N</mi><mi>P</mi></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mtext>=</mtext><mfenced><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><msub><mi>P</mi><mn>0</mn></msub></mfrac></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mi>k</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>3</mn><mfenced><mfrac><mrow><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub><mo>-</mo><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><mn>9</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>220</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>10</mn></mfrac><mi>ln</mi><mo> </mo><mn>9</mn><mo>,</mo><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>ln</mi><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>An extremely tricky question even for the strong candidates. Many struggled to understand what was expected in parts (b) and (c). As the question was set all with pronumerals instead of numbers many candidates found it challenging, thrown at deep water for parts (b), (c) and (e). It definitely was the question to show their skills for the Level 7 candidates provided that they did not run out of time.</p>
<p>Part (a) Very well answered, mostly correctly referring to the rate of change. Some candidates did not gain this mark because their sentence did not include the reference to the rate of change. Worded explanations continue being problematic to many candidates.</p>
<p>Part (b) Many candidates were confused how to approach this question and did not realise that they<br>needed to differentiate implicitly. Some tried but with errors, some did not fully show what was required.</p>
<p>Part (c) Most candidates started with equating the second derivative to zero. Most gave the answer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>omitting the other two possibilities. Most stopped here. Only a small number of candidates provided the correct mathematical argument to show it is a local maximum.</p>
<p>Part (d) Well done by those candidates who got that far. Most got the correct answer, sometimes not fully simplified.</p>
<p>Part (e) Most candidates separated the variables, but some were not able to do much more. Some candidates knew to resolve into partial fractions and attempted to do so, mainly successfully. Then they integrated, again, mainly successfully and continued to substitute the initial condition and manipulate the equation accordingly.</p>
<p>Part (f) Algebraic manipulation of the logarithmic expression was too much for some candidates with a common error of 0.33 given as the answer. The strong candidates provided the correct exact or rounded value.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p>In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.</p>
<p>Find the number of different orders in which these 18 examination papers may be taken.</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><strong>METHOD 1</strong></p>
<p>consideration of all papers</p>
<p>all papers may be sat in 18! ways <strong><em>A1</em></strong></p>
<p>number of ways of positioning “pairs” of science subjects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3! \times 17!">
<mo>=</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>17</mn>
<mo>!</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>but this includes two copies of each “triple” <strong><em>(R1)</em></strong></p>
<p>number of ways of positioning “triplets” of science subjects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3! \times 16!">
<mo>=</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>16</mn>
<mo>!</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>hence number of arrangements is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18! - 3! \times 17! + 3! \times 16!">
<mn>18</mn>
<mo>!</mo>
<mo>−</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>17</mn>
<mo>!</mo>
<mo>+</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>16</mn>
<mo>!</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( = 4.39 \times {10^{15}})">
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>4.39</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mn>15</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong>METHOD 2</strong></p>
<p>consideration of all the non-science papers <strong><em>(M1)</em></strong></p>
<p>hence all non-science papers can be sat in 15! ways <strong><em>A1</em></strong></p>
<p>there are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="16 \times 15 \times 14{\text{ }}( = 3360)">
<mn>16</mn>
<mo>×</mo>
<mn>15</mn>
<mo>×</mo>
<mn>14</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>3360</mn>
<mo stretchy="false">)</mo>
</math></span> ways of positioning the three science papers <strong><em>(M1)A1</em></strong></p>
<p>hence the number of arrangements is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="16 \times 15 \times 14 \times 15!{\text{ (}} = 4.39 \times {10^{15}})">
<mn>16</mn>
<mo>×</mo>
<mn>15</mn>
<mo>×</mo>
<mn>14</mn>
<mo>×</mo>
<mn>15</mn>
<mo>!</mo>
<mrow>
<mtext> (</mtext>
</mrow>
<mo>=</mo>
<mn>4.39</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mn>15</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>consideration of all papers</p>
<p>all papers may be sat in 18! ways <strong><em>A1</em></strong></p>
<p>number of ways of positioning exactly two science subjects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3! \times 15! \times 16 \times 15">
<mo>=</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>15</mn>
<mo>!</mo>
<mo>×</mo>
<mn>16</mn>
<mo>×</mo>
<mn>15</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p>number of ways of positioning “triplets” of science subjects <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3! \times 16!">
<mo>=</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>16</mn>
<mo>!</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>hence number of arrangements is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="18! - 3! \times 16! - 3! \times 15! \times 16 \times 15">
<mn>18</mn>
<mo>!</mo>
<mo>−</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>16</mn>
<mo>!</mo>
<mo>−</mo>
<mn>3</mn>
<mo>!</mo>
<mo>×</mo>
<mn>15</mn>
<mo>!</mo>
<mo>×</mo>
<mn>16</mn>
<mo>×</mo>
<mn>15</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="( = 4.39 \times {10^{15}})">
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>4.39</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mn>15</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) \equiv {z^4} - 6{z^3} - 2{z^2} + 58z - 51,\,\,z \in \mathbb{C}">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>≡<!-- ≡ --></mo>
<mrow>
<msup>
<mi>z</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>58</mn>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mn>51</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>58</mn> <mi>x</mi> <mo>−</mo> <mn>51</mn> </math></span>, stating clearly the coordinates of any maximum and minimum points and intersections with axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, state the condition on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> such that all roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = k"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> </math></span> are real.</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><img src="data:image/png;base64,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"></p>
<p>shape <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis intercepts at (−3, 0), (1, 0) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis intercept at (0, −51) <em><strong> A1A1</strong></em></p>
<p>minimum points at (−1.62, −118) and (3.72, 19.7) <em><strong> A1A1</strong></em></p>
<p>maximum point at (2.40, 26.9) <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Coordinates may be seen on the graph or elsewhere.</p>
<p><strong>Note</strong>: Accept −3, 1 and −51 marked on the axes.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>from graph, 19.7 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> ≤ 26.9 <em><strong> A1A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> for correct endpoints and <em><strong>A1</strong></em> for correct inequalities.</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">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>Use mathematical induction to prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^n} > 1 - na">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msup>
</mrow>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mi>n</mi>
<mi>a</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {n\,{\text{:}}\,n \in {\mathbb{Z}^ + },\,n \geqslant 2} \right\}">
<mrow>
<mo>{</mo>
<mrow>
<mi>n</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</mrow>
<mo>}</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < a < 1">
<mn>0</mn>
<mo><</mo>
<mi>a</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{P}}_n}">
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mi>n</mi>
</msub>
</mrow>
</math></span> be the statement: <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^n} > 1 - na">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msup>
</mrow>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mi>n</mi>
<mi>a</mi>
</math></span> for some <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{n \in {\mathbb{Z}^ + },\,n \geqslant 2}">
<mrow>
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < a < 1">
<mn>0</mn>
<mo><</mo>
<mi>a</mi>
<mo><</mo>
<mn>1</mn>
</math></span> consider the case <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 2{\text{:}}\,\,\,{\left( {1 - a} \right)^2} = 1 - 2a + {a^2}">
<mi>n</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" > 1 - 2a">
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> because <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} < 0">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo><</mo>
<mn>0</mn>
</math></span>. Therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{P}}_2}">
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
</math></span> is true <em><strong>R1</strong></em></p>
<p>assume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{P}}_n}">
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mi>n</mi>
</msub>
</mrow>
</math></span> is true for some <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = k">
<mi>n</mi>
<mo>=</mo>
<mi>k</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^k} > 1 - ka">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>k</mi>
</msup>
</mrow>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mi>k</mi>
<mi>a</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Assumption of truth must be present. Following marks are not dependent on this <em><strong>M1</strong></em>.</p>
<p><strong>EITHER</strong></p>
<p>consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^{k + 1}} = \left( {1 - a} \right){\left( {1 - a} \right)^k}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>k</mi>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" > 1 - \left( {k + 1} \right)a + k{a^2}">
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>a</mi>
<mo>+</mo>
<mi>k</mi>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" > 1 - \left( {k + 1} \right)a \Rightarrow {{\text{P}}_{k + 1}}">
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>a</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> is true (<strong>as</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k{a^2} > 0">
<mi>k</mi>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
</math></span>) <em><strong>R1</strong></em></p>
<p><strong>OR</strong></p>
<p>multiply both sides by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - a} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (which is positive) <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^{k + 1}} > \left( {1 - ka} \right)\left( {1 - a} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>k</mi>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^{k + 1}} > 1 - \left( {k + 1} \right)a + k{a^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>a</mi>
<mo>+</mo>
<mi>k</mi>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - a} \right)^{k + 1}} > 1 - \left( {k + 1} \right)a \Rightarrow {{\text{P}}_{k + 1}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>a</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> is true (<strong>as</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k{a^2} > 0">
<mi>k</mi>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
</math></span>) <em><strong>R1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{P}}_2}">
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
</math></span> is true <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{P}}_k}">
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mi>k</mi>
</msub>
</mrow>
</math></span> is true <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {{\text{P}}_{k + 1}}">
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> is true so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{P}}_n}">
<mrow>
<msub>
<mrow>
<mtext>P</mtext>
</mrow>
<mi>n</mi>
</msub>
</mrow>
</math></span> true for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 2">
<mi>n</mi>
<mo>></mo>
<mn>2</mn>
</math></span> (or equivalent) <em><strong>R1</strong></em></p>
<p><strong>Note</strong>: Only award the last <em><strong>R1</strong> </em>if at least four of the previous marks are gained including the <em><strong>A1</strong></em>.</p>
<p><em><strong>[7 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 3rd term of an arithmetic sequence is 1407 and the 10th term is 1183.</p>
</div>
<div class="question">
<p>Calculate the number of positive terms in the sequence.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>1471 + (<em>n</em> − 1)(−32) > 0 <em><strong>(M1)</strong></em></p>
<p>⇒ <em>n</em> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1471}}{{32}} + 1">
<mfrac>
<mrow>
<mn>1471</mn>
</mrow>
<mrow>
<mn>32</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mn>1</mn>
</math></span></p>
<p><em>n</em> < 46.96… <em><strong>(A1)</strong></em></p>
<p>so 46 positive terms <em><strong> A1</strong></em></p>
<p><em><strong>[3 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">
<p>Consider the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {2 + x} \right)^n}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 3"> <mi>n</mi> <mo>⩾</mo> <mn>3</mn> </math></span> and <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>
<p>The coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}"> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> </math></span> is four times the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span>. 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>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempt to find coefficients in binomial expansion <em><strong>(M1)</strong></em></p>
<p>coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}\,{\text{:}}\,\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) \times {2^{n - 2}}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>:</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></span> ; coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}\,{\text{:}}\,\left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right) \times {2^{n - 3}}"> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>:</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone terms given rather than coefficients. Terms may be seen in an equation such as that below.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right) \times {2^{n - 3}} = 4\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right) \times {2^{n - 2}}"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mo>=</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>attempt to solve equation using GDC or algebraically <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} n \\ 3 \end{array}} \right) = 8\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>8</mn> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n{\text{!}}}}{{3{\text{!}}\left( {n - 3} \right){\text{!}}}} = \frac{{8n{\text{!}}}}{{2{\text{!}}\left( {n - 2} \right){\text{!}}}}"> <mfrac> <mrow> <mi>n</mi> <mrow> <mtext>!</mtext> </mrow> </mrow> <mrow> <mn>3</mn> <mrow> <mtext>!</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>!</mtext> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>8</mn> <mi>n</mi> <mrow> <mtext>!</mtext> </mrow> </mrow> <mrow> <mn>2</mn> <mrow> <mtext>!</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</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="\frac{1}{3} = \frac{8}{{n - 2}}"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mo>=</mo> <mfrac> <mn>8</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = 26"> <mi>n</mi> <mo>=</mo> <mn>26</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>