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<h2>HL Paper 2</h2><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" 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>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2\,{\text{ln}}\,x + 1}}{{x - 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 3.</p>
</div>
<div class="specification">
<p>Draw a set of axes showing <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> values between −3 and 3. On these axes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the coordinates of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </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>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</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>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f^{ - 1}}\left( x \right)"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>finding turning point of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> or finding root of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f''\left( x \right)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</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="x = 0.899"> <mi>x</mi> <mo>=</mo> <mn>0.899</mn> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {0.899048 \ldots } \right) = - 0.375"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mn>0.899048</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>0.375</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>(0.899, −0.375)</p>
<p><strong>Note:</strong> Do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.9"> <mi>x</mi> <mo>=</mo> <mn>0.9</mn> </math></span>. Accept <em>y</em>-coordinates rounding to −0.37 or −0.375 but not −0.38.<br> </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><img src="data:image/png;base64,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"></p>
<p>smooth curve over the correct domain which does not cross the <em>y</em>-axis</p>
<p>and is concave down for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> > 1 <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>-intercept at 0.607 <em><strong>A1</strong></em></p>
<p>equations of asymptotes given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 3 (the latter must be drawn) <em><strong>A1A1</strong></em><br> </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><img 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"></p>
<p>attempt to reflect graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p>smooth curve over the correct domain which does not cross the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and is concave down for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> > 1 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept at 0.607 <em><strong>A1</strong></em></p>
<p>equations of asymptotes given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 3 (the latter must be drawn) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> For <em><strong>FT</strong></em> from (i) to (ii) award max <em><strong>M1A0A1A0</strong></em>.</p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = x"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </math></span> to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0.372 <em><strong>(M1)</strong></em><em><strong>A1</strong></em></p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 0.372 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>FT</strong> </em>marks.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">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">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<p>The region enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to form a solid of revolution.</p>
</div>
<div class="specification">
<p>Pedro wants to make a small bowl with a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math> based on the result from part (a). Pedro’s design is shown in the following diagrams.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The vertical height of the bowl, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BO</mtext></math>, is measured along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The radius of the bowl’s top is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></math> and the radius of the bowl’s base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>. All lengths are measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cm</mtext></math>.</p>
</div>
<div class="specification">
<p>For design purposes, Pedro investigates how the cross-sectional radius of the bowl changes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the solid formed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi></mrow><mn>34</mn></mfrac></math> cubic units.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> that satisfies the requirements of Pedro’s design.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of a suitable derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, find where the cross-sectional radius of the bowl is decreasing most rapidly.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the cross-sectional radius of the bowl at this point.</p>
<div class="marks">[2]</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>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mi>a</mi><mi>b</mi></munderover><msup><mfenced><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></munderover><msup><mfenced><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mi>V</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></munderover><mfrac><msup><mtext>e</mtext><mi>x</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mn>2</mn></msup></mfrac><mo>d</mo><mi>x</mi></mrow></mfenced></math></p>
<p><br><strong>EITHER</strong></p>
<p>applying integration by recognition <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></mrow></mfenced><mn>0</mn><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></msubsup></math> <em><strong>A3</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>attempt to express the integral in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> <em><strong>(M1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>2</mn></math> and when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>17</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>2</mn><mn>17</mn></munderover><mfrac><mn>1</mn><msup><mi>u</mi><mn>2</mn></msup></mfrac><mo>d</mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mi>u</mi></mfrac></mrow></mfenced><mn>2</mn><mn>17</mn></msubsup></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>u</mi><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>attempt to express the integral in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> <em><strong>(M1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>1</mn></math> and when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>16</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>1</mn><mn>16</mn></munderover><mfrac><mn>1</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>u</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo>d</mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>u</mi></mrow></mfrac></mrow></mfenced><mn>1</mn><mn>16</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent working with indefinite integrals and original limits for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>17</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so the volume of the solid formed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi></mrow><mn>34</mn></mfrac></math> cubic units <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)(A0)(M0)(A0)(A0)(A1)</strong></em> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>15</mn><mn>34</mn></mfrac></math> is obtained from GDC</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>a valid algebraic or graphical attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>300</mn><mo>×</mo><mn>34</mn></mrow><mrow><mn>15</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>14</mn><mo>.</mo><mn>7</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mn>680</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Candidates may use their GDC numerical solve feature.</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>attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mi>k</mi><mn>2</mn></mfrac></math></p>
<p>with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>14</mn><mo>.</mo><mn>712</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mn>680</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext><mo>=</mo><mn>7</mn><mo>.</mo><mn>36</mn><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext><mo>=</mo><mi>f</mi><mfenced><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><mi>k</mi></mrow><mn>17</mn></mfrac></math></p>
<p>with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>14</mn><mo>.</mo><mn>712</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mn>680</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext><mo>=</mo><mn>3</mn><mo>.</mo><mn>46</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>8</mn><mn>17</mn></mfrac><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><mfrac><mrow><mn>8</mn><msqrt><mn>10</mn></msqrt></mrow><msqrt><mn>17</mn><mi mathvariant="normal">π</mi></msqrt></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">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>recognising to graph <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to use quotient rule or product rule differentiation. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></math></p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"><br>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> graph decreasing to the local minimum <em><strong>A1</strong></em></p>
<p>before increasing towards the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>recognising to graph <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo mathvariant="italic">=</mo><mi>f</mi><mo mathvariant="italic">''</mo><mfenced><mi mathvariant="italic">x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempting to use quotient rule or product rule differentiation. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>-</mo><mn>6</mn><msup><mtext>e</mtext><mi>x</mi></msup><mi>+1</mi></mrow></mfenced></mrow><mrow><mn>4</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mn>3</mn></msup></mrow></mfrac></math></p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" 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">for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>, graph increasing towards and beyond the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <em><strong>A1</strong></em></p>
<p>recognising <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> for maximum rate <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>76</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note</strong>: Only award <em><strong>A</strong> </em>marks if either graph is seen.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>1</mn><mo>.</mo><mn>76</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>the cross-sectional radius at this point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>20</mn><mo> </mo><mfenced><msqrt><mfrac><mn>85</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mfenced><mo> </mo><mfenced><mtext>cm</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.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">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>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has a probability density function given by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>The median of this distribution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mfenced open="|" close="|"><mrow><mi>X</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo>≤</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>m</mi></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo> </mo><mtext>arccos</mtext><mo> </mo><mi>m</mi><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>m</mi><mn>2</mn></msup></msqrt><mo>-</mo><mfenced><mrow><mn>0</mn><mo>-</mo><msqrt><mn>1</mn></msqrt></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>360</mn></math> <em><strong>A1</strong></em></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>METHOD 1</strong></p>
<p>attempts to find at least one endpoint (limit) both in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> (or their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>m</mi><mo>+</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>-</mo><mi>a</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mi>x</mi><mo> </mo><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>-</mo><mi>a</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>+</mo><mi>a</mi></mrow></msubsup></math></p>
<p>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> The above <em><strong>(M1)</strong></em> is dependent on the first <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><menclose notation="left"><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><menclose notation="left"><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></menclose></menclose></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><menclose notation="left"><mi>x</mi><mo>-</mo><mi>m</mi><menclose notation="left"><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></menclose></menclose></math>.</p>
<p><br>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><strong>EITHER </strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>+</mo><mi>m</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>2</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo>-</mo><mi>a</mi></mrow><mrow><mn>2</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p><strong><br>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mi>a</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><br><strong>THEN</strong></p>
<p>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> The above <em><strong>(M1)</strong></em> is dependent on the first <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</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="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form a solid of revolution that is used to model a water container.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the container is empty. Water is then added to the container at a constant rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, clearly indicating the coordinates of the endpoints.</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>Show that the inverse function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain and range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</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>Show that the volume, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup></math>, of water in the container when it is filled to a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></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>Hence, determine the maximum volume of the container.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time it takes to fill the container to its maximum volume.</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>Find the rate of change of the height of the water when the container is filled to half its maximum volume.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</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>correct shape (concave down) within the given domain <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mfenced><mrow><mo>=</mo><mfenced><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>.</mo><mn>73</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The coordinates of endpoints may be seen on the graph or marked on the axes.</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>interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> (seen anywhere) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><msqrt><mn>3</mn></msqrt></math> OR domain <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo>,</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mfenced><mrow><mo>=</mo><mfenced open="[" close="]"><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>.</mo><mn>73</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>≤</mo><mn>2</mn></math> OR range <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>1</mn><mo>,</mo><mn>2</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">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math> into the correct volume formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>π</mi><munderover><mo>∫</mo><mn>0</mn><mi>h</mi></munderover><msup><mfenced><msqrt><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mfenced><mn>2</mn></msup><mo>d</mo><mi>y</mi><mo> </mo><mfenced><mrow><mo>=</mo><mi>π</mi><munderover><mo>∫</mo><mn>0</mn><mi>h</mi></munderover><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>π</mi><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mi>y</mi></mrow></mfenced><mn>0</mn><mi>h</mi></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Award marks as appropriate for correct work using a different variable e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><munderover><mo>∫</mo><mn>0</mn><mi>h</mi></munderover><msup><mfenced><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>732</mn><mo>…</mo></mrow></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>8828</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>9</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>time <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>10</mn><mo>.</mo><mn>8828</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>27</mn><mo>.</mo><mn>207</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>27</mn><mo>.</mo><mn>2</mn><mfenced><mrow><mo>=</mo><mn>5</mn><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced><mfenced><mi>s</mi></mfenced></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>attempt to find the height of the tank when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>4414</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced><mo>=</mo><mn>5</mn><mo>.</mo><mn>4414</mn><mo>…</mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>1818</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempt to use the chain rule or differentiate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>V</mi></mrow></mfrac><mo>×</mo><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">π</mi><mfenced><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>×</mo><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p>attempt to substitute <strong>their </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow><mrow><mi mathvariant="normal">π</mi><mfenced><mrow><mn>1</mn><mo>.</mo><mn>1818</mn><msup><mo>…</mo><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>053124</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0531</mn><mo> </mo><mfenced><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 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;">
<p>Part a) was generally well done, with the most common errors being to use an incorrect domain or not to give the coordinates of the endpoints. Some graphs appeared to be straight lines; some candidates drew sketches which were too small which made it more difficult for them to show the curvature.</p>
<p>Most candidates were able to show the steps to find an inverse function in part b), although occasionally a candidate did not explicitly swop the <em>x</em> and <em>y</em> variables before writing down the inverse function, which was given in the question. Many candidates struggled to identify the domain and range of the inverse, despite having a correct graph.</p>
<p>Part c) required a rotation around the <em>y</em>-axis, but a number of candidates attempted to rotate around the <em>x</em>-axis or failed to include limits. In the same vein, many substituted 2 into the formula instead of the square root of 3 when answering the second part. Many subsequently gained follow through marks on part d).</p>
<p>There were a number of good attempts at related rates in part e), with the majority differentiating <em>V</em> with respect to <em>t</em>, using implicit differentiation. However, many did not find the value of <em>h</em> which corresponded to halving the volume, and a number did not differentiate with respect to <em>t</em>, only with respect to <em>h</em>.</p>
<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.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">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</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>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</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="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</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">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</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>3, −3 <em><strong>A1</strong></em><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>stretch parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis (with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis invariant), scale factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>translation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 0.003} \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.003</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (shift to the left by 0.003) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Can be done in either order.</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><img src="data:image/png;base64,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"></p>
<p>correct shape over correct domain with correct endpoints <em><strong>A1</strong></em><br>first maximum at (0.0035, 4.76) <em><strong>A1</strong></em><br>first minimum at (0.0085, −1.24) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> ≥ 3 between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.0016762 and 0.0053238 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.011676 and 0.015324 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for either interval.</p>
<p>= 0.00730 <em><strong>A1</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}} = \frac{1}{{0.007}}\int_0^{0.007} {6\,{\text{sin}}\left( {100\pi t} \right)} {\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right){\text{d}}t">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</mfrac>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</msubsup>
<mrow>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 2.87 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>in each cycle the area under the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis is smaller than area above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis <em><strong>R1</strong></em></p>
<p>the curve begins with the positive part of the cycle <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{{4.76 - \left( { - 1.24} \right)}}{2}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</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="a = 3.00">
<mi>a</mi>
<mo>=</mo>
<mn>3.00</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = \frac{{4.76 + \left( { - 1.24} \right)}}{2}">
<mi>d</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1.76">
<mi>d</mi>
<mo>=</mo>
<mn>1.76</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{2\pi }}{{0.01}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>0.01</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 628\left( { = 200\pi } \right)">
<mi>b</mi>
<mo>=</mo>
<mn>628</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>200</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.0035 - \frac{{0.01}}{4}">
<mi>c</mi>
<mo>=</mo>
<mn>0.0035</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>0.01</mn>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.00100">
<mi>c</mi>
<mo>=</mo>
<mn>0.00100</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">g.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, OQ = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> and OR = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, such that 0 < <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">
<mi>q</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> < 10.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
</math></span> be the angle APO, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _q}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
</math></span> be the angle AQO and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> be the angle ARO.</p>
</div>
<div class="specification">
<p>Consider the case when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\theta _q} + {\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> and QR = 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{{q^2} + q - 1}}{{2q + 1}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> as a function of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>, determine the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> for which there are possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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>use of tan <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,{\theta _p} = \frac{1}{p}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\text{arctan}}\left( {\frac{1}{p}} \right)"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>AP <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {{p^2} + 1} "> <mo>=</mo> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </math></span> <em><strong>(A1)</strong></em></p>
<p>use of sin, cos, sine rule or cosine rule using the correct length of AP <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\text{arcsin}}\left( {\frac{1}{{\sqrt {{p^2} + 1} }}} \right)"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\text{arccos}}\left( {\frac{p}{{\sqrt {{p^2} + 1} }}} \right)"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>p</mi> <mrow> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <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>QR = 1 ⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = q + 1"> <mi>r</mi> <mo>=</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> This may be seen anywhere.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,{\theta _p} = {\text{tan}}\left( {{\theta _q} + {\theta _r}} \right)"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>θ</mi> <mi>q</mi> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>θ</mi> <mi>r</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>attempt to use compound angle formula for tan <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,{\theta _p} = \frac{{{\text{tan}}\,{\theta _q} + {\text{tan}}\,{\theta _r}}}{{1 - {\text{tan}}\,{\theta _q}\,{\text{tan}}\,{\theta _r}}}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>q</mi> </msub> </mrow> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>r</mi> </msub> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>q</mi> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>r</mi> </msub> </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="\frac{1}{p} = \frac{{\frac{1}{q} + \frac{1}{r}}}{{1 - \left( {\frac{1}{q}} \right)\left( {\frac{1}{r}} \right)}}"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>r</mi> </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="\frac{1}{p} = \frac{{\frac{1}{q} + \frac{1}{{q + 1}}}}{{1 - \left( {\frac{1}{q}} \right)\left( {\frac{1}{{q + 1}}} \right)}}"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{1 - \left( {\frac{1}{q}} \right)\left( {\frac{1}{{q + 1}}} \right)}}{{\left( {\frac{1}{q}} \right) + \left( {\frac{1}{{q + 1}}} \right)}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </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="\frac{1}{p} = \frac{{q + q + 1}}{{q\left( {q + 1} \right) - 1}}"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>q</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for multiplying top and bottom by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{q\left( {q + 1} \right)}"> <mrow> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{{q^2} + q - 1}}{{2q + 1}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p> </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><img src="data:image/png;base64,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"></p>
<p>increasing function with positive <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>-intercept <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept curves which extend beyond the domain shown above.</p>
<p> </p>
<p>(0.618 <) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> < 9 <em><strong>(A1)</strong></em></p>
<p>⇒ range is (0 <) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> < 4.68 <strong>(A1)</strong></p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> < 4.68 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \,{\text{cot}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</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>Hence, or otherwise, show that the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span>.</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>Using the results from parts (b) and (c) find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b\sqrt 3 ">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</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{Z}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>stating the relationship between <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cot ">
<mi>cot</mi>
</math></span></span> and <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan ">
<mi>tan</mi>
</math></span></span> and stating the identity for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2\theta ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{1}{{{\text{tan}}\,2\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2\theta = \frac{{2\,{\text{tan}}\,\theta }}{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <em><strong>AG</strong></em></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><em><strong>METHOD 1</strong></em></p>
<p>attempting to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and using the result from (a) <em><strong>M1</strong></em></p>
<p>LHS = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}\,\theta + 2\,{\text{tan}}\,\theta \left( {\frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}} \right) - 1">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}\,\theta + 1 - {\text{ta}}{{\text{n}}^2}\,\theta - 1 = 0">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>+</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>(= RHS) <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfies the equation <em><strong>AG</strong></em></p>
<p>attempting to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \,{\text{cot}}\,\theta ">
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and using the result from (a) <em><strong>M1</strong></em></p>
<p>LHS = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{t}}^2}\,\theta - 2\,{\text{cot}}\,\theta \left( {\frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}} \right) - 1">
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>t</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{{\text{ta}}{{\text{n}}^2}\,\theta }} - \left( {\frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{\,{\text{ta}}{{\text{n}}^2}\,\theta }}} \right) - 1">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \frac{1}{{{\text{ta}}{{\text{n}}^2}\,\theta }} - \frac{1}{{{\text{ta}}{{\text{n}}^2}\,\theta }} + 1 - 1 = 0">
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>1</mn>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>(= RHS) <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \,{\text{cot}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfies the equation <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = {\text{tan}}\,\theta ">
<mi>α</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta = - \,{\text{cot}}\,\theta ">
<mi>β</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span></p>
<p>attempting to find the sum of roots <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha + \beta = {\text{tan}}\,\theta - \frac{1}{{{\text{tan}}\,\theta }}">
<mi>α</mi>
<mo>+</mo>
<mi>β</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{ta}}{{\text{n}}^2}\,\theta - 1}}{{{\text{tan}}\,\theta }}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"> </span><em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><strong>A1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2\,{\text{cot}}\,2\theta ">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</math></span> (from part (a)) <em><strong>A1</strong></em></p>
<p>attempting to find the product of roots <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha \beta = {\text{tan}}\,\theta \times \left( { - \,{\text{cot}}\,\theta } \right)">
<mi>α</mi>
<mi>β</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>= −1 <em><strong>A1</strong></em></p>
<p>the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and the constant term in the quadratic are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2\,{\text{cot}}\,2\theta }">
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</math></span> and −1 respectively <em><strong>R1</strong></em></p>
<p>hence the two roots are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = {\text{tan}}\,\theta ">
<mi>α</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta = - \,{\text{cot}}\,\theta ">
<mi>β</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\frac{\pi }{{12}}">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - {\text{cot}}\frac{\pi }{{12}}">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> are roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\frac{\pi }{{6}}} \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>6</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>if only <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\frac{\pi }{{12}}">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> is stated as a root of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\frac{\pi }{{6}}} \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>6</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 2\sqrt 3 x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to solve <strong>their</strong> quadratic equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \sqrt 3 \pm 2">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>±</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} > 0">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>></mo>
<mn>0</mn>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{cot}}\frac{\pi }{{12}} < 0">
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo><</mo>
<mn>0</mn>
</math></span>) <em><strong>R1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>attempting to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{{12}}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> into the identity for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2\theta ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{6} = \frac{{2\,{\text{tan}}\frac{\pi }{{12}}}}{{1 - {\text{ta}}{{\text{n}}^2}\frac{\pi }{{12}}}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}\frac{\pi }{{12}} + 2\sqrt 3 \,{\text{tan}}\frac{\pi }{{12}} - 1 = 0">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to solve <strong>their </strong>quadratic equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = - \sqrt 3 \pm 2">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>±</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} > 0">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>></mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <em><strong>AG</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> is the sum of the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\frac{\pi }{{12}}} \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}} = - 2\,{\text{cot}}\frac{\pi }{{12}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2}}{{2 - \sqrt 3 }}">
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to rationalise <strong>their</strong> denominator <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 4 - 2\sqrt 3 ">
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an even function.</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 considering limits, show that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a horizontal asymptote and state its equation.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and the result <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfenced open="|" close="|"><mi>x</mi></mfenced></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>.</p>
<p> </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>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, justifying your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</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>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>a sketch graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> with line symmetry in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis indicated <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is an even function. <em><strong>AG</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>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>→</mo><mtext>arcsin</mtext><mo> </mo><mn>1</mn><mfenced><mrow><mo>→</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so the horizontal asymptote is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></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>attempting to use the quotient rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><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"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempting to use the chain rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arcsin</mtext><mo> </mo><mi>u</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>u</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mi>u</mi><mn>2</mn></msup></msqrt></mfrac></math></p>
<p><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><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mstyle displaystyle="true"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mn>1</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mn>1</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 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>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p>
<p><br><strong>EITHER</strong></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo><</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>R1</strong></em> for stating that in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>, the numerator is negative, and the denominator is positive.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Do not accept a graphical solution</p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><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"><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⇒</mo><msup><mi>y</mi><mn>2</mn></msup><mo> </mo><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p>domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math> and so the range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>≥</mo><mn>0</mn></math></p>
<p>hence the positive root is taken (or the negative root is rejected) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong></em> is dependent on the above<em><strong> A1</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><msqrt><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow></mfrac></msqrt></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The final <em><strong>A1</strong></em> is not dependent on <em><strong>R1</strong></em> mark.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>≤</mo><mi>x</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept correct alternative notations, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⌊</mo><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>,</mo><mo> </mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>⌊</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⌊</mo><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>,</mo><mo> </mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>)</mo></mstyle></math>.<br>Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>57</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>57</mn><mo>[</mo></math> if correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> s.f.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:<em> A1</em></strong> for correct domain and correct range and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn></math><br><em><strong> A1</strong></em> for asymptotic behaviour <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math><br><em><strong> A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math><br> Coordinates are not required. <br> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>57</mn></math> or other inexact values.</p>
<p> </p>
<p><em><strong>[3 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;">
[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">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="specification">
<p>The height of water, in metres, in Dungeness harbour is modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo> </mo><mi>sin</mi><mo>(</mo><mi>b</mi><mo>(</mo><mi>t</mi><mo>-</mo><mi>c</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the number of hours after midnight, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> are constants, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The following graph shows the height of the water for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> hours, starting at midnight.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The first high tide occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>04</mn><mo>:</mo><mn>30</mn></math> and the next high tide occurs <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> hours later. Throughout the day, the height of the water fluctuates between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>m</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo> </mo><mtext>m</mtext></math>.</p>
<p>All heights are given correct to one decimal place.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</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 smallest possible value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</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 height of the water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>:</mo><mn>00</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of hours, over a 24-hour period, for which the tide is higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> metres.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> minutes earlier than at Dungeness.</p>
<p>Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</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>12</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>b</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>12</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><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>8</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mtext>max</mtext><mo>-</mo><mtext>min</mtext></mrow><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mfenced><mtext>m</mtext></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>d</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mfrac><mrow><mtext>max</mtext><mo>+</mo><mtext>min</mtext></mrow><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>8</mn></math> for example into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math></p>
<p>attempt to solve their equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using horizontal translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>12</mn><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo>-</mo><mi>c</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mfenced><mi>cos</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempts to solve their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>'</mo><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <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"><mfenced><mrow><mn>2</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mfenced><mi>cos</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>12</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, graphically or algebraically <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>87365</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>87</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p>times are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>91852</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>08147</mn><mo>…</mo><mo> </mo><mo>,</mo><mo> </mo><mfenced><mrow><mi>t</mi><mo>=</mo><mn>13</mn><mo>.</mo><mn>9185</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>t</mi><mo>=</mo><mn>19</mn><mo>.</mo><mn>0814</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>total time is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mfenced><mrow><mn>7</mn><mo>.</mo><mn>081</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>919</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>3258</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10</mn><mo>.</mo><mn>3</mn></math> (hours) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>11</mn><mn>3</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>8</mn></math> into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> and attempts to solve for <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"><mn>6</mn><mo>.</mo><mn>8</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mfrac><mn>11</mn><mn>3</mn></mfrac><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn><mo>⇒</mo><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong><br>uses their horizontal translation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>12</mn><mn>4</mn></mfrac><mo>=</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>11</mn><mn>3</mn></mfrac><mo>-</mo><mi>c</mi><mo>=</mo><mn>3</mn><mo>⇒</mo><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</math></span> where <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="b">
<mi>b</mi>
</math></span> are positive integers.</p>
</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="{x^2} - 1">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> is a factor of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</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> and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> into a product of linear factors.</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>Using your graph state the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = c">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>c</mi>
</math></span> has exactly two distinct real roots.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(1) = 0 \Rightarrow a + b = 8">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mn>8</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g( - 1) = 0 \Rightarrow - a + b = - 6">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo>−</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = 7,{\text{ }}b = 1">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mn>7</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^4} + 7{x^3} + {x^2} - 7x - 4 = ({x^2} - 1)(p{x^2} + qx + r)">
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>q</mi>
<mi>x</mi>
<mo>+</mo>
<mi>r</mi>
<mo stretchy="false">)</mo>
</math></span></p>
<p>attempt to equate coefficients <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 3,{\text{ }}q = 7,{\text{ }}r = 4">
<mi>p</mi>
<mo>=</mo>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>q</mi>
<mo>=</mo>
<mn>7</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>r</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^4} + 7{x^3} + {x^2} - 7x - 4 = ({x^2} - 1)(3{x^2} + 7x + 4)">
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>7</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = (x - 1){(x + 1)^2}(3x + 4)">
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any equivalent valid method.</p>
<p> </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="c > 0">
<mi>c</mi>
<mo>></mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6.20 < c < - 0.0366">
<mo>−</mo>
<mn>6.20</mn>
<mo><</mo>
<mi>c</mi>
<mo><</mo>
<mo>−</mo>
<mn>0.0366</mn>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct end points and <strong><em>A1 </em></strong>for correct inequalities.</p>
<p> </p>
<p><strong>Note:</strong> If the candidate has misdrawn the graph and omitted the first minimum point, the maximum mark that may be awarded is <strong><em>A1FTA0A0 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c > - 6.20">
<mi>c</mi>
<mo>></mo>
<mo>−</mo>
<mn>6.20</mn>
</math></span> seen.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graphs <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x"> <mi>y</mi> <mo>=</mo> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1 + {\text{cos}}\,x"> <mi>y</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> on the following axes for 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 9.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0"> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo><</mo> <mn>0</mn> </math></span> in the range 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 9.</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><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct curve, showing all local max & mins.</p>
<p><strong>Note:</strong> Award<em><strong> A0A0</strong></em> for the curves drawn in degrees.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 1.35, 4.35, 6.64 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt to find points of intersections between two curves.</p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 1.35 <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 1.35.</p>
<p>4.35 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 6.64 <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct endpoints, <em><strong>A1</strong></em> for correct inequalities.</p>
<p><strong>Note:</strong> Award <em><strong>M1FTA1FTA0FTA0FT</strong></em> for 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 7.31.</p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 7.31.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</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>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.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="{x^2} - 1 > 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>></mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></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> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 1">
<mi>x</mi>
<mo>></mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_15.40.09.png" alt="M17/5/MATHL/HP2/ENG/TZ1/12.b/M"></p>
<p>shape <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <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> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-intercepts <strong><em>A1</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><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is symmetrical about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f( - x) = f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is not one-to-one function <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>horizontal line cuts twice <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any equivalent correct statement.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1 + \ln \left( {\sqrt {{y^2} - 1} } \right)">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = {y^2} - 1">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}(x) = \sqrt {{{\text{e}}^{2x + 2}} + 1} ,{\text{ }}x \in \mathbb{R}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{1}{{\sqrt {{x^2} - 1} }} \times \frac{{2x}}{{2\sqrt {{x^2} - 1} }}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}} = 0 \Rightarrow x = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p>which is not in the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> (hence no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = \frac{{{{\text{e}}^{2x + 2}}}}{{\sqrt {{{\text{e}}^{2x + 2}} + 1} }}">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} > 0 \Rightarrow ({g^{ - 1}})'(x) > 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
</math></span> so no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept: equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> has no solutions.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>q</mi></math>.</p>
</div>
<div class="specification">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has exactly one point of inflexion.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></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>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></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>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the point of inflexion.</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>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></math>, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equations of all the asymptotes on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, or otherwise, solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo><</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> e.g. by factorising <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> or vice versa <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>attempt to use quotient rule or product rule <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>3</mn><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>8</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></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 term in the numerator with correct signs, provided correct denominator is seen.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>8</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></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 term.</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 find the local min point on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> OR solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>60</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1A1A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for both vertical asymptotes with their equations, award <em><strong>A1</strong></em> for horizontal asymptote with equation, award <em><strong>A1</strong></em> for each correct branch including asymptotic behaviour, coordinates of minimum and maximum points (may be seen next to the graph) and values of axes intercepts.<br>If vertical asymptotes are absent (or not vertical) and the branches overlap as a consequence, award maximum <em><strong>A0A1A0A1A1</strong></em>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>667</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p>(oblique asymptote has) gradient <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>33</mn></mrow></mfenced></math> <em><strong> (A1)</strong></em></p>
<p>appropriate method to find complete equation of oblique asymptote <em><strong> M1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mover><menclose><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>0</mn><mi>x</mi><mo>-</mo><mn>1</mn></menclose><mrow><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>8</mn><mn>9</mn></mfrac></mrow></mover></math></p>
<p style="padding-left:60px;"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>8</mn><mn>3</mn></mfrac></mstyle><mi>x</mi></mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>8</mn><mn>3</mn></mfrac></mstyle><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></p>
<p style="padding-left:60px;"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>8</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>16</mn><mstyle displaystyle="true"><mfrac><mn>9</mn><mstyle displaystyle="true"><mfrac><mn>7</mn><mn>9</mn></mfrac></mstyle></mfrac></mstyle></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>8</mn><mn>9</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>33</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>889</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Do not award the final<em><strong> A1</strong></em> if the answer is not given as an equation.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find at least one critical value <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>568729</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>31872</mn><mo>…</mo></mrow></mfenced></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo><</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>569</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo><</mo><mi>x</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>1</mn><mo>.</mo><mn>32</mn></math> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only penalize once for use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≤</mo></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML"><mo><</mo></math>.</p>
<p> </p>
<p><em><strong>[4 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;">
[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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 3x\arccos (x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 \leqslant x \leqslant 1">
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>1</mn>
</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="f">
<mi>f</mi>
</math></span> indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.</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>State the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</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>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| > 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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="images/Schermafbeelding_2017-03-01_om_06.12.12.png" alt="N16/5/MATHL/HP2/ENG/TZ0/05.a/M"></p>
<p>correct shape passing through the origin and correct domain <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Endpoint coordinates are not required. The domain can be indicated by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1">
<mo>−</mo>
<mn>1</mn>
</math></span> and 1 marked on the axis.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0.652,{\text{ }}1.68)">
<mo stretchy="false">(</mo>
<mn>0.652</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1.68</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>two correct intercepts (coordinates not required) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>A graph passing through the origin is sufficient for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}0)">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="[-9.42,{\text{ }}1.68]{\text{ }}({\text{or }} - 3\pi ,{\text{ }}1.68])">
<mo stretchy="false">[</mo>
<mo>−</mo>
<mn>9.42</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1.68</mn>
<mo stretchy="false">]</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>or </mtext>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>π</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1.68</mn>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1A0 </em></strong>for open or semi-open intervals with correct endpoints. Award <strong><em>A1A0 </em></strong>for closed intervals with one correct endpoint.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to solve either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| > 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>></mo>
<mn>1</mn>
</math></span> (or equivalent) or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| = 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> (or equivalent) (<em>eg</em>. graphically) <strong><em>(M1)</em></strong></p>
<p><img src="images/Schermafbeelding_2017-03-01_om_06.22.47.png" alt="N16/5/MATHL/HP2/ENG/TZ0/05.c/M"></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 0.189,{\text{ }}0.254,{\text{ }}0.937">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.189</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.254</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.937</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 \leqslant x < - 0.189{\text{ or }}0.254 < x < 0.937">
<mo>−</mo>
<mn>1</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>0.189</mn>
<mrow>
<mtext> or </mtext>
</mrow>
<mn>0.254</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>0.937</mn>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A0 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < - 0.189">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>0.189</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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 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 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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="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mn>2</mn><mi>x</mi></msup><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>x</mi></msup></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an odd function.</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>Solve the inequality <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≥</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to replace <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><msup><mn>2</mn><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mrow><mo>-</mo><mi>x</mi></mrow></msup></mfrac></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>x</mi></msup></mfrac><mo>-</mo><msup><mn>2</mn><mi>x</mi></msup><mo>=</mo><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfenced><mrow><msup><mn>2</mn><mi>x</mi></msup><mo>-</mo><mfrac><mn>1</mn><msup><mn>2</mn><mi>x</mi></msup></mfrac></mrow></mfenced><mfenced><mrow><mo>=</mo><mo>-</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1A0</strong></em> for a graphical approach including evidence that <strong>either</strong> the graph is invariant after rotation by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo></math> about the origin <strong>or</strong> the graph is invariant after a reflection in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and then in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis (or vice versa).</p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an odd function <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find at least one intersection point <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>26686</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>177935</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>06167</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>27</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>178</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>06</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>.</mo><mn>27</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mo>-</mo><mn>1</mn><mo>,</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>178</mn><mo>≤</mo><mi>x</mi><mo><</mo><mn>3</mn><mo>,</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mn>3</mn><mo>.</mo><mn>06</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="specification">
<p>A scientist conducted a nine-week experiment on two plants, <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>, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was given fertilizer regularly, while Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was not.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>+</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>27</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>B</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>32</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
</div>
<div class="specification">
<p>Use the scientist’s models to find the initial height of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> correct to three significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></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>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, prove that Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>, find the total amount of time when the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was greater than the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn></math> (cm) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>sin</mi><mfenced><mn>6</mn></mfenced><mo>+</mo><mn>27</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26</mn><mo>.</mo><mn>7205</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26</mn><mo>.</mo><mn>7</mn></math> (cm) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>0074</mn><mo>…</mo><mo>,</mo><mn>4</mn><mo>.</mo><mn>7034</mn><mo>…</mo><mo>,</mo><mn>5</mn><mo>.</mo><mn>88332</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>01</mn><mo>,</mo><mn>4</mn><mo>.</mo><mn>70</mn><mo>,</mo><mn>5</mn><mo>.</mo><mn>88</mn></math> (weeks) <em><strong>A2</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><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>+</mo><mi>t</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn><mo>,</mo><mo> </mo><mi>t</mi><mo>-</mo><mn>5</mn><mo>></mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>and as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>≥</mo><mo>-</mo><mn>1</mn><mo>⇒</mo><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p>so for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>t</mi><mo>-</mo><mn>6</mn><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>hence for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> are required <em><strong>(M1)</strong></em></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></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>18879</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>23598</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>33038</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>37758</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>47197</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>51917</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award full marks for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo> </mo><mfrac><mrow><mn>10</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>11</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><em>Award</em> subsequent marks for correct use of these exact values.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>18879</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>2</mn><mo>.</mo><mn>23598</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>33038</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>5</mn><mo>.</mo><mn>37758</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>47197</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>8</mn><mo>.</mo><mn>51917</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempts to calculate the total amount of time <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mrow><mn>2</mn><mo>.</mo><mn>2359</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>1887</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mfenced><mrow><mfenced><mrow><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>.</mo><mn>14</mn><mo> </mo><mfenced><mrow><mo>=</mo><mi mathvariant="normal">π</mi></mrow></mfenced></math> (weeks) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) In general, very well done, most students scored full marks. Some though had an incorrect answer for part(a)(ii) because they had their GDC in degrees.</p>
<p>Part (b) Well attempted. Some accuracy errors and not all candidates listed all three values.</p>
<p>Part (c) Most students tried a graphical approach (but this would only get them one out of three marks) and only some provided a convincing algebraic justification. Many candidates tried to explain in words without a convincing mathematical justification or used numerical calculations with specific time values. Some arrived at the correct simplified equation for the difference in heights but could not do much with it. Then only a few provided a correct mathematical proof.</p>
<p>Part (d) In general, well attempted by many candidates. The common error was giving the answer as 3.15 due to the pre-mature rounding. Some candidates only provided the values of time when the rates are equal, some intervals rather than the total time.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{cot}}\left( {\frac{\pi }{4} - x} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>cot</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {\text{tan}}\,x">
<mi>t</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α<!-- α --></mi>
</math></span>, <em>β</em> be the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right) = k">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>k</mi>
</math></span>, where 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> < 1.</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 = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}"> <mo>−</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>8</mn> </mfrac> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mfrac> <mi>π</mi> <mn>8</mn> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph, explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> is a function on the given domain.</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>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> has no inverse on the given domain.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> is not a function for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}"> <mo>−</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iv.</div>
</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="g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}"> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( t \right)"> <mi>y</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span> for <em>t</em> ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> and <em>β</em> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</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="\alpha "> <mi>α</mi> </math></span> + <em>β</em> < −2.</p>
<div class="marks">[2]</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><img src="data:image/png;base64,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"> <em><strong>A1A1</strong></em></p>
<p><em><strong>A1</strong> </em>for correct concavity, many to one graph, symmetrical about the midpoint of the domain and with two axes intercepts.</p>
<p><strong>Note:</strong> Axes intercepts and scales not required.</p>
<p><strong>A1</strong> for correct domain</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>for each value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> there is a unique value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept “passes the vertical line test” or equivalent.</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>no inverse because the function fails the horizontal line test or equivalent <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> No <strong>FT</strong> if the graph is in degrees (one-to-one).</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the expression is not valid at either of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}\,\,\left( {{\text{or}} - \frac{{3\pi }}{4}} \right)"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>or</mtext> </mrow> <mo>−</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iv.</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="f\left( x \right) = \frac{{{\text{tan}}\left( {x + \frac{\pi }{4}} \right)}}{{{\text{tan}}\left( {\frac{\pi }{4} - x} \right)}}"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mi>x</mi> </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{{\frac{{{\text{tan}}\,x + {\text{tan}}\,\frac{\pi }{4}}}{{1 - {\text{tan}}\,x\,{\text{tan}}\,\frac{\pi }{4}}}}}{{\frac{{{\text{tan}}\,\frac{\pi }{4} - {\text{tan}}\,x}}{{1 + {\text{tan}}\,\frac{\pi }{4}{\text{tan}}\,x}}}}"> <mo>=</mo> <mfrac> <mrow> <mfrac> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mfrac> </mrow> <mrow> <mfrac> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>−</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </mrow> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}"> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>AG</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="f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{tan}}\left( {\frac{\pi }{2} - \frac{\pi }{4} + x} \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>−</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> <mo>+</mo> <mi>x</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=" = {\text{ta}}{{\text{n}}^2}\left( {x + \frac{\pi }{4}} \right)"> <mo>=</mo> <mrow> <mtext>ta</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right) = {\left( {\frac{{{\text{tan}}\,x + {\text{tan}}\,\frac{\pi }{4}}}{{1 - {\text{tan}}\,x\,{\text{tan}}\,\frac{\pi }{4}}}} \right)^2}"> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <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( {\frac{{1 + t}}{{1 - t}}} \right)^2}"> <mo>=</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>AG</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> </p>
<p><img src="data:image/png;base64,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"></p>
<p>for <em>t</em> ≤ 0, correct concavity with two axes intercepts and with asymptote <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 1 <em><strong>A1</strong></em></p>
<p><em>t</em> intercept at (−1, 0) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> intercept at (0, 1) <em><strong>A1</strong></em></p>
<p><em><strong>[3 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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span>, <em>β</em> satisfy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\left( {1 + t} \right)}^2}}}{{{{\left( {1 - t} \right)}^2}}} = k"> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>k</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {t^2} + 2t = k\left( {1 + {t^2} - 2t} \right)"> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>t</mi> <mo>=</mo> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {k - 1} \right){t^2} - 2\left( {k + 1} \right)t + \left( {k - 1} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>t</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>A1</strong></em></p>
<p>attempt at using quadratic formula <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span>, <em>β </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{k + 1 \pm 2\sqrt k }}{{k - 1}}"> <mo>=</mo> <mfrac> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>±</mo> <mn>2</mn> <msqrt> <mi>k</mi> </msqrt> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> or equivalent <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="\alpha "> <mi>α</mi> </math></span>, <em>β</em> satisfy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 + t}}{{1 - t}} = \left( \pm \right)\sqrt k "> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mo>)</mo> </mrow> <msqrt> <mi>k</mi> </msqrt> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t + \sqrt k t = \sqrt k - 1"> <mi>t</mi> <mo>+</mo> <msqrt> <mi>k</mi> </msqrt> <mi>t</mi> <mo>=</mo> <msqrt> <mi>k</mi> </msqrt> <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="t = \frac{{\sqrt k - 1}}{{\sqrt k + 1}}"> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t - \sqrt k t = - \left( {\sqrt k + 1} \right)"> <mi>t</mi> <mo>−</mo> <msqrt> <mi>k</mi> </msqrt> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>+</mo> <mn>1</mn> </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="t = \frac{{\sqrt k + 1}}{{\sqrt k - 1}}"> <mi>t</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>so for <em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = \frac{{\sqrt k - 1}}{{\sqrt k + 1}}"> <mi>α</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span>, <em>β</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt k + 1}}{{\sqrt k - 1}}"> <mo>=</mo> <mfrac> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mi>k</mi> </msqrt> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> + <em>β </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\frac{{\left( {k + 1} \right)}}{{\left( {k - 1} \right)}}\,\left( { = - 2\frac{{\left( {1 + k} \right)}}{{\left( {1 - k} \right)}}} \right)"> <mo>=</mo> <mn>2</mn> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + k > 1 - k"> <mn>1</mn> <mo>+</mo> <mi>k</mi> <mo>></mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha "> <mi>α</mi> </math></span> + <em>β</em> < −2 <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept a valid graphical reasoning.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iv.</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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right) = 2{x^4} - 15{x^3} + a{x^2} + bx + c">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>15</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</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">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}">
<mi>c</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span></p>
</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="\left( {x - 5} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is a factor of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right)">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, find a relationship between <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>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 5} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> is a factor of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right)">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P'\left( 5 \right)">
<msup>
<mi>P</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 5} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> is a factor of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( x \right)">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, and that <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>, find the values of <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>.</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>attempt to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 5">
<mi>x</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> and set equal to zero, or use of long / synthetic division <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times {5^4} - 15 \times {5^3} + a \times {5^2} + 5b + c = 0">
<mn>2</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>5</mn>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>15</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>5</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { \Rightarrow 25a + 5b + c = 625} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mn>25</mn>
<mi>a</mi>
<mo>+</mo>
<mn>5</mn>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>625</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></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>0 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P'\left( 5 \right) = 0">
<msup>
<mi>P</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>5</mn>
<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=" \Rightarrow 8 \times {5^3} - 45 \times {5^2} + 4 \times 5 + b = 0">
<mo stretchy="false">⇒</mo>
<mn>8</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>5</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>45</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mo>×</mo>
<mn>5</mn>
<mo>+</mo>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{x^2} - 10x + 25} \right)\left( {2{x^2} + \alpha x + \beta } \right) = 2{x^4} - 15{x^3} + 2{x^2} + bx + c">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>10</mn>
<mi>x</mi>
<mo>+</mo>
<mn>25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>α</mi>
<mi>x</mi>
<mo>+</mo>
<mi>β</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>15</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>
<mi>b</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong> (M1)</strong></em></p>
<p>comparing coefficients gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> = 5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta ">
<mi>β</mi>
</math></span> = 2</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> = 105 <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore c = 625 - 25 \times 2 - 525">
<mo>∴</mo>
<mi>c</mi>
<mo>=</mo>
<mn>625</mn>
<mo>−</mo>
<mn>25</mn>
<mo>×</mo>
<mn>2</mn>
<mo>−</mo>
<mn>525</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> = 50 <em><strong> A1</strong></em></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="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 the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 \leqslant x \leqslant 5">
<mo>−<!-- − --></mo>
<mn>3</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>5</mn>
</math></span>.</p>
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Sm6o0Zjv6WFUYsV8egZgOjqOcviDIju8Gnut7Qw/Kw4c1QYgOiOSiaVcbSuHqcXv7yh/R5pY/wZmry0uGYGiO4aOlQH/ZYWVBPBeCQZgOiOZFoLgvr0x/Ta/p/SxRYRtS7S2ee30NjY0zS1yB3LG0RXmNC1WFrQ8ZWjNUQ3u6zfoPkjb9KFVX0lWpyiA2Ofp/HTv+uwAdHtUKHawdKCiq4sjWspulZPf7lCLLBjY7bm99E9dx6m6ZurSpxSdGPH1+vKjoE7yNJCDNxeMbr9ueG6sVdpH6KrzIZF4cbDXKD5N/fR17c8Q1NX/7iGCYjuGjoGOhh0aSFePvu7mRtufzbsRiG6Su4tCjcK5uIUPdVsLD9Ia+yg8RPz9InDBUTXIWPA3UGXFqLkcwAfc8MdgBITkyCiK8mU1heJjPlarb3M4Z4nfb7WtZN9n530iY3bytjx48eJ/8kxt77NHe/e19rL+e550udrXTvZ99lx38V/OkTHJprUaNxDW6c/pqWVE0R0fefJnG7rs5M+1072ZYxbC07ZD/HB55OMSSs2bitj0t599910//33t+d17WRf7EYlXomHW98m46ni9flQxT6I7uxs5+KTIvElSsZSFZDgua3rl9vfve/ayX63jXtMrRk6tn2Mxg6coYWVEyC6a+tCeHRbl8OjR4+2/2o4ePAgRHeFJOEn1TXj5qbK+0FEN3WAkszUuIxngR0f8wb95shmr+im4Dh+fP4oQuLu3bu3LbrXr1/3gzm9IXGdaQt3c8MtJMTIAKKrJN6icONjfkQzE03v8oKSnqHM48fndysk7saNG2nnzp1+oK7ekLhdU/c9zA23LxmGg7UUXUO+6g/dOkdvfOUOuvPJ79L0FX5j4RZdO/0wNXceo5lPbV4ZqzupJ0+ebN/l8p/R2MBAEQMQ3SKGRm58geZf3UrNxsqbC809tOt7c8ufTnNilTVdpwu7PRjQLC30mALdGTEA0c0o2ZpQIbqDs6VZWhh8VliOKgMQ3VHNbMm4ILqDEYilhcF4gtUqAxDdVS6w5zAA0XXI6LOLpYU+5GDIy0AtRdfqKSwzaIFtgZlSdC3iC5XLYZYW6hyvV0UKOq3iLXDLbBiiq6TeooAsMCG6xYUx7NKCRT45mtxwizNoYwHRVfJuUbgWmBDd4sIYdmnBIp8cTW64xRm0sYDoKnm3KFwLTIhucWEMs7TAs1rkM0fc4gzaWEB0lbxbXDAWmBDd/oUx7NICz2qRzxxx+2fQbhSiq+Te4oKxwITo9i+MYZcWeFaLfOaI2z+DdqMQXSX3FheMBSZEt39hDLu0wLNa5DNH3P4ZtButpeja0ZUPckrRrRurZZYW6hYr/A3PAEQ3PKcjMSNEt3cayywt9J4VI7kwANHNJdPKOCG6vQnjpQUWXmxgYBgGILrDsJbBORBdf5JlaYFbbGBgGAYgusOwlsE5EF1/kvkOl+90sYGBYRmopehaPf1lki2wLTBTiq5FfMPkkn+Kh3kpu7RQl3iHFZXu86zi7fajKscQXWUmLArIAhOiu74w+JchmJeySwsW+eRocsNdn8Fq9EB0lXmwKFwLTIju+sLg30ALsbRgkU+OJjfc9RmsRg9EV5kHi8INj7lAH57aTXuaKz/ZM/4U7T97jZYcLiC6DhlEFGppgWcNn8+1vvY6yg23Fw/W/RBdZQYsCjcs5i26+v3HaPepD+kTjr11kc4+v4WajUdo19zqz4dDdNcWRqilBZ41bD7X+tnvKDfcflxYjkF0lexbFG5QzNYv6ciRy2vuaqk1Q8e2j9HYgTO0sMIHRHdtYYRaWuBZg+ZzrZt9j3LD7UuG4SBEV0m+ReHGx/yIznzrDohuj1oIubTAEPHz6Q8kN1w/C/a9EF1lDiwKNz7mRzQz0aSt0x937oBxp7taGCGXFnjW+Plc9d3dyw3Xjb1K+0FEV5IprS9AGfO1sF+9EE34WZyiieZhmr7Z6qRCRNfCH8b0bT5fpC+m/f333+99a0GwfW1MfxjPt/n8kL4c7H0xVrEPojs727nzyKlAJdYLF6bp3KH71jxE40KF6C7XxalTp3p+IEI49LW+i91nJ32wL3/j4eOwin1BRLeKgcGnQRhYoptzT9D4KxeW32RwThHRdbqy3A29tJAliQh6DQMQ3TV05HXQunKYdr/43jrBZRYgusu1EPKthbyqC9H2YgCi24uZEe9vXZ2i3bvfooudZdwlunnxEO17+3ftyCG6YT8QMeLlhPAUDNRSdPutgyliH8rUAjss5hLdvDRJBx4Y69zNisA2Grto3/zv27xI31AkKU8KG9/g4EW4sZYWinAHj0BnmRuujp101hBdJdcWhRsSs3Vlkh6Xj/82Vj4GLO2OYzS7cucL0SWKtbQQMp+a8s0NV8NNSluIrpJti8K1wMxddEN/IMItM4t8Mn5uuC7nVdqH6CqzYVG4Fpi5i64sLZw/f15ZIcXmFvlkr3LDLc6EjQVEV8m7ReFaYOYuury0sGnTJmV1DGZukU/2LDfcwbKR3gqiq+TconAtMHMW3cuXL7cfMk5MTCirYzBzi3yyZ7nhDpaN9FYQXSXnFoVrgZmz6L700ktt0Y2xtJCj+FnUr/KyTmpeS9FNylCmYClFt2oUb9u2LdrSQtVihT/pGYDopue8Foi5im7spYVaJB9ORmUAohuV3vpOnqvoxl5aqG9FwPNQDEB0QzE5YvPkKrpYWhixQq5gOBDdCialCi7lKLpYWqhC5Y2+D7UUXcunoRbYFpgpRdciPr60u3FTLS1046aSmdxwU/GqxYHoKhmzKFwLzBxFN9XSgkU+ucxzw1Ve2snMIbpKqi0K1wIzN9FNubRgkU+IrvJCj2gO0VWSa3HBWGDmJrqplhZyFD+L+lVe1knNIbpKui0KyAIzN9FNtbQA0VVecCNoDtFVJtVCAC0wcxLdlEsLEF3lBTeC5hBdZVItBDAO5gJdeuM5OvHsFrr9wBla6OIhJ9FNubTANMfJZ1cCPYe54XooqERXLUW3EszV2okb9OGrW2l8x59Qs9GgMWPRtaYy5dKCdazAt2cAomufAzsPWjN0bPtY1qKbemnBLtlArgoDEN2qZMLCD4guydICiy82MJCCAYhuCparigHRJV5a4H/YwEAqBiC6qZiuIk7moitLC3y3iw0MpGKglqJr9RSWk2KBHQ2zIqIbLb6Cq+jRRx9t/0JE6qUFq3hzwy1Iv9lwENGVZErri0bGfK3WXuZwz5M+X+vayb7PTvrExm1ljH8llv/JMbe+zR3v3tfay/nuedLna1072ffZffD+i/T3993W90Ga7zyZ0219dtLn2sm+jHFrwSn/xPrdd99NmzdvFpfW5NT1j/d9W7eNe9zP3iJe9kf8E9/k2NeKjdv67KTPtZN9GUsVr+BWvYXozs52ilGKxJc0GUtVQILntq5fbn/3vmsn+902fJyr6J46daq9jsvvIj/33HNC0bo6cDnrGDk77nj3vmPW2RUbixpiJwRfHJJjXys2buuzkz7XTvZlLFW8glv1Nojopg5Skpkal/EssKNhZri8wHe4/OCMBffgwYPtfKauo2j5LAgkN9wCOsyGIbpK6i0KNxpmZqLrCi7ffUXjtaCmgFtA0IgPQ3SVCba4YMJjLtGNd7a3P40mH/dtNHbRvvnfd9iQ/k5HxJ3w8a13tltw2SIF7npPgOvjJKc+iK4y2xYXqgXmKImuT3A57Ra8Ald5wY2geS1FdwTzULmQUopuzOB7CW5MTMwNBvoxANHtx07GY6MguhDcjAu4wqFDdCucHEvXRkF0d+7c2X5LAZ84s6wkYHczANHtZgTHbQbqLrp79+5tCy632MBAlRiA6FYpGxXypc6iC8GtUCHBlXUM1FJ0rZ46M3sW2BaYKUU3ZHwawQ2Ju+7K6tMB3D7kZDAE0VUm2eKCscCso+hqBJfTbsErcJUX3AiaQ3SVSbW4UC0w6ya6WsGF+CkLv4S5Rf2WcDf6qRBdJcUWBWSBWSfRHUZwOe0WvAJXecGNoDlEV5lUiwvVArMuojus4EL8lIVfwtyifku4G/1UiK6SYosCssCsg+iWEVxOuwWvwFVecCNoDtFVJtXiQrXArLroTk5Oln4P14JXiK7yghtB81qK7gjmoXIhpRRdbfD8tYzsH38vLn/UFxsYqBMDEN06ZSuhr1UVXQhuwiIAVBQGILpRaK3/pFUUXQhu/esKERBBdFEFXgaqJroQXG+a0FlDBiC6NUxaCperJLoQ3BQZB0YqBmopulZPnTkpFtgWmClFt198MQW3H27MCxC4Mdmt/twQXWWOLC4YC8wqiG5MweW0W/AKXOUFN4LmEF1lUi0uVAtMa9GNLbgQP2XhlzC3qF/XXX6ne/DtXfrhX93efiWx0XiEds1dJ2pdpLPPb2n/kOvt3/kV/WHwybyWEF0vLb07LQrIAtNSdE+ePJnkPVwLXrmygNv7+ooxwrW8adMm4roadGtdmaTHmw0aO3CCfnLwYdr3weKgpxbaQXQLKVprYHHBWGBaie758+dp48aN7X+XL19eS37gIwteOQTgBk5kwXRSy9w++OCDNFhdfURnvnUHNRr30Nbpj2mpAEMzXFp03YCw31j5swQtagE1UOUamJiYKPg04xLdeGc7NRu7aN/87zWaWmgL0W3g4qjyxQHfUJ+xaqD/D5beoqs/+CI1G01qvjZfrTvdQlmHQS0ZkEJP4by7pMD72MBASAaklqXlX4kuWmLgNd0dW79KTz4wRmMHztBCQIdK3+kG9AVTVYgBKdDYLkFwYzOM+aWW+WEar6cXbq1z9IN7H6PJK4v0myObqdF8hqZ++zZ9e/s0XWgVnl1oANEtpChPAynUmNFDcGOyi7mFAX4w238pYcVyaZpe2NygxvgzNHmJ31ZYoptzu2m80aQ7n/whzX4aQHEJ370geUHbxUBs0YXgdhGOw2gMVO3rP3GnGy3V9Z44puhCcOtdG/C+HAO1FN2LMwfp4CP/avn1rM6fAuWIKDz701l6+9ktdFf7bYe76M4nv0vTV/5YeFoIg/fe+yH96IXH6cSzW+j2wIv6vfyLJbrrBXehHduLf7MpWWxEC/Thqd20866VNwPGn6L9Z68FfUK9ntdbdO3nj9GeJmPeRf9mz8vB/lxdj+XpaZ2jl//iX9O/e/atyHEKtrxy5bx9seMYzYb5C11AatnWT3Q//TH9j788Tsffe7/z8byxsadpajFiNlvn6PXdh+jlDxbbC/Es+i9+eQM1xv+Rpm9GxG2X1A1669v/gb74n/99+2OIoZ+k9qraGKLLf+bxrz3w3MsPNG7Qh69ubcfG/5mlie0WXf3+Y7T71Id08r336P33jq98xHPlI5+9CCnVz2uDz9H+ny0Lu9w0pImXHV9+/emuxl3pRLd1jt7Ys8F5b/2u4K9elUqJ4ck1E90bNH/kTTp24f3Vp5CLU3Rg7PM0fvp30WhszR+nI79dvquVTxO15vfRjsYXouJKQIz5wfsv0rHt4V9fEYzuNrTouoLL36vgbhzb3993WxrRbf2Sjhy53L7bk1xSayYutw4mxz03N0P/cvRein6z0CaZBf8rtOWxx+gvbrstkeguP4BqTsy0X7Xq8OwmPeP9monucqbcJLL43XPn4QR3nF3YCcRe6rLuottPcDnGpKIrpK75OO7yRz5T3Xm+++6b9PbffY7u/Ic5+sTxJ8ruzdfp2S3foZ/+9mX6WjLRlY/Q7qDxV07Q374xt3qTFCXIek1aW9Hldc75N/fR17c8Q1NX06ytcmo7gs+ie/t/o8kE67p1Ft0iwWVOqyC6MxPN4J+x90rBp7N09OC9tOnL/4cuxl6Zoit07tB9y9+UtTiVTHSX/wp01nIbD9CfvfBm/P9gvIRXr7OWojs/97f01/IQpLGDxk/MJ0soCyD/eXjj9EPUfD7BncqK0NdxeWEQwa2E6C5O0UQz9l9LXQ+W/vwbK++CxhKFlWWFZ88uXxsJRVcial2ZpulXxunP2w+fx5fFXwYzbmspunK32dAIFa8AABEkSURBVLr6Oh2baEb5JqBeNcHYl84dav/JFv8h2rIXdbzTHVRwOULLO90LF6ZX7wZ7JT1o/y36f9/bRV/7wm3UaH4z3l9KN19f+wkqA9EV2j448zf0jS8kWrMX0Aq3FRDdP9DCD+51nnK6f5Y4+5tfoGnf96sN/RDE/bJiB6frC3DWf2nxFfrnUt+vOWS8Q8c5XPWVeZCmEdy2d4ljW2WE7wafoPFXLiT7S0mw4z6IvUJzT35praC3n0GMGb1BsHyXn+bBoTBc3bYColuWnBvtz0eneQhyi6587wna/8GNsk7rz08sTMOKrlpwmYnEsQn5rSuHafeL7yUX3DZ+a4aO/qd/G+ftl8Upeqr9PnCvm4nwX1conPZq+T+ZZvM1vKc7Gh8D/ojSPAS5Rdd+8gTtfuuq83L5An2w/+/oSMx3hKWKEwvTMKI7lOByfIlja0NenaLdu99yHmYt0c2Lh2jf2/FePZRUttuED2I7eGOGd7qnv0Tb30nE7Rqiq3dQrztdfuH6K3c4nwa7RddOP0zNncdoJtCXUfhTtEAfnthO411LD21hSvUpm8TCNIzo8lfm8XkDfbmIS3TS2Jbo5qVJOvDAmGdJK84dYPunX8a20/irZ5dFvnWBZvZvpi3/+9fp7rKTLS/coms/O0zP/Vw+4XeLrv3iCRrf8xPnPzg3+fnt10t0aYHmX93a/mRWWxSae2jX9+YiJ1O+zNj3p1r4LzheX4JdT73bwh9HHFxsreju3bu3LWLcDr6lj01++0riW9PG+g/05pt0lD/BuPKf9thXD9PkL0SUBmerlGVK0T39cOcGhWM99KNL6f5zKUVSmpNrJrrLpMjbC2koWotigW2BKQKxNnr/0XCCuzqXRXyMDtzVHMTcs+I5Zkxl5oboKtmzKCALzEFFt6zgMv0W8QFXWfglzK3yW8LlqKdCdJX0WhSQBeYgohtCcJl+i/iAqyz8EuZW+S3hctRTIbpKei0KyAKzSHRDCS7TbxEfcJWFX8LcKr8lXI56KkRXSa9FAVlg9hPdkILL9FvEB1xl4Zcwt8pvCZejngrRVdJrUUDhMRfo0hvP9f1S9F6iG1pwmf7w8Q2WVOAOxlNZKyuey/od63yIrpJZiwIKi7n8xeHjO/6k75ei+0Q3huAy/WHjGzyhwB2cqzKWVjyX8TnmubUU3ZiEZDN3wQcSukV3cnJyiPdws2ETgYKBgRmA6A5M1YgZKkSXf+mBRZh/aoc/6osNDICB4RmA6A7PXb3PHFB0Ibj1TjO8rx4DEN3q5SSNRwOKLu5w06QDKPkwANHNJ9drIx1QdLGksJY2HIGBsgwEEV15OimtzykZ87Vae5nDPU/6fK1rJ/s+O+kTG7eVMf5zm//JMbe+zR3v3tfay/nuedJ37tw79LMX/7TzZSryAGxd+7mnaPLXlzp+y6813Pa1/9v+xVZ3bllS4DlOnTrVOUcwXVvZlzFfKzZu69pZcMq+iA/ilxz7WrFxW5+d9Ll2si9jiDfONSM8V72F6M7Odi4+uSh8SZMxiwtGsF2/pM/Xunay323XS3RdwWXR7T6Pj32bz076iuwtOGWfuv2TY19bFEP3Of3sEW/4GmL+67IFEd3UwUqBp8ZlPAvsKJie5QURXF5SkDvlFBxHiW8Ax4E7AEkBTKx4DuB6lCkgukpaLQooCmaX6LqCy6+FQXSVhaEwj5LPAfBzwx2AEhMTiK6SdovCDYvp++Lw/7LuPVyIrrIwFOZh8zk4cG64gzOT1hKiq+TbonBjYnbf4QodEF1hInwbM5/9vM0Ntx8XlmMQXSX7FoUbC/P8+fPtO9yNGzeu+6QZRFdZGArzWPksciE33CI+rMZrKbpWZI0SLgsuiy3/4/3uLaXodmPjGAyMMgMQ3VHObo/YigSXT4Po9iAP3WCgJAMQ3ZIE1u30QQSXY4Lo1i2z8LcuDEB065KpAH4OKrgMBdENQDimAAMeBiC6HlJGsUsjuBw/RHcUqwAxVYGBWoqu1VNYTpgFdllMreBynClFt2x8w15IwB2WOd15VjzrvExnDdFVcm1RQGUwhxFcpgSiqywMhXmZfCpg1pnmhruOgIp0QHSVibAo3GExhxVcpgSiqywMhfmw+VRAeE1zw/WSUIFOiK4yCRaFOwxmGcFlSiC6ysJQmA+TT8X0PU1zw+1JhPEARFeZAIvC1WKWFVymBKKrLAyFuTafiqn7muaG25cMw0GIrpJ8i8LVYIYQXKYEoqssDIW5Jp+KaQtNc8MtJMTIAKKrJN6icAfF5K9k5I/1smCy+JbZILpl2Ot/7qD57D+LfjQ3XD1Dac6opeimoaZeKCy48uXj/M1hZbeUolvWV5wPBurEAES3Ttnq4WtowWUYiG4PstENBkoyANEtSaD16TEEl2OC6FpnFvijygBEt8aZHU5wF+jDU7tpT7OxLKzjT9H+s9doqYsHiG4XITgEA4EYgOgGIjL1NMMJ7i26+v3HaPepD+kTdrh1kc4+v4WajUdo19z1NSFAdNfQgQMwEIyBWoqu1VNYZt0CuxtzOMFlkf0lHTlyee1dbdcPVEplpRTd7vjEh9gtcGMzvDy/Fc9potOjQHSVnFkUkIs5tOD2jPMjOvOtO2jswBlacGwgug4ZgXfdfAaeuu90ueH2JcNwEKKrJN+icAUzvOBy8B/RzESTtk5/vOYOGKKrLAyFueRTcUoQ09xwg5AWYRKIrpJUi8JlzFOnTgV9D7cT9uIUTTQP0/TNVqeLdyC6a+gIemBRQxxAbrhBkxZwMoiukkyLwmXB3bx5c1sIQ3zwYTXkK3Tu0H3rHqLxOER3laXQexY1xDHkhhs6b6HmCyK6kkxpfc7JmK/V2ssc7nnS52tdO9n32Umf2LitjLHo8T855ta3uePd+xr7EydOeAVX5jx37h362Yt/2hFJEct17eeeoslfX+r4PTc3Q+++/jD92dM/Wn6TocspOV9w3LbLtH3ojnfvF9mn5pT94038FP/k2NeKjdv67KTPtZN9GUO8ca4Z4bnqLUR3drZz8clF4UuajKW6YFhsH3rooY6YHjx4cI1b4o+vXWO4ctBt98GZb9D2J0/Syffe85l3cLvP42Pf5rOTviL7VJyKPxKDHIt/cuxrxcZtfXbS59rJvowh3vA1xNzWZQsiunUJtg5+Xr58mfbu3dsRPd4v++U13XG3rk7R7t1v0cXOMu4S3bx4iPa9/buOqdzpdjqwAwbAQBAGILpBaCw/iU9suS/stkQ3L03SgQfGOqIu4tpo7KJ987/vwEl/pwM7YAAMBGEAohuExuEnSSO2y/61rkzS4/Lx38bKx4Cl3XGMZjt3vmkfpA3PHs4EA/VjAKJrlLOUYjtMiLjTHYY1nAMGihmA6BZzFNSi6mIrwUJ0hQm0YCAsA7UUXXkKHJaKwWYbFruM2A6LOVhEfquUomsRH0cNXH/uQ/da8Rw6jlDzQXSVTGoLqIzYimtaTDmvTAvRLcNe/3Mt8ske5YbbPwt2oxBdJfeDFm4IsRXXBsUU+xAtRDcEi/45LPLJnuSG62ffvheiq8xBUeGGFFtxrQhT7EK2EN2QbK6dyyKf7EFuuGtZr84RRFeZi16Fy/0PPvhg5/1X/lADC3CIrRdmiLl7zQHR7cVM+X6LfLLXueGWz1ScGSC6Sl67C5ePY4mtuNaNKf0xW4huPHYt8snR5IYbL4PlZoboKvmTwuU2ttiKa4IpxylaiG48li3yydHkhhsvg+VmrqXolgu53Nmu2G7cuJEmJiaCLSOU8yzs2SlFN6znmA0MVJsBiO6A+ekW28nJSeJfchjVDaI7qplFXNYMQHQLMpCb2AodEF1hAi0YCMsARLcHn7mKrdAB0RUm0IKBsAxAdLv4zF1shQ6IrjCBFgyEZaCWosvCyP9CboOKbQzsojgsMFOKrkV8zDlwiyovzLgVz2G8Dz9L9qLLBSGvfvHbCEUPyCwKyAITohv+YpMZLfLJ2LnhCt9Va7MVXS5AjdhK4iwK1wIToisZD99a5JOjyA03fObCzJid6HLhDSO2QrdF4YbGbF09Ti9+ecPyR5bHn6HJS4sSXqeF6HaoCL4TOp+DOpgb7qC8pLbLRnT5F1jLiK0kxqJwg2J++mN6bf9Pl3+UsnWRzj6/hcbGnqapRee3eijtz/UEjU8SNUAL3AFICmBixXMA16NMMfKiy2K7adOm9l3dIGu2RSxbFFA4zBs0f+RNuuDq6+IUHRj7PI2fXv0lYOYAd7pFlTD8eLh86nzIDVfHTjrrkRXd0GIrKbEo3JiYrfl9dM+dh2n6pqvEEF3Jd4w2Zj77+Zsbbj8uLMdqKbr9CIsltv0w6zm2QPNv7qOvb3mGpq7+cV0IKe9014GjAwyMMAMjI7qu2PJyAh9j68HA4hQ91fkp9h00fmKePukyheh2EYJDMBCIgVqJLn/BjDwMY1Hg/ampqc6aLcRWVxWtq6/TsYkmNRr30Nbpj2nJOR2i65CBXTAQkIFaiS7/GoOIgdvmLbZ/oIUf3OvlxeWosfkFmnZVVYqoNUPHto/R2IEztCB9iR+kObDYBQMjz0AQ0ZUFeml9rMmYrx3Uftu2bevE5TOf+Uz7dN+80jfo/Dnav/vum/SbI5t7iq5w4rax+WQs3+b60L0P+9UPP3RzkwufvhqoYl/tRZeFmDdfoUmfj3gZ87X97HmtmP+55/Wzd+1kX2vvO0/6fK1m/l//+vs0M9HsubxQdn4+37e581pwyj6JD+KfHPtasXFbn530uXayL2OIt7gmhCtphUO3lTFp3bEq7wcR3VQBnjx5ct2dLvel3CwSHAyzdY7e+ModdOeT36XpK/zGwi26dvphau48RjOf4pWxVHUULJ9Kh3PDVdKTzLxWosuscOHw2i7/4/3Um0XhhsNcoPlXt1Kz0Vj+z6u5h3Z9b27502ldRMp6cFd3lMNw8encA66Or2GtrXge1t/Y59VOdJkQyyRaYFtgQnTjXXoW+bS8bqzijZfBcjNDdJX8WRSQBSZEV1kYCnOLfLJ7ueEqUpLUFKKrpNuicC0wIbrKwlCYW+ST3csNV5GSpKYQXSXdFoVrgQnRVRaGwtwin+xebriKlCQ1hegq6bYoXAtMiK6yMBTmFvlk93LDVaQkqWktRTcpQ5mCpRTdTClG2JkyANHNNPFFYUN0ixjCOBgYjgGI7nC8jfxZEN2RTzECNGIAomtEfNVhIbpVzxD8qysDEN26Zi6y3xDdyARj+mwZqKXoWj2F5SqxwLbATCm6FvFZ5RK42WptJ3CIboeKwXYsBMICE6I7WD0MY2WRT/YzN9xhcpPiHIiukmWLwrXAhOgqC0NhbpFPdi83XEVKkppCdJV0WxSuBSZEV1kYCnOLfLJ7ueEqUpLUFKKrpNuicC0wIbrKwlCYW+ST3csNV5GSpKYQXSXdFoVrgQnRVRaGwtwin+xebriKlCQ1hegq6bYoXAtMiK6yMBTmFvlk93LDVaQkqWktRTcpQ5mCpRTdTClG2JkyANHNNPFFYUN0ixjCOBgYjgGI7nC8jfxZEN2RTzECNGIAomtEfNVhIbpVzxD8qysDEN26Zi6y3xDdyARj+mwZqKXoWj2F5SqxwLbATCm6FvFZ5RK42WptJ3CIboeKwXYsBMICE6I7WD0MY2WRT/YzN9xhcpPiHIiukmWLwrXAhOgqC0NhbpFPdi83XEVKkppCdJV0WxSuBSZEV1kYCnOLfLJ7ueEqUpLUtJaim5QhgIEBMAAGAjIA0Q1IJqYCA2AADBQxANEtYgjjYAAMgIGADEB0A5KJqcAAGAADRQxAdIsYwjgYAANgICADEN2AZGIqMAAGwEARAxDdIoYwDgbAABgIyABENyCZmAoMgAEwUMTA/wcvCQw2eHyifQAAAABJRU5ErkJggg=="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ f} \right)\left( 1 \right)"> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo>∘</mo> <mi>f</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( a \right) = 3"> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>3</mn> </math></span>, determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">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="g\left( x \right) = 2f\left( {x - 1} \right)"> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span>, find the domain and range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0"> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = - 1"> <mi>f</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>1</mn> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = f\left( 3 \right)"> <mi>a</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mn>3</mn> <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=" \Rightarrow a = 4"> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2 \leqslant x \leqslant 6"> <mo>−</mo> <mn>2</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>6</mn> </math></span> <em><strong>A1</strong></em></p>
<p>range is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 6 \leqslant y \leqslant 10"> <mo>−</mo> <mn>6</mn> <mo>⩽</mo> <mi>y</mi> <mo>⩽</mo> <mn>10</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> that satisfy the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</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>The triangle ABC is shown in the following diagram. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B < \frac{1}{4}"> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span>, find the range of possible values for AB.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(k - 4)(k + 3) < 0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 < k < 4"> <mo>−</mo> <mn>3</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>4</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B = \frac{{{2^2} + {c^2} - {4^2}}}{{4c}}{\text{ }}({\text{or }}16 = {2^2} + {c^2} - 4c\cos B)"> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <mi>c</mi> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mrow> <mtext>or </mtext> </mrow> <mn>16</mn> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>c</mi> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{c^2} - 12}}{{4c}} < \frac{1}{4}"> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>12</mn> </mrow> <mrow> <mn>4</mn> <mi>c</mi> </mrow> </mfrac> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {c^2} - c - 12 < 0"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>c</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span></p>
<p>from result in (a)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{AB}} < 4"> <mn>0</mn> <mo><</mo> <mrow> <mtext>AB</mtext> </mrow> <mo><</mo> <mn>4</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 < {\text{AB}} < 4"> <mo>−</mo> <mn>3</mn> <mo><</mo> <mrow> <mtext>AB</mtext> </mrow> <mo><</mo> <mn>4</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>but AB must be at least 2</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 2 < {\text{AB}} < 4"> <mo stretchy="false">⇒</mo> <mn>2</mn> <mo><</mo> <mrow> <mtext>AB</mtext> </mrow> <mo><</mo> <mn>4</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \leqslant {\text{AB}}"> <mo>⩽</mo> <mrow> <mtext>AB</mtext> </mrow> </math></span> for either of the final two <strong><em>A </em></strong>marks.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{ax + 1}}{{bx + c}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>b</mi>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne - \frac{c}{b}">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>c</mi>
<mi>b</mi>
</mfrac>
</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">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{Z}">
<mi>c</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<p>The following graph shows the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\left( {f\left( x \right)} \right)^2}">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>. It has asymptotes at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = p">
<mi>x</mi>
<mo>=</mo>
<mi>p</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = q">
<mi>y</mi>
<mo>=</mo>
<mi>q</mi>
</math></span> and meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at A.</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>On the following axes, sketch the two possible graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> giving the equations of any asymptotes in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<p><img src="data:image/png;base64,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"></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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{4}{3}"> <mi>p</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = \frac{4}{9}"> <mi>q</mi> <mo>=</mo> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </math></span> and A has coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{1}{2},\,\,0} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </math></span>, determine the possible sets of values for <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>.</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><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAf8AAAHiCAYAAADmucX7AAAgAElEQVR4AexdCfxWw/4et2hBkdKifV+INmtc2UqhcO1ku0R2l8u1bxf3hqtc+y6UyJ61BSkiKtFGSUWlUiqp1O3/eebvec17Ou/v9+7vWZ75fM47c2b5zneeOe95ZubMssWmTZs2GRkhIASEgBAQAkIgNgj8KTYlVUGFgBAQAkJACAgBi4DIXw+CEBACQkAICIGYISDyj1mFq7hCQAgIASEgBET+egaEgBAQAkJACMQMAZF/zCpcxRUCQkAICAEhIPLXMyAEhIAQEAJCIGYIiPxjVuEqrhAQAkJACAgBkb+eASEgBISAEBACMUNA5B+zCldxhYAQEAJCQAiI/PUMCAEhIASEgBCIGQIi/5hVuIorBISAEBACQqCiIBACpUDAPVJiiy22KIUKsciTOBcLY+YHcIuVZywqUoUUAnlGQD3/PAMqcekhcPfdd5tTTz3VRgZh/O9//zOw/S5E8vqnl4tikYAffPBBc9JJJ5mNGzdarPONDOtv1apVZtCgQWbcuHFJdZbv/CRPCAiB3BAQ+eeGn1JnicDnn39uRo4cmUhNkqKHS/YgFhr68152+QgA2y+//NKMGjXK/OlPfypIjxx5oG623npr8/HHH5spU6ZYxbz1Wr62iiEEhEAxEBD5FwNl5bEZAqlI4bfffjMzZ840EyZMSJAUSAU9yWXLlhWMvDZTMIIewLGQBnWKxkXTpk3NggULbFZuw62QeUu2EBACmSEg8s8ML8UuAAJsCICchg4daq6++mrTq1cvs379etsA+PXXX83xxx9vBg4caHuXBVAhFiILSf6u7LZt25qlS5ca1JuMEBACwURA5B/MeomtVqeccoq55557zIoVK8z06dMtDlWrVjV33XVXYiQgtuBkWXAQ84477miaNWtmJbhEnaXIzZKhx09Tr14988svv5iff/5ZdUZQZAuBgCHwxz82YIpJnfghgBEAXHXq1DHbbrut+eGHHywI8Fu+fLnp06dP4j5+6GRXYhL9NddcY8aPH29HTjjSkp1E/1TIh3J32mkns2HDBrNw4cKC5eevhXyFgBBIFwGRf7pIKV5REOA34rp165p58+bZPFeuXGlWr15tOnToYO9JaEVRKEKZAFsQdKHwYwNghx12MJUqVTLffvutRY91GiEoVRQhEHoERP6hr8JoFQDkhAsEsmjRIktU9957r+nbt2/BSCtaCCaXhr1x+JL4Xb/k2LnfoQGw/fbbm+22285+tkFehcwvd40lQQjEEwGRfzzrPXCl9hJE7dq17ex+LAfs1q2bqVmzZmKmvzdu4AoTUIUKScSUTbtNmzbmm2++MVi9UaiRhoDCLLWEQCgQEPmHoprioyTIA2SBnj8m/E2bNs3sueee6j3m+Ahcf/31plOnTkUj4latWpm1a9fa0Rt3MmCOxVByISAE8oSAyD9PQEpMbgiwdwgbZAHyx4Yx5513Xm6CldoigAmT33//fdEaUS1btrT5zp8/v2gNDlW1EBAC6SMg8k8fK8UsEgIYKgbxP/nkk6ZiRR0/kSvsHIrPVU466dmIQ+OtevXqZvbs2ekkUxwhIASKjIDIv8iAK7tkBEBMMCQNuLHv/3HHHWfJw/VPTqm7TBAAjrwySZdpXNZn5cqVDZb8ffbZZ5mKUHwhIASKgIDIvwggK4vUCJCQXn75ZbsJTf/+/c0BBxxgWrRoYQ+g4RyA1BIUUh4CwHifffYxp512WlGG/TnS0Lp1a7vWf8mSJeWpqHAhIASKjIDIv8iAKzt/BGrVqmVJ/+KLLzYdO3a0vVR8+wdxsTfpn1K+ZSEA/GAwknL77beXFTXvYZ07d7Yyv/7667zLlkAhIARyQ0AfVHPDT6lzRIDE3rVrV4MLxiV8hueYjZL/vs7fi28hgcGwP9b746AmjDzICAEhEBwERP7BqYtYakKid0nedYcVFO6ml0r/XMvIHj3l454jJfSjTYxx77oZXih7q622Mg0bNrTLNakv9SxUnpIrBIRAegho2D89nBRLCGSNAIkvawFlJIRsXGhM+G2jy7xxwh62SM610VGGKr5BmLuBbZrx3R9546JOvgnkKQSEQFEQEPkXBWZlEjcEQHKLFy82b7zxhvnggw9s8eE3adIk8/zzz+cFDpI+5GINPw7uGTJkiJX92muv2ZMQeSwyNvnZdddd85JvJkKwQRPMxIkTfRsnmchSXCEgBPKHgMg/f1hKkhBIQmDy5MmW6Pv162fPtn/zzTftVsXnnHNOUjzcZNMbBunDoEc/YcIEc8QRR1iSfeihh+ypevfdd585/PDDrRu77SFesU2DBg3sccLYrZH6FlsH5ScEhMDmCOib/+aYyEcI5AWBQw45xOy2226mefPm5sMPPzTjxo0zX331lZk6dWpiqB4ZgfgxLH/qqafafEGSfo0BbJqDPRCwAZJrttlmG7sFMmQsWLDAzurfcsst7al61113nR2BQPxSkC/ybNu2rR3xgA4crXD1l1sICIHiIyDyLz7myjEGCJC8d9xxR7t/wZVXXmnefvttgyWNmAXvEjHcOAIXnwOQDpd3P3zKow0IkY73kL1u3TpzzTXXJHZFXLNmjXVj4l2pDPTbZZddzHvvvWemTJliPz3Azy1/qXRTvkIgzgho2D/Ota+yFwwBkhvsgw46yJI5Tib0m5QHJVwiL0spykUcEj/sF154wZx00kmWaBk2atQo065dO3siYvfu3c1ll11mRTNdWfnkKwz6YvQD2zR/8skn+RIrOUJACOSIgHr+OQKo5EIgFQIgvlWrVtnh9xkzZpilS5daIvaLj4l55W3Cg+F9HHRUpUqVBPFD1k8//WS/+T/++OPWH/niON2PPvrINgpw37NnT3v55V0oPzYyoHf79u3tKY0bNmxIjEwUKl/JFQJCoHwERP7lY6QYQqBcBEh0jMh7EDK+u7/zzjv2ez82u0EjoG7duoxqbfSMDz30UOsGWXsN5GH4HvEom3Gwfz4OQ9pjjz0Sw+n33HOPnQDYo0ePRIOA8YthowxuObBrI7BYtGiRqV+/fjFUUB5CQAiUgYDIvwxwFCQEMkGApIwd7W655Rb7bb9bt252wlujRo3sN33Mej/++OM3I2R84+/SpUuCMF3ihA6QjQv+tPEJAelwLkKrVq3srHqEPfvsswZb6j733HNJjQWm88rOpIzZxt15553tqgNMfET5ZYSAECgtAvrmX1r8lXuEEACp4kIPHaS87777GvS80Vv/xz/+YTZu3Gh69+5tt7z1Fptp4Q+S9l7wJ2nDJpFjhj/IH42LRx55xLz44ouWZEeMGGGqVatms0F8NAawD0CpDEY6WrZsab/7Y+hfRggIgdIisMUmvEVkhECREejbt68ZOXKk3ZyGpFZkFfKand/fiASNjFw37r1ldtOT3F0F/fwQjuWDWFKIOQVYU5/KXHjhhWbYsGF22B1xvPmnSpcvf4xSjBkzxjz66KPmtttus42VfMmWHCEgBDJHQD3/zDFTCiGwGQIgU++F3j/9XLcf8TIew9x7Pz8qgGH0Nm3a2DkE3gaEnwykc+NRTjFsHPELnTBHAY0B6MGrGPkrDyEgBP5AQOT/BxZyCYFQIYB1/JhEh010KlSokOjN+5E7/EC8tEtRUAz9Y3QCIwBc8uinayl0U55CIG4IiPzjVuMqb+gRAGHioBys7T/hhBPs2n58QqHhSAHvaYNw2QCgXzFt5H3wwQebH3/80cyZM8dmDb9U+hZTN+UlBOKGgGb7x63GVd5IIICdAjFvgqa8HjTOE8De/zClIFvkCR07dOhgdzP89NNPDU78g4F/KXSymetHCMQUAfX8Y1rxKnZ4EQBReq/y5hTg0wB2Giylgc7bbbedXZY4evRoux1xKfVR3kIgzgiI/ONc+yp7bBAob2SgGECwwYJGCE4YxMQ/mCDoVozyKw8hECQERP5Bqg3pIgQKhABGBmBKRbRuvhj6r1Gjht2SGP5oFMgIASFQXARE/sXFW7kJgZIggP0ABg8ebPN2ibhYyqDxwXxx3HCnTp3MxIkTzbJly4qlgvIRAkLAQUDk74AhpxCIKgJDhw61p/px6L0U5XTz7tWrl13uh2V/bBSUQiflKQTiioDIP641r3ILgRIhgEZAnTp1DPb7xyZFGvYvUUUo21gjIPKPdfWr8HFDIEhE27VrV7vdMI4eVu8/bk+iyltqBET+pa4B5S8EioCAO+RehOzKzQIbDoH8MfEPxx2L/MuFTBGEQF4REPnnFU4JEwLBQgCkiuuOO+4wc+fODQzJojGCLYlxKNGsWbPsAU/Qk3v+BwtFaSMEooeAyD96daoSCYEEAuzxY4Z9lSpVEv6ldkAvkD16/7BxLDFs+pdaP+UvBKKOgMg/6jWs8gmB3xEAuQbJgOh32GEHu98/tvv96aefEg2AIOkpXYRAFBEQ+UexVlUmIZACARBukAzW/x9wwAFmw4YN5t1331XPP0iVI10ijYDIP9LVq8IJgf9HYPjw4eaaa66xPesgYIJGCK/GjRub3Xbbzbz11lvm559/1tK/IFSQdIg8AiL/yFexCigEjHn//ffNww8/HBjy99bJkUceaX777TczduxYb5DuhYAQKAACIv8CgCqRQiCoCARt2J84NWvWzJ7299JLL9lDf+gvWwgIgcIgIPIvDK6SKgQCgwAn+vFwn8Ao5lHkmGOOMb/++qvBcb8yQkAIFBYBkX9h8ZV0IVByBPhtPcjkDx1btWpl2rVrZ5f9rVy50n6iQMOFjZeSAykFhECEEBD5R6gyVRQhkAqBgQMHmvnz56cKDoQ/GifHH3+8WbNmjXn99ddF+oGoFSkRVQRE/lGtWZVLCHgQALkG9Zs/e/hNmzY1u+++u93yFzP/ZYSAECgMAiL/wuAqqUIgkAgEeQidnyc483/w4MGBxFBKCYEoICDyj0ItqgxCoBwE2LMOKvlzVAINgCZNmtiNfyZMmGD3/EfRXP2DWoZyqkDBQiBQCIj8A1UdUkYIFAYBHOzTo0cPKzyoQ/8sOfTr06ePPfjn0UcftYf9MEy2EBAC+UFA5J8fHCVFCAQaAUz2mzx5cmi2z8We/5j8N336dPP5559r8l+gny4pF0YERP5hrDXpLASyQIA9/jAMm+No3/3339/UrVvXPP7444mNf8KgexZVoyRCoOgIiPyLDrkyFALFRYCESZuNgOJqkVlu0LFy5crmjDPOMMuXLzdvvvmmev+ZQajYQqBMBET+ZcKjQCEQfgRApK1bt7Y9aTYAwlAq6N22bVu79A/b/s6dOzcMaktHIRAKBET+oagmKSkEckOgf//+ZsiQIYFd5+9XOjRU0AA4/fTTzTbbbGOeeOIJTf7zA0p+QiALBET+WYCmJEIgjAgEeXvfVHiC/KtVq2aOPfZYM3PmTPPqq6+miip/ISAEMkBA5J8BWIoqBKKAAAg16AY6srEC94EHHmg6dOhgXnnlFTNv3jyt+w96BUq/wCMg8g98FUlBIZAfBDiMjpn0YTEgfjYEMPmvYsWK5sEHHzQbNmywRQjTHIawYC4944GAyD8e9axSxhQBkCOuq6++2uyyyy5m48aNofru71ZbrVq1zGmnnWa++eYbM3ToUPv9Hw0DNQBclOQWAukhIPJPDyfFEgKhRIA951WrVpklS5YkhtJDWRhjTNeuXc2f//xnu/Tvyy+/DGsxpLcQKDkCIv+SV4EUEALFQSAKvWT08k855RRTs2ZN89///tcsXbrUjmRwhIN2cRBVLkIgvAiI/MNbd9JcCKSNAIkfdhiNqzeW/V166aVm3bp15r777jO//fZbGIsknYVASREQ+ZcUfmUuBIqDQPfu3c3ll1+emCVfnFzzl4s7SRENgUaNGplTTz3VTJs2zX7/V48/f1hLUjwQqBiPYqqUQiDeCPTs2dMceuihiSHysKHBnj9sXCD7bt262bX/I0aMMDvttJM9Bjhs5ZK+QqBUCIj8S4W88hUCRUQAZOkSaBGzzktW1J3CUB7sA/DXv/7VTmTE7n877rij2XnnnX1n/3vTU45sIRBXBDTsH9eaV7ljgQCHw7lhDu6jYNj733LLLc0FF1xgcAQwJgB+//33mxUvKmXerGDyEAI5ICDyzwE8JRUCYUFg8eLFZvbs2Ynef1j0Lk9PEPv2229vLrnkErvu/7bbbjMrVqxIJEO4ev0JOOQQAgkERP4JKOQQAtFFAKS4//77+w6Jh7nUJPaGDRvaBgD2M/jnP/9pVq5cGeZiSXchUHAERP4Fh1gZCIHSIQByxIXZ8uvXr49ML5jlog2E27RpYy666CKzaNEiM2DAALN69epEeTECwBUD+gxQuudROQcHAZF/cOpCmgiBgiHA4e+oEh/L17FjR4Pji7EF8L///W/bAACoDI9q+Qv24EhwZBEQ+Ue2alUwIZCMAHq+nPiXHBL+O45uoCR77bWXOeuss8ycOXPsCMAvv/ySVEA1AJLg0E1MERD5x7TiVex4IACiw/WXv/zF3H777Ymh7yiWHg0AGNjYAwDLANEAuPXWW83y5csT8x0YL4oYqExCIF0ERP7pIqV4QiCECJDo9ttvP3P66acnvoGHsChlqoxyei8cAHT++eebBQsWmJtuuslgxQMMG0SuXaZwBQqBCCIg8o9gpapIQsCLAIgO5EjC84ZH7R7lxCeOPfbYw25rjFUA1157rZ0LgLIiXEYIxBkBkX+ca19ljw0CJH72jqNecJYX5WzXrp254YYbTKVKlewIwNixYxPkr0ZA1J8ElS8VAiL/VMjIXwhECIHPPvvMvPLKK7ZEcSE8NABgYDdo0MDcfPPNpnHjxvYkwCeffDLS8x8i9OiqKAVCQORfIGAlVggEAQES/VNPPWX69etniTCqM/5dvF3ipxs7AV5zzTXm4IMPNm+99ZZtDGBPABjihBURdLvy5BYCUUNA5B+1GlV5hICDAIiP5Od4x9ZZsWJFc+aZZ5pzzz3XzJ07184DGD9+vB0FAPELq9g+GrEruMg/dlWuAguB+CGA3jwukDtsrATAZ4BatWqZQYMGmYceeijpTID4IaQSxw0BkX/calzlFQIxRMAdAcFnD9zXr1/f3HjjjaZPnz5m3Lhx5oorrrD2xo0bY4iQihw3BLbYpA9ccavzQJS3b9++ZuTIkfYIVg21FrZK8BdfuHChwXK3li1b2p5vHL77p4sq8MFmQOj941NA69at7acBTBKEQTgbD3xd6plNF13FCyoCIv+g1kzE9RL5F6+CSVgkMeQs8voDf+CCa926debDDz80gwcPtnMAsDHSsccea6pVq5aIjEaTi2MiQA4hEDIEKoZMX6krBIRAlgiA8DmpTeT/B4jAAoRepUoVc9BBB5kOHTqYF1980YwZM8ZMmDDB9OzZ0xx66KE2nA0F4fcHfnKFEwH1/MNZb6HXWj3/4lUhCMtrRF7JiBAj2MQGJwMOHz7cTJ482WCZ4BFHHGEwGlC1atVEnGQpuhMC4UFA5B+euoqUpiL/4lQnSe399983s2bNsqfdwU/f/P3xJ14IpfvLL780w4YNs1sDb7311ua4444ze++9t4GbhqMH7j3dsoVAEBEQ+QexVmKgk8i/uJV8wQUXmOeeey5xuA17t8XVIjy5kfhd220EYB5Ar1697OmBcCMeGwCwhW946jqumuqbf1xrXuWOFQIgI01Wy7zKSeKwd9llF9O2bVszY8YMOydgyJAh5rXXXjN77bWXXS5Yo0YNmwHTZJ6bUgiB4iEg8i8e1spJCJQMAfRMMdmPDYCSKRKSjNmTB5HDjQumQoUKZuedd7aHBX399dfm3XffNe+884557733bOOgR48epn379ur5h6Se46ymyD/Ota+yxwIBl7xIYrEoeA6FdOdEoAHg15tv0aKFwXXUUUfZlQFYHYADlJo2bWq6du1qJwduu+22iU8CUIcNMLhRF5RLOweVlVQIZISAvvlnBJci5wsBffPPF5LlyyHhi2zKxyrdGFwyifjEde3atbYRgL0CsGnQlltuaRsAe+yxh2nTpo3BuQKsCzYo3Pt081Y8IZAPBET++UBRMjJGQOSfMWQ5JXDJCoLU08wJzsR+CSR+SKMbNpYJ4uTAL774wqxcudLUrFnTThDcddddTb169ZI+vyC+O9KQm2ZKLQTSQ0Dknx5OipVnBET+eQY0hTgQCwxtkj7tFMnkXQ4CxJN2KjzXrFljPv30Uzs3AFsHI36TJk3MgQceaHbbbTe7fwCySpW+HDUULASyRkDknzV0SpgLAiL/XNDLLC0Jiqlwr54m0Si8Dbwx8vLjjz+aTz75xJ5psXTpUrPVVluZxo0b2/kB2DcAOwyiEYD45TUGygsvfKmUQ9gREPmHvQZDqr/Iv7gVh0Nr8C36iSeesMQi8ige/t7G12+//Wa+++47M2nSJNsQWLFihalcubI9dAmjAZgsWL169YSCTI86w5VO4yCRWA4hkAIBkX8KYORdWARE/oXF1ysdm/xgl7pFixZZ8lDP34tQYe9J4MyF97CnTZtmtxCeOHGiPX0REwNx+iL2FejYsaMdHSDpM70ab0RCdrYIaKlftsgpnRAIEQLsNUJlEUfpK86tg3bt2tm9A04++WQ7URCNAOwm+MILL9hdGevWrWs6depk42Bp4TbbbFP6AkiD0CMQefLnEBlnO7PF7e35MJw16v456SdbCIQVATz3vPRsF78WU2Hu9efeAair5cuXG5zJgIbABx98YF5//XU7VwOfBjp37myaN29u0DDA3AHXIK1XLsPhX1Y448mOPgKRJ39UIR72xYsXm48++sj+ebxEzzhoECAM23Tuv//+0a99lTA2CGCpWaNGjVKSQmyACFFBcZJgnz597IVVAzNnzjQYFZg6dapdQoj3Gg4XwvJBHEOMxgDeXdiFMBXJI42MEAACkf/mz4f9nnvusftxo2WdqlXMRgF26PrHP/6hJ6SACOibfwHB9RHN/wFsPP+p/gM+SeVVAgRYX8zavd+4caNZsmSJwfbC2FFwypQpBhsMgfTRyEMjAFsQY4QAhw7B3zWQ5R35dMPljgcCke/58yU3fvx4889//tMewuH38uOfCzbTxOMRUCnjhABHt/SMB7/WUUd8L7n1hQmBderUsUP+++23n1m3bp1ZsGCBmT17tm0ITJgwwa7swA6DHPFp1qyZnTew0047+b7f9N4L/vOQbw0j2/Pnwwx7w4YNdvYsWskYSoNx/0z5BlXyykdAPf/yMcpnDIxqgfj9yCSf+UhWaRBAvfKdt379eoMNhTAygAbBV199ZbCcEAYjAWgIYH8BjHC2bt3a+vlpTXluI8Qbz43jDdN9sBGIRc8ff4DtttvOeA/ZCHbVSDshkF8E8KKG4Qs7v9IlrdQIsEODCYBYKohPnGzwYWQASwrRKPj222/t5EHsNwBTv359G79hw4amQYMGdm4I3pWUx+fGr3yM4xcmv2AjEHnyx4OLlm+rVq3swRrBrg5pJwTyjwD+AzfccIM9e/7zzz9PvNTzn5MklhoBkjEbeCRuEDwuGPhhNBTvRTYIYGNlAeYTwGAVASYSokGAswh23HFHO2rKxgTzsZF//2FefmFuPLmDgUCkyZ8PI5bKYLMMGj2cREJ2XBDAsrHvv/9evf6IVjjeaXjf8Z3nLSbD6Y/5ACB3TAqEwSgAVhTMmDHD7jWAg4k+/vhjM2rUKNtQwDwD7C+ATwUYVUCjAI0J+FWtWjUpX+igdyyRDq6dN/L3PnTeB6CUDwMe4ttuuy1RC9TV1dF1I2Ip9U0oKocQyAMCepbzAGIIRJRXz2443O49GgPYUhjHD++5556WzDGRcPXq1bbRiM8F+HyKxgGWGqKxgPT4PIB5VJhIiAYBGgew0SDA5wfvexX3HD1gmOvnhZk6Io5r6O/6yZ0ZAnknf1QKKoqVxUry3memZvaxMbyFIS1slek10I3L+xhG3b3LYxguWwiECQH+76Aznne8eL3PfJjKI11zR4DvZFeS1w/3lSpVstcOO+xg2rdvb9/peHZ+/vlne0gRlhtiNGnOnDl2/4Fx48ZZkXjG0JBAOqw2qF27tv10gImGGC3gM4k86OZ7FwJcXRhelq5umNzpI5A38keWbqXhAYBB5bn+3vv0Vc08JvLCedpoibo6UBLCqSd1he0Xl2lkC4EwIoBT49C7wzMvIwTKQ4Dvafd5wXsRnSL09HFhHhX80CBAvGXLlpkffvjBNgywqRomGeLzAZYeUh4+H9SqVcvOIcA8AjQOcKFRwN0KGZe2V9dU/t54ui8bgbyRP4aFhg4dau69914zZMgQs++++9oH48orrzSYZPTuu++W5MWDs7TR6/cSOjbFeOCBByz5owGAdbM4+QyfB9z5AWXDp1AhEA4Ejj/+eHPccceFQ1lpGQgESLKwceE9ST9XQXagSOpuGNx414IfMEqAY41xnDFGDbBbIeaiwFAuGgJoBOBiIwO7FnIUAaMRMvlBIG/k36RJEwOiB8m//PLL9lhKqIgJJThNDJULQxLm/eTJk+3Oe/RnsRgOu02bNuakk05iUJKdaggT6fBQYsgfQ1Yw8EM+v/zyi8HLsHv37ua8886z/tjOFy1V6OvVJSlD3QiBECHAZ5nPPlSnX4iKIVWLjID7jMDNe9p+6qQKq1Klin2H4z2O59A1aBjgeGM0DjBagNEDNAgwYou9CRCONLjwPscEQ2xTjcYBPidgzgEaB/DHhX0M0EDw6oJ7yvGGufogjjecad14dPuFpfJjmqDYeSN/FAiFRgX/9NNPtnwA8phjjjHz5s3zLS+IG/EvvfTSzQBnRaHCMVTkNQiHQZ6ucf3hxqhDv379bMWzUv72t7/Z4av+/fvb9PyOdeKJJ262FaYrW24hEFYE8F/AfylVYzms5ZLe4UUAz2TlypXt5wOsIIDhOxpLDjFfCysQ8CkBowY4jhqNA4waoNOIhgEuTD7Es40Ln7YgE58UcHHUAKMImIeAC+GYjIiL3OLlDehBXWB9KyUAACAASURBVBjmRZrhXv9U8b3xSn2/OavmqBGGZ+bPn5+Qgs8APXv2TNy7DlaW31COC6wfmPBDHK9hOoTj4cGDg12sGB+zVZ9++mkzYsSIROWix4+NL7BVJuN55epeCIQZAez6hpckekcyQqBUCPD9jPy972/wAd+/IGVcGDUAgeOsAoYhLd7tv/76q93aeNWqVXa0AJ8U2DjApwWMHuC5R1w0JtjwBd+Q/LEqAaMF2AQONi40FJAnbNyjQYG5DrigMyeDQx/qRNsPV4bB9is307jhrpvh+bbzSv5QGC0rbiW5cOFCCx7Oq/ZWNAqC+Kxw2K5h4V0/1015iIdhoieeeMJccsklSflgiV/btm1tHkiLuDjgB9+mMAEKMuA3cOBAW6G77757Uno3P7mFQJgRuPbaa+3nNWz5yv9OmMsj3cOLAN+7KAHdtFkq3rs8AD8aNAww5I8L8wTw2RkG8ZkWNhoIWK6IEQI0EnCtXLnS2li1gMYCPjeg84fPwUhPTmJeaCzgwogBbZymCDcaD7xwz0YEP0EgHsLdclAu/aiv198tD92wXRzc8nrDcF+WyRv5swBoLQFUgP3YY48ZDLG7yrrKwB8E/eSTT24WxwUG3+zPPfdcN2nC/cwzz5j//Oc/9sxrTOw7+OCDrSzIxhnY2MgChiBNnz7dbnvJ4R7s9//aa6+ZAw880DZcWI5EBnIIgQggwJdfqv9iBIqoIoQIAfc5dN1uEVL5u3Ho5vudaWiTmBnPa5Nn4I+RMTQO0FhAQwA2PjvADZtuNB5wYaSB8ZGeOsCmm3qgEYBGBRorsKEXbPjDwB9p0MDASAPSIQ5sxkc83CMOwmiYDv4YlcC5DemYvJE/CwvyB2iY8Y9JdRi2YZirEPxgsKEENpZAAV3DcNfP60YcFBhD+JCDU/tA4pCFIR5M9jv88MOTkqFFhh3/MC8BxI8zslGRxx57rAU2nXyTBOpGCIQAATzXfBGFQF2pKAQyQiDbZ5vp8P8A6XKOAP8v5fEB0iMOGwqwcWGDJDQM8NkBbjQc8OkBDQnYGJEARyEuDEYfcM85DJCJuPAjnzEv6ox0Xv3Avw8++GBa2OWN/KkYwMP3FiiBTR2gnKsstaJfqnDEQxwUnm6mde+PPvpo63355ZfbYX+sNDjyyCNtOpD/jTfeaMOp3/nnn29OPvlk+zkAabAkEZWDxoOMEIgqAnz+y/q/RbXsKpcQSBcB/k8YH/cwLsl64yAMnUpcmGSYjuH/kLZfGpf40VllvmgUoPHgGjQ0wGOYy5CuyRv5ExwMp1988cWWgKEE/MsqIFo1TOtVuqx03rCzzz7b3H///eamm26yEwzRukIjBA0Q12B5H5aVAEgMndx99932MwC2p/TKdNPJLQTCjECPHj1SbnYV5nJJdyGQDwRI8q6ssnjJjeeX1g2nm+SNe7pd2xuPo+GwMZGe+WCOA91MQ11p078sOyvy92aAeyiDyRM4Ger666+3ecLPqySVcf1dt184/VybaWhjyOb222+3jQ58cgCZY/Y+w2HTjRYadMb10ksvmd69e9uhFYa7+cgtBKKAAFbcpFp1E4XyqQxCIF8IZMoDmcR349JN29Xf6+e9d+PCzXDa3nC/++QP7X4xfPxInAjCtwwMo7/55ptmwIABttcPBTJRwieLjL2gE3o3+OaPb/+vvPJKYqe+VLpg7SgmHPbq1avo+mZcQCUQAjkggP+HjBAQAkKACGRF/kyMFwomLODYx1dffdVcddVVdgJeKrJlukLZ+ORw880327X9gwcPThzmw3kDbr5YJXDWWWfZuQGIi+WJbqPGjSu3EAg7AhxCDHs5pL8QEAL5QWCLTVl0CUimZZF8WWH5UT1ZilsMjEQMHz7czvbn+k+vPiR6+uOebkh23ck56S4fCPTt29eMHDnS7twlrPOBaNkysL4fn+WwlwWMGgNl46VQIRB1BLL65h/UFwcbAP/6179MgwYN7MYPXlJnhYJwXNJx3YwjWwhEAQH8BwYNGmSee+45u6QoCmVSGYSAEMgNgZyG/XPLOv+pSeD16tWzp/MhB/rlPzdJFAJCQAgIASEQTgQiRf7s5YPwOQoQzmqR1kIgvwioEZxfPCVNCIQdgciQP8je/RyhBkDYH03pnw8E2Aimjf+FGgL5QFYyhEC4Ecjqmz9fJH5FRxiJ1+8lw3BvWr+4bhxMMnTjuHLgj6ssvSDLL9yV4+YHdyqZSOM2NLzpUuXlFy9TP+qbSjfIc8Pg5k5Rbl5eLCiXcXjvyvKG8b4sm/lATirDOG64N75b/4gP/KmjX3qGuTL93OXF8yu/nxyvvq5OqfJwZbtuyPfeM0/K8oZ77xmfNjbBwpJWpE8lg/lyQi/uiTPlMC3vy7PTje/qjzTu/wv3fobx3GcD8VxZfuky8aP+kAnjzYuyGM57V2fqQ1mMk47NNJTBNN57+vvZlIEwpoMNQz1x71c2hPMZoBzGTSWPsm0G5fxQvl/ekOPq54ryi++Gp3K7MhGHZfK6/e69ad08yisz82F53LSu282DadxwPzfisY7K08NNn1XPH5m5hWCGrrL0czPzc7ty/MLT8fOTQV2oB+55uTIZ7vrB7caFuzzD+K5dVhrG84uTTn54+F3jpmEYygZ/2m78ssqNeEzjyqU7VVqkYxw3Lz6YDKdOvHfjpnJTH4TDzTLSdmVBvjdPpvfqR39vvtTRG9/Nj7p40/KeaZEHDO8ZDlnw815+cennleWXFnG98XCyJna39AtzZUMnpAV+9HdtyoWf67aRfX7cON7yMzr8mS/9aDMNbdcfsuHPuvbGQVw/P8pgOOPQZjh1hz8v1w/xqAPT0KYshLOemZY2ZTKNn50qLmUyH6b13tOfNvOkTX/siOrmBX/EgR9lesOZlv64d+PznvG8NuV765750Xble2X43SMd03rDXX+4XdlwE1ekc8NwX5Zcbz5+934Ye+W6+nnz98ZlHojHdLQZVpadVc8fe+bD4PQ+HsoDBXDNmjXLLFq0KJEnjjlEHBrEwcY6OPCASmPrQpzX7BqcyOeapk2b2l376If9jT///POEDPi3adPGHu9IXbCJz+zZsy0w9MNSJ+gEkOCHoxwXLFiQAA87Be61114JuYiDc6FRVhikw7HFHTp0oCrW9uqL1QaNGjVKvERxpvS4ceMSLyokatmypalTp05CzpIlS+zyRORJg3xw4hPNd999Z+bNm2dvEQ8vPh5PDE/ohzrAiVO8x+FKXbp0SZQZ/uPHj7fnXPNhwZ7UrVu3tuVmurFjxyZwgF/z5s0NJlPSYD/pSZMm8dbaIBnUJw10hc4wLBfqgPnCH3Uwf/78RF7Ynxp14JrJkyfbOuCLHnWAExspE/JcfeHfsGFDWwdwIxynduHZo4E/ysQ6QBzghsOeYCgbdYBdIenHOkB8XNAXdeAa/EewtA7hMH518NFHH9lDPhindu3aplWrVq4YezKl6+GtAzyXeD5hWE6cbomzNWigLy6WBxjieeD/APnPmTPH7o+BNCwTzrvAyxDxkRYHYWFTL+qLPJAX0yCO93+A/239+vVtHITj4JJPPvkkoSsCUGaUnXJxFDj+t5SL/FEHPAEN/igPnhmkgVycZrbPPvvYe8qZOnWq3b+D5IL0nTp1snL5g/8BXspIg3zwLLRo0cIGIx2M97nCluF169ZN6Isjxb/66iuKtDZwwfGu1MXVFxGgL/B1z4b/5ptv7LuT+eKZ4dJMpEE5J0yYYPdw5z3OUsGx5QgjFthlFWVhmXDKG+sA6VCHqEukocF/H0edw8Af70T8L13TuXPnxD4uiMPnimXEIWt417typ0yZkjjiHf54l+22226uWIsv84Us7M6K55zlwf8W/xUY5oU6Qh3Q4N05Y8aMpEYk3g94T1AOtnXnuxPp8K4nf1EOlsTikB0alIl1QDl4h0AnGPgBN3APyw0//A94j3jeZwb744C/UP98PslfzBvPN+qA+UIe+Ytx3DIhnPzF8DJtrPPP1PTq1WsTrurVq29auXLlpv/9739WxMaNGzedeeaZm7bYYovE1bhx403wx4V4uJo0aZIIB35HH310kgqIA3/3GjRoUCI9widNmmRluHGGDRtm4zCvAQMGJPKBTpUrV970zTffJMm5/vrrE/kgTp06dRLhUAqy9t1330Qc5Lf33nsn6YsbVw+4//73vyfkQMaSJUuS9EVeDz/8cCIOyvT8888n9IUMxPnoo4+S4rj6Is7222+/ae3atUn4nnTSSUn6NG/ePCGDiqOcrs4nnnhiIg50wYX8GQfu//73v0lxxo8fnxQHcV966aWk+v7nP/+ZkIHwqlWrbpo3b96mU045ZVO9evWsvCuvvDIpTt26dalmwu7SpUtSnD//+c9JukBf6kr7hhtuSIqzYMECG8ct1yOPPJKE3ZNPPpkU509/+tOmzz77zMrhc3Xdddcl5VWjRo2EnnTgmXbzadeuXVI+0He77bZLknPGGWck6eutA5TrvvvuS+iCvMaMGZMkA3HeeOONJDl8ZqAPrq233nrT4sWLk+JcdNFFCTmIU79+/aR6hC677LJLUpzu3bsnyQA+LDPzuvXWW5Pi4P/HMMYF5pCPCzIefPDBRD4oT8WKFTdNnTrVQst4qFvWM+xatWolZDBOz549bRzm06FDh0QejIN3AtIzTv/+/ZPirFu3LikccR944IFEXtAXeLu6wD169OiEHMRx6wDheHf+9NNPCTnQ55xzzknogTgNGzbc7Jlp2bJlkj69e/dO5AOAkJdXl4EDByblAyy9cYYMGZJU30jjxtlqq602zZw5M0kflonxateunSQDZTrggAOS5Oy5555JMlx9WQeXXXZZkr4///xzkgzk99BDDyXJwXsH6SkDccaOHZskB88Mw2Hj//fLL78kyenbt29CDuKAq1AO6Ekb9QL5lHXcccfZMP73EY+Y0Hb5C7ImTpy4WZzhw4dbERs2bLD53X777Yk8IAfP6ty5c5PKdPXVVyfFwXs9XZPVJj/Tp0+3DQr0/NAix856NGixo/cEPkTrE72ijh07MtjaaPGgB8CWEXqKbo8Had0eGhKhF80eGu7RekWrEjJwIQ160mgJU+4PP/yQ1OtEHOiCHg8NWldo5dJQX8qEP8rLnj/u0Xplj4fpvC1TtF7R+6d+aCm6PR74o1fEHg/u0VNEr901GBFxe/6uvkiDliNa5HDDoIzoQUAWW5RovaKeaBB34sSJtgfBHgIOi2CPh/FYJt5jwyToy7xQ/zgeGQb5wh89CPQIeY/RF7e1jfzQ8sc36Hfffdf2NhGOeEgPnaGvt4eGunZ7nWjRo6UMQ328+gJ/1APDceoVRiqgGw1a5O5JXDgMCq1/pGG89u3b214ny4TnBfVAffHMuHUA2Rg9wElcSIMLh0i5IxWI8+mnnyZ6nbiHHt6DqNAzpf6Qg3DUAQzucRw1nk+4qTN6gm6PB/oOGzbMPhennHKK/V/ieWDPH7LQg8D/hc8D/wcIY/6sA+aDPJAXdYE/6oDh0Am9To4W4Z51gDjs4eJ/i+ePcvD+YI8HfoiLOnDPMEeZ8NwwL+iLZ4b3sNETRK+c2CA95CCMeaEnDcP/CrDF/5LhsN13EdLif+A+M3g3IC/X4NmsVq2a9UIaV1944n/Ldyf1QZkxakp9WSbcwyAenl+8O2Hgj3cn8KMM+LMOWCbWAeVg1JSjRVaQMfa/zzqAH/TAM8E0eC6AHf6bNPjPolyMg14n/7fQB/7edydG0PjuZDovvujR433PMuHdifcVDNPgf4AeN59X/Nc4Yod4SItnk3UAP+iK/y0NeAt8gLqgwbsTowjIBzLwHyF/MW/wF55j5u337nTLBNmoA/AXZaAOMDLlGpe/EA//R/fdCX34v2U6lw+QBs8MRvXSMVmRPwuADOBmJfGeGdPfjcO0DGMaAOkabzy+KJDOleemoZtpce/Nh+kZBrnMm3JpUx5s+tFmem8c3iMeDGRTd9z76cM0sCmfttfPT4ab3pXPtN40bny6mZ/XdmW4chgP+bF83rxTpYX/qaeemtjhz60DhMF4ZSE/5kn796jWcv3o9rO9spE38sLljY+49KPbm57+lIF7GKSjzbLQz5sX4/E5tAmdvL06UB7joQx4gVE+/SkX9kUXXVTuJj9MT/nM18+feXjLwrRu3vRz5dGP8cq698ZBXAzVAy9XJuIhzPXzk+uWxy+uN5xyqQdsGOYFt7fu+H2X+UMm3wVM59r/LzH519UjOeSP5wv+bh6USf+yZLgyEY9y4J8qHf+rtL0ymC9t/r9cmUyDPLy4MV/WLeXQ303r6kv59HPjExPaXplMQxluONMwDuW6/vBjONLCeO+JgxvXz035v4uxchgPNp8h5sEwxqe/e5/Kncy4qWKV4e8WkorDDxfv3eQM8/q593S7sl03w/1sb57uvVcGwrwPH+O46ZAP/b3xqYM3PtOw0pGeMhiX94jr9UOY6+fnZh6U7crzhuGexo1HN2zk4d4jPv3gzzj0YzhspnP9EI9/YoQDC9cgHIayXRluPLoZTpv+lAGburk24zEf3sOmftQFfq58uhHOuqdshvnJhRzGY36p4rn+SIOLenhlUF+Gw6Yf09nEPj/Ixy8O/GnodvN19UM83sNmPMql7cajH/Fz82C+rs1wPz/myTLjnvEZ5t57ZVAXbxzeIxxu3tPtpqMfZDMewt3LLStlUoabjuldPel283HT0u0Npyzarhw3T9ffGzeVbKZnuYg/ZTHclYf/uxuPYalkMBy2qwfcDGN+bhzXD27GRxz3cmVSX6Z1beZFm2GuXDe9G4/5Ia57IT7zp+1NB3+mZ57efBiH8hCf8tw06bhzJn83E7cw8Od9qgJRadquLLjhz8sb5nfPfPzydfNww1058Gc8xkk3HPG9F9KmkuPKRZ6Ix7wZxj8J7ymLthvfz+36IQ0v6kU5uEdc955x/HTwi0cdmY42XwC498rCPfP16urKc93evN0wuBnOvHBPP6+N+MjXjeuVx3ukpY50855xvDbz8/rjnmGwedGfYbx307MxCT9v/m46pnFlI757zzi0mR7xWDcMg81wPz/K9cZhnkjj6u6V4VcWr0zce+NBjpuHK9fPTZl+YfBz9Xfdfnggfqq8mY8bTrcr13X76cRw2tTRvaefm57hfLYRhvxpGI57uHlPm/GoM+9dm3Fdud5wNz3z8fohDfxwufoiPvwY7sqGm/m7/szD9aObYd78KYd5MT5thsN207rxKZtpvDbDKcsbjns3jPG98dw4xIp+tL1pUt3/8bE+VQwf/7IyySTMq7yblZ8c+tF249PNMNpe/1T39IftTZtuWDbx3Pz88qUf7VR5uOGu242fyo34ZaUpKyyVTPqXlZZ/JDeO66YM2K6/63bjeN1+8eiXyvbK8N4zndffe59uPKZLN3668SjXtfl/cxtkDHfl0s34jAObYa5fWW43Pt1eOx256aZhvLJ0yiTMlZcpHm5abxndMNftp1t54X5pXL900qeK4/X33nvL5ebr5/ZL75XhxnHdfvL8/PzSuH5002b+7r2fXMajXVb8bMNS5ev6Qzbl02a4957+fnZee/5+GchPCAiB0iMwYMAAO+EPmmTygii95tJACAiBQiCQVc+/EIpIphAQAoVDALOA3VUuhctJkoWAEAgDAur5h6GWpKMQyBEB9PZxud8pcxSp5EJACIQYAfX8Q1x5Ul0IlIeAS/acY1FeGoULASEQfQTU849+HauEMUaA3/dfeuklc/3119uev9sgiDE0KroQiDUCIv9YV78KH3UESPRjxowxDz74oB36Z4Mg6mVX+YSAEEiNgIb9U2OjECEQegRcomdDIPSFUgGEgBDIGYGsyD/VSwT+7svGqx3TIY43HsO8adx7yoeNNbe8R5x00iOeN19XvtxCIMoI6NmPcu2qbGFCIF2+Yjy/PSZyLW9W5I+jaWFw9CYMDsxwj9vE4Q88AhX+CMeLB4fukLRJ3F7bCnR+GA4vvrxgExTaTpIyna68MiMqUAhEAAH+P/jfiUCRVAQhEHoEyEO0WSD3HptxpTqzg/FzsbMi/w8//NDmidONYKAwSd0lZlcxt1BYb8x1xzgJCvc4YQkNBshBYwF+aETADRv3OB8bJ9y5l5sHX3B+Orh+7jajTOPKkVsIRA2Bu+++2wwcODBqxVJ5hEBoEQAnwrz88stm7NixZv369eaee+6xfPrqq6+ap59+2uB/i5NJC2GyIv/zzz/f6oKjFmFwxCSP6EWBUAgeO4ljWHEPg+NHQe44zhAEjDgbNmywR7VC1urVqw1sHOn566+/2mMT4UeCJliQBTdaRWgwoDEAG0eMwsZoA2xe8MfRjrgQBnmUaRXTjxCIOALu8+66I15sFU8IBBIB9z/Yu3dvU79+fXPggQeaY445xur7wAMP2KOZcWwyjsR24+erQFmRP0i10MYlepyXjUYAGg04txlu3sMPDQzYOLsbZyDTD2cuwxA4NhhwDjTOwcaZ3Dh7Ho0DNCDYSICbow9Iw/SuLI4eeMO9uCCtN4733ptG90IgXwjw+aM8PHsw7jPNMNlCQAgUBwH3/wd3p06dTM+ePW3Pv02bNgY9f4yO8/9aCK2yIv9CKFKWTJKyC5hffACFC8QMG+S/bNky22BYsmSJbRxg9AENiFWrVpnp06fbOBxlQBqMTPDCZ4a6devaVhkaCzVr1rTzFjCCwM8R+FwBg7TUz88NPxrG471sIVAsBPTsFQtp5SME0kcA/8suXbqYW265xdx3330Gn8NdzkhfUvoxQ0H+6b6wGI+EDAAxStGoUaMEOZOYYaORgE8SvNAoQCMBNj49wA179uzZ9nMEPlEgDT43YB4C5GMUoXbt2raRgIYChmjQUECrDRcaEt5KpA7pV5NiCoHsEOCzd9ddd5l33nnHvPXWW9kJUiohIAQKigB4ip/IXY4gr+U780CTPwoNEPgC8xae4bQR7gWKIHr9cQ8Sr1Klir2QFkTeqlUrmw3T4YbzEfA5gSMHGFFA4+Cbb74xn3zySVKloVHAOQYYLYBcNApatGhhJ2/4teq8+lkl9CMEckQAzxWe5blz55rJkydbt561HEFVciGQJwTIbRilnjRpkv2E/fHHH5vDDz88TzmkFhNY8ucLinaqIjCctjee1997743PezceevBYpogLhhUGN+JhNADzEjBigEbB0qVLrY3JGnjpTpw4MfHSxUgAGwQYMUDDoHHjxvbC/APXuA0Q19/PDT28evnFk1/8EHCf5fiVXiUWAsFDgO9qvuMHDRpkzj77bDvr/80337Tf/2fMmGHatWtn+cXvP8x3PmSAVzI1gSX/TAtSzPjeisA95gfgatasmSVhVipsDOWgIbBo0SL7GYE2lkqi0YDGAwzSY1kHZn7iEwLdaBR482R5WfGwZYRAWQjwmSwrjsKEgBAoDgJ497dt29Z06NDBnHfeeaZhw4amT58+ZsCAAbYzePrpp1tF+O73/n/5zmd4plqL/DNF7Pf4rAjXRpC3QlAx2KMAjQI2DNwsMfEQnw6+++47u2kS5higUYDRAxg0DNAoaN26tWnSpIltENSpU8c2DvD5gHGyfQCsAP1EHgF8ctp3330jX04VUAiEBQGsKrvzzjvNzjvvbHv40PvUU0+1n6Fhgzdg8G4nr+Cebr7zyUE2cgY/W2yipAwSKermCLgVQLcXWm8lUgrjsTI3btxoVyFg2AeTDefNm2eXMGIpI+YdYIgH8xWaNm1qHxrYGCXAZwk8MJBDWcwjaHbfvn3NyJEj7YhI0HUNGnaZ6sPni88l0gvzTFFUfCGQPwTc/6Qr1f1fMg79eI+OIQ7puvjii+2EdqRnHFdWeW71/MtDKM1wF3y6absiUvmxYhGO1Qq4sPYTFwwaBNjPAPMKMFKAa+bMmQbfhzhDFCsb8LkA34kwcRGNAkxohCzI91t5QH0Qjov3yNO9d/3d8sgdbARQhzJCQAgECwG+T2n7aeeG8V0M+/HHHzc33HCDGT9+vHnuuefsHjXwx5XJt3/1/P1QD6AfKhYGDwQrGvcYCcCqgwULFthRAswYXbx4sY2DiYqYXIhvSRj2xTpSbGwE420IUD5lIx/3QXIfRCsgxx/1/HMEMIPkrFskgRt1me/6zEAdRRUCQiBDBNz/MD4F33bbbebaa681u+++u12+i0/DMIjnvrfLykbkXxY6AQpzKz+VWnyxr1ixwsyfP9/OI5gzZ45d4oVGAgw+DWDuAK727dub5s2b208ISPvss8+aO+64w35mQDgeriOOOKIgZCHyT1WL+fV3nxs+H7DTfUHkVxtJEwJCIBsEQPhosLv/5/vvv99ccMEFdqQXI8D49JvJf1vkn01NlCANK509Ntzj4j1V8vNDGOYNYA4Blh7OmjXLjhTAH58KMJkQGxhdffXVSQ8XRg5GjBhh95z25sP8srVF/tkil126q666yrzyyitm6tSpBWnMZaeVUgkBIZAOAnyvw3bN4MGDTb9+/ew+Mm+88YZp2bLlZpzgxnfdWX3zRyYwIBKQAi6vUm4mcgcLAfT28TkAqwrQKMAyRDQMvAZzCa677jrbWHDrmG7YMHwwvenLusd8BcxhQOuVcsqKr7DcEMD3QdQ1JgrJCAEhEB0EDjnkEPPaa6/ZThrOBOjYsWNahcuq5/+Xv/zFCsd2oVyjnlZuihRIBEC+PGERCrpkjK2KMWdg4cKFlqyx2yGGjDkMhbh0Z1I4NCww2oAzEiBDjcdM0Ms8LvHGBFDUn/DOHEOlEAJBQ4D/Y3SkYLA3wGOPPZaWmlmRPzYngOExvmnlpEiBQwAPDokA50bffvvtVkc+ULjp1q2bJX8+XNiVEJNMsAoB8wew5BAGadhocNPbQJ8ffKt677337DC0T7C88oCA26j6xz/+YYf9p02blgfJEiEEhECpEcD/G/vE/PWvf7Xv0jPOOMOeCogGfjomK/J3X+5QDsV1DAAAIABJREFUIJueXzrKKU7xEMDaUQwfYf93mr322ssuJUQdYxOiL774wnzwwQcGEwpB+tiJEBtU7L///taNeOkabGLBdf7pplG87BHApzoQ/9/+9jcrJJO6yj5XpRQCQqAQCICD0Qk/7rjjzLvvvmuuvPJKu/wP87TS/W9nRf6FKIxklhYBPEyrV6+250h/++23duJIjx497IRA92HCfgP4dgwimTBhgt1rAOFoCOyyyy52OSG2rCzPuOTvyi8vncIzR4CNddjEmnbm0pRCCAiBYiPg7WBji/hjjjnGrvW/9dZbzaWXXmpPkXX/4+XpKPIvD6GYhJMgMi0uDjLCqYYYFfjyyy/tCYj4NLDHHnvYTwPYX4DLypgHiEfknynS2cUn5m5q+LFOXH+5hYAQCCYC/B/DRucLZwCgA4al2eeff37i8y20T7dhn9Vs/2DCI60KgQAfJDx0JA0+iMgPEwK7d+9uDj30UPv96cMPP7RHU7799tt2FKFBgwama9eutiGA0QFvC7YQOkvm5gj41d3mseQjBIRAUBHge/ell16yO7w+8MAD5rTTTrPqIgzvasZJpwzq+aeDUgzipCLlTB4o9wHEpkJjxowxOJsa8wUwsx/7CaChgM8DOMWK3/zZwIgBzCUpIuoFczowOQjHRwNvYV6SqlCmQiArBEjqtPFObdSokf0f493tjuSl+98W+WdVFdFLxIcKJcNmMIcffnjigUr3YXJRoTzY2G547NixdkYqzibA8j5sSYzNhjB05T64kIE0yJM2/LLRwdUn7u5LLrnEPP/883bnR+EZ96dB5RcCxoj89RRYBEC0uEDOmLCH7SLdzSIyJV/IoiGJr1u3zg5XocePyYKYPMjRgD333DPldyuk9zYQKFt2eghgaSUOAUFDDCbT+kwvF8USAkIgLAj8KSyKSs/CIkCyfuaZZ2yvfOjQoXnNEPJx3DBOHLzwwgvtSVQ4dwCNjYEDBxr0TF988UW7kRB7/VRAREUksrdZv9lLUEohIASihIDIP0q1mUNZQLDYyAfbQ+62227m4Ycftuv5cxCZSOolb9xjg6gffvjB3HXXXXa2KjYMGjZsmG0Y4IAhfBYQYSUgzNmhkZOcIZQAIRApBDTbP1LVmVthxo0bZ2ftV69e3Zx99tnmhRdesLtHZSPVj/D95Gy55ZZm3333tRe+/2MzGlz47IB9Bg466CB7DDEaAq5M1+0nV37JCBx55JF21IUNKuGXjI/uhEDcENA3/7jVeIry4vs79oW+88477ZB8s2bNTNOmTe2OfiCMfPcc3VP9oBLzgB44RwCfHbB3AO6xjBC7D9aqVWsz7UFi3obBZpHkkRhFIVYifz0UQiDeCGjYP971nyj9lClTrBvr9vFt/ogjjjAYCcBpcIU0ICESEZes4FxqbEP773//2+yzzz52JAD3Tz/9tFm5cqVVx01TSP2iIhukD0PcolIulUMICIHsEBD5Z4db5FLhe/tFF11kd+jDCXDnnHOO7S0+/vjjRSMMEBNJCgBjp0Do8a9//ctuEvT666+byy67zE4MxOmCMCKz9B5F4ZQeToolBOKCgIb941LTnnK6JIsZ94cddpjBcjuSBMJxYAQm5eHCaAAN4/A+G9sd9k9HHkYFvv76azspENsI16tXz+BoaRw+5Jfezy8bPaOS5vPPPzcLFiyw+zegTMInKjWrcgiB7BAQ+WeHW+hTgdxxgQSw1A7D/SeffLItF/2ffPJJc+aZZ5rbbrvNXH755Yn4CM91DkCm5E99YU+cONHOCfj+++9N586dzYknnmjwqYC7FKJMIrc/HlFgxnX+P/74ow0QPn/gI5cQiCMCGvaPY607u+hh293Ro0eb448/PjHkTmJAz7pmzZoGa/+xQQ/8QSSlMMgX+aPR0aVLF/sp4NhjjzUzZsywDRPsd41PAdS9FDoGPU9gV6r6Czo20k8IxA0BkX/cavz38oIkQZY4FapixYoG3/m9xFmhQgXTpk0bM3XqVLvuHw0FGG+8YkDoHWnAudVHHXWUGTBggNl1113t54CrrrrKbhlcDH3ClAfqC5eIP0y1Jl2FQGEREPkXFt9AS8c3fZA+NvXBxjokB9pY59+tWzdz3XXXmeXLl9sNgEpB/ASRJMaGAO6xORA+SWCHwF9//dXceOON5oknnrBulAOfAnDRzbJRZhxslBkX8YsjBnGoZ5VRCGSCgL75Z4JWhOKCEGFACCQG3JMY6Idwr0EYCdgblu59pt/8vXKpJ/yhI8qDU+uwfz0+Y2AOAFYKNG/e3JvUxt/MM+IemB+xYsUKu9EPsMq1/iIOl4onBCKPgHr+ka9i/wKyF4hQEjwJ3/WD2/X3huG+lAa6U79tt93WTlDEcsDVq1ebm266yWAuAAwbO3C7DYdS6l7MvLE6Agc2AQfWdzHzV15CQAgECwFt7xus+iiqNiROZkpSoA1/EmtQCdPVC3rj6tSpk2nSpIl55JFH7KoATArEKMD222+fKA/LHCebdRmnMqusQkAI+COgnr8/LpH3JcGTMFPZGB72Cys1QH46uX4g+r///e92+eL06dPN1VdfbVcGUG8QoXvRP8o28HEbS1Euq8omBIRA2QiI/MvGR6EhRwCbF1177bV2RQM+A4waNSrxCYBESDvkRS1T/ffff9+OhKgBUCZMChQCsUFA5B+bqo5XQUlysFu0aGFuueUW+80bnwIee+wxu8qBiCBO1M3w4cPt6EccGjpRr0uVTwjkAwGRfz5QlIzAI1CtWjX7GaBnz55m5MiR5vbbbzdr164NvN75UBCE75J+HBo7+cBNMoRAlBEQ+Ue5dmNcNhCcO18BUGBjoFNOOcWcffbZZubMmfZzwOLFi+2cBsyCJ0m6RBkVCEn4tKNSLpVDCAiB7BAQ+WeHm1KFEAEQH64DDjjAHhmMA41uvvlmM2fOHOuPIon4Q1ixUlkICIGMERD5ZwyZEoQVAZfYO3bsaK655hp7ZsGtt94a+W2BcXjTokWLwlp10lsICIE8IyDyzzOgEhceBLAXAHr+OK4YKwGwH0BUjTvc77qjWl6VSwgIgbIREPmXjY9CI4IAh/xpo1hw165d237732677ezRxdOmTbMljvIcAHe3w4hUr4ohBIRAhgiI/DMETNGjhQAbADi8qHr16vaUwy+//DIxByBKpeWERpRZRggIgXgjIPKPd/2r9L9P8ttxxx3NlVdeaXA+wL333mtmz56dmPznzhUIK2CPPvqoXeUA/aNQnrDWg/QWAkFBQOQflJqQHiVDAD1hEGLdunXNFVdcYd133XWXWbhwodUp7D1llG3KlCmJI5nDXp6SPSjKWAhECAGRf4QqU0VJHwEQoHvxiFscBXz55ZebNWvWGDQAVq5cmegpg0R5pZ9T6WOS7GmjDDJCQAjEGwGRf7zrX6X/HQEQIsmxadOmdh8ALI0bNGiQXQ4YZsKk7rRZTlW+EBAC8UVA5B/fulfJHQRAiCRHeLdr186cfvrp5quvvjJDhgyxhwEhPKzEWaNGDdOgQYNQjlw41SSnEBACeUJA5J8nICUm3Ahg+RsbAPgEgAs7AR5yyCHmrbfesqcBMjxsJYXeN9xwg5k4cWJoGy9hw1z6CoGgIyDyD3oNSb+iIMBzAPjtn6MAffv2NTvvvLN5+umn7SZA9C+KUnnOhI0X2DJCQAjEGwGRf7zrX6UvAwGQZIUKFUz//v0NTgV84IEH7Pf/MpIEOijMny0CDayUEwIhREDkH8JKk8rFQYC9/B122MGcddZZBicA3n///Ym5AQhnnOJolJ9cwqhzfkouKUJACBABkT+RkC0EPAi4w+Pt27c3Rx11lJkwYYL56KOPPDGDe8sGCr75d+rUyTZW3HIFV3NpJgSEQCEREPkXEl3JDi0CIEiXJEGivXv3No0aNTKPPfaYWbp0aSh6/SwHji9esGBBUplCWzlSXAgIgZwREPnnDKEERBkBkifsrbbaypxzzjn2u/8TTzyRKHYYhtHdhkxCcTmEgBCILQIi/9hWvQqeKQJYDogNgPr06WM+/fRTu3QuDMSPckJPNQAyrXHFFwLRRaBidIumkgmB/CKAZYAg0cMOO8x+9x88eLBp27atqVq1auCJdY899rB7F6gRkN9nQtKEQFgRUM8/rDUnvYuKgDv8X7lyZYP1/8uWLTMjRoywxA9Sda+iKpdGZieffLIZOHBg4BspaRRFUYSAEMgDAiL/PIAoEfFDABv/dOjQwZL/kiVLEpP/0AAIonH10vB/EGtIOgmB4iIg8i8u3sotIgjgE8BJJ51kNm7caIYPH54oFYjVJdpEQAkd1Mdrl1AlZS0EhECJERD5l7gClH04EQCR1q1b13Tr1s18+OGHZuHChQnSD2LPev369faY4nCiLa2FgBDINwIi/3wjKnmxQIA9/COOOMKS/tChQwNbbuh6zTXXmF133TXRQAmsslJMCAiBoiAg8i8KzMokigiAVGvWrGm6d+9uJk+ebDf+QTk5vB6kMq9bt86sWrVKE/6CVCnSRQiUEAGRfwnBV9bhRcAl+AMPPND89ttv5qWXXrLEH8Rhf1ff8KIuzYWAEMgXAiL/fCEpObFCgAQPe6eddjL77LOP3fd/9erVgez5u5VD3V0/uYWAEIgXAiL/eNW3SpsnBECg7nXwwQfbCXXY+S9oBr3+Qw891Pztb3+zqmkUIGg1JH2EQPEREPkXH3PlGDEEQKbNmjUz9erVM2+88Uagev4k+h49epjLL788sJ8lIvZIqDhCIPAIiPwDX0VSMAwIVKhQwU78w8l5c+fODZzKGKXA2QSw2SAInJJSSAgIgaIhIPIvGtTKKKoIcPi/c+fOplKlSmbSpEmBKSp0o+HZBLyXLQSEQHwREPnHt+5V8jwjsP3229tT/9577z2zYcMG28NGLzsIPe05c+bYCYnq+ee50iVOCIQUAZF/SCtOagcLAfb+O3XqZLDXP4b+g0D6QAm63X333aZ3796B0SlYtSdthED8EBD5x6/OVeICIECix5I/fP+fNm2aPUK3AFnlJJKNlJyEKLEQEAKhR0DkH/oqVAGChED16tXtnv8zZ860E+yCpBsbKEHSSboIASFQGgRE/qXBPfa5gojQC4WJEilhUl2jRo3MrFmzAlfHxDtwikkhISAEio5AxaLnqAyFgDGmSZMmZuedd7bED8KMisFyuubNm5uffvrJrF271lSpUsWWsdTEe+aZZ9qNfqKCs8ohBIRAbghssSlK3a7csFBqISAEhIAQEAKxQCA6Xa5YVJcKKQSEgBAQAkIgdwRE/rljKAlCQAgIASEgBEKFgMg/VNUlZYWAEBACQkAI5I6AyD93DCWhyAhgmgom1sH2m7LihsENw7hM68Ypsvp5zw67Ca5cuTKBSd4zMCYhG7gRu40bNxYiK8kUAkKgCAiI/IsAsrLIHwIke86eJ6m7OTAMfnB747jhbrqwur/77jtz4YUXmkIdJwyyJ44zZswwjz76qPnll18SfmHFTXoLgTgjIPKPc+2HtOwgo1GjRpk77rgjsVeAX1GWLVtm3n33XfP888/bYDQC0Et+4IEHzOrVq31HDfzkBN1v3rx5Zs2aNfZYYTaO8qkzG0tYkvn+++/bhsaKFSvKxD6f+UuWEBAC+UdA5J9/TCWxgAiAiEBCo0ePNgMGDLA5kZy82S5dutRMnDjRnHbaaZYcEe+HH36w5IXea6p0XjlBv8cxwjhUqEaNGgUtExoWxIx20LGRfkJACPgjIPL3x0W+AUbAJR7X7VW5VatWpn///naznW+//dYGN2jQwDz++OP29D14lJXeKy9I9/yUAfubb74xjRs3LlhZUmGUyj9IOEkXISAE/BEQ+fvjIt+IIFCtWjXbK160aJEd5gdh4cS97t272xIWYpi8mNBhF0GQPxo1hTLEiA0O5kN/3ssWAkIgPAiI/MNTV9I0AwRA8iAn2DvuuKMB+cOMGzfOdOjQwVSqVCkDacGNOn36dPPbb7+ZNm3aFFRJYnnOOefYyX5169YtaH4SLgSEQGEREPkXFl9JLxECJCvY+B6+ZMkSu9/+hx9+aHr27FkirfKTrVs29PrRkME5CTAIy7dBAwoXDe9dP4bJFgJCIBwIiPzDUU/SMksEQFC1a9c2y5cvN0899ZS5+OKLE5JIYgmPEDlI8jg6uGXLlmarrbZKjHTkuxjMi3LZ+MCqCxkhIATCiYDIP5z1Jq3LQYCEBbt69ermjTfeMHvvvbftJYeZ9FlslAHD/SD/Zs2aWW/4FdIQU+TBtf+FzE+yhYAQKBwCIv/CYSvJJULA7ZmCELEE7qKLLjK77757iTTKf7Yg3y+++MKsX7/etGvXLv8Z+EgElkOGDDHt27e3n1F8olgvt5GQKk66/n6y4Ode6cpSPCEgBP5AoOIfTrmEQDQQAEmBHLAfwOzZsw2W/J188snRKJyza+GUKVNMlSpVEpP9SJSFGgGAfGwjDEyxWRLz8wMWYYXSg/WLfAuVh1+Z5CcEooSAev5Rqk2VxSKA3fueeeYZM2vWLLu7Hzb5QUMARBEFsiCxYqvdjh072u/9KBfKiKsQxjvMj339b731VnPQQQeZhx56yGb5448/mh49epirrroqqWcOfd3Lqx/DOIeA97Dnz59vrrvuOnP44YebG2+80X6+wTLN/fbbz4YhjowQEAKZI6Cef+aYKUXAERg6dKg577zzzAUXXGBuvvnmxGQ4qA2yCHsDAGUA0YIYDznkkKLUBjGjXaFCBXP11VebbbbZxtx5553m9NNPN5dddpnBvgqLFy+2GBNrpMF2zGPHjvXVlTKPOeaYxCgGI9avX98cddRR5pZbbrFLNFu0aGGGDRtmdt11V3PDDTeYxx57jFFlCwEhkAECIv8MwFLUcCBw0kknmb322su0bt3agKRgQDAko3CUomwtscc+zG677VZ2xDyFkqAx0kDSh+gzzjjDXH755ebSSy+1PfTmzZtvliNwx6cXNBRSGcTBqgwYuGmQ7+TJk+2IBkYZKB/7DGDXxijVKcssWwgUAwGRfzFQVh5FRQDfwbHu3SURKhAVspg2bZpd4lerVi0WreA2sOvSpYvp1KlTYvQEhI4GSMWKFS0xE3OQNhsMsOvVq2d22mknWyfeTxNMwwIg3P3MMHLkSHPwwQebrl272vRY5YB9G8K+XwPLK1sIlAKBwnwgLEVJlKcQ+B0Bl3S8JMSwMIEFcnSvVatW2Ul3IOJiGmDnkjJJu2rVqvbAJOiCOCBv4k68YSM+78vS242HCYavvPKKnbDJtOjx4zAjfA6QEQJCIDsE1PPPDjelEgJFQwBkCEIl2WKXQizxw5K7YhkSL/JzyXnw4MGmc+fOBvMsaNxwxn/xxRfN66+/zii+NuZpYFTBNfi8gR0Mjz766ET5//Of/9i5DhgJkBECQiA7BET+2eGmVEKgaAi4xA9i/fTTT+1BPg0bNiyaDmx44EhknCeAYX7OpwBpg5A/++wzs2zZMrsCAI0FXlASvfQjjzwyyY/KUzbvYbMBgSH/fffdN3EWA85mmDBhgnnzzTcT+bvp5BYCQiA9BET+6eGkWEKgZAi45LhixQp7it8JJ5xgiZQkWQzlkNerr75qV1KA/Pv162cGDRpks8YGShdeeKG9d0mfeiGt24ihP2yWD+HuZwW4X3vtNXtc8QsvvGD3Fhg/frwlfkwOZDp3VMKVK7cQEAKpERD5p8ZGIUIgMAiA4EB26Pligx1OaCwW8blEW7lyZfP1118bnuwHHT7++OOUWCGcetJ2I4P0adhAQDz08L///nu7RJCTBY8//viELKaRLQSEQOYI/PGvyzytUggBIVAkBEC+2FgH5I/hflwuqRZSDeQNUiZxuw2BQuWLPDC0jyWbWCkAw4ZBofKUXCEQJwRE/nGqbZU11AgsXLjQzJs3z+6ih4KAjEnEhSwY83HzopsNgnznD/lvv/22/d5P2Wj8yAgBIZAfBET++cFRUoRAwRF47733bO8Xs/xBjiTggmfsTMDDXANsK1ynTh2bbSF0wFI+7BaI5Xyff/65effdd21eaGgUqrFRDAyVhxAIEgL65h+k2pAuQiAFAuvWrTNY4ocNdXbYYYeikyB7/zgeGVv4FtI0bdrUrh7ACgI0Lkj4tAuZt2QLgbggoJ5/XGpa5Qw1AtjRDzP9sZd/sUmQvXs2AIoJZLHLWsyyKS8hUEoERP6lRF95C4E0EcCStxo1atg98tNMkrdoIGAvCbNBkLdMJEgICIGiIiDyLyrcykwIZI4AlrvheOIDDjgg6YTCzCVlnwJr7mHGjBljzj77bDsKkb00pRQCQqDUCIj8S10Dyl8IeBBAr9q9sLENlrl169atqJP8qBZ0Ye8fjRBs6fvrr7+WRBfqJFsICIHcEBD554afUguBvCPgDqn/8ssv5o033jC77LJLYqKfdwg+7wqkEMjeP/J3dUwRXd5CQAgEGAGRf4ArR6rFEwGXXCdOnGjWrl1r+vTpUzIwqA9tjgSUTCFlLASEQM4IiPxzhlAChED+EQDRoqeNvfSbNGlimjVrlv9MMpBI4seWvvvss4+dewA/GSEgBMKJgMg/nPUmrSOOAIh18uTJdm/7Hj162G/+pSoySR7zDnr37m1w0l7NmjU3WwFQKv2UrxAQApkjIPLPHDOlEAIFRwBb2eJbP3vaQRhq937nD4JOBa8IZSAEIoqAyD+iFatihRuB6dOnm6lTp9rlfRUqVLC9bC/5lqKEHAWgLrRLoYvyFAJCIHsERP7ZY6eUQiCvCIBISaYjRoyws/u7d+9u86A/7bxmnKYwEP/69esTy/zYEEgzuaIJASEQIARE/gGqDKkSTwRc0gcCX331lZk0aZI56KCDEhPreKQu7GIbNjhgY41/48aNzY8//phoqBRbH+UnBIRA7ggU/02Su86SIAQihQBIFb1okix6/dttt53p1atXwi8oBUbPf9WqVXYlQlB0kh5CQAhkjoDIP3PMlEII5BUBEj9sfOv/7LPPDGb4b7XVVnnNJ1th7vC+685WntIJASFQegRE/qWvA2kgBBI9/OHDh9sDfA4++ODALKXjyIQf8fv5qTqFgBAIPgIi/+DXkTSMOAIc7sd3fszwP/LII83WW29tSx0UcsWGQ9Bz3333NQMHDjTVqlWz+lH3iFeRiicEIodAxciVSAUSAiFDAAQPch02bJipU6eO2X///RMjAUEpCnQE0bdt29a0bt3abjqE+6A0ToKCk/QQAmFBQD3/sNSU9IwsAiDRUaNGmTlz5pgTTjghMN/6Cbi3dy/CJzKyhUB4ERD5h7fupHlEEMDxuC+//LJp1aqV2WOPPRKlCgrJstfvZyeUlUMICIFQISDyD1V1SdkoIICeNHvTGO5/4YUXzLJly8wpp5xiiweSJfHTLnW5SfzTpk0zDz/8sFmzZk2pVVL+QkAI5ICAyD8H8JRUCOSCABoACxcutAflYEOfFi1a5CKuYGmhJxskY8eONRdeeKH5+eefC5afBAsBIVB4BET+hcdYOQgBXwRweM+jjz5qKlasaI4++mg7GlCKHfx8lXM8QfwYofAzHMHwC5OfEBACwUVA5B/cupFmEUdgwoQJBsPoJ598st3Rj0PrQSs2e/7QiyMA8HP9g6az9BECQqBsBLTUr2x8FCoEckbA2zvGPYbNn3jiCTvU37Vr18TSuZwzK4AANkpgw7ABwPsCZCmRQkAIFBgB9fwLDLDECwH2kDl0DtJ8+umnzdq1a81ZZ51ll/axgRBUQqVe/fr1M6tXrzb16tVTxQoBIRBiBET+Ia48qR4OBECcIHd8z4f9+eefG0ycO+yww0yDBg2SCsFGQJJnAG6gF3VjQyAAakkFISAEskRA5J8lcEomBLJBYMWKFeahhx6yx+JiG1+vCTKxurqxIeDVX/dCQAiEAwGRfzjqSVqGGAESJezBgwcbbOpz7rnn2ln+KBZIlVdQi+kSf1B1lF5CQAikj4DIP32sFFMIZIUAiXP06NFm3Lhxpnfv3qZRo0aW8NkwyEpwkRJRf2T3zDPP2L39lyxZUqTclY0QEAKFQEDkXwhUJVMIeBCYP3++7fXvuuuu9tQ+EiptT/RA3WKiIhspq1atMvPmzTMbNmxI+AVKWSkjBIRAWghoqV9aMCmSEMgegfXr15t7773XVK5c2fTv3z97QSVK6TZQUrlLpJqyFQJCIEsE1PPPEjglEwKpEGBPGb1luLGLH3rLZ599tt3MB7P+QaK8UskJsj/KxtGAIOsp3YSAEPBHQOTvj4t8hUDWCHBJH8jx7bffNh988IHp06eP6dixY+gJs3Pnzuaqq64y1apVyxofJRQCQqD0CGjYv/R1IA0ihgB6++jVT58+3QwZMsSSPvbuh3GHzcNSbPbwoTsaMLjCWpawYC49hUChERD5FxphyY8NAiRJFHjp0qVm0KBBplatWna4v0KFCqHFAaTPsrmNFz+/0BZSiguBmCGgYf+YVbiKm38EQIIkQkjHtr0DBgywZ95ffPHFiUN7QJwueeZfk8JIdHV2yxnW8hQGJUkVAuFCQOQfrvqStgFHYN26dbbHj6V9l19+ualfv75tGLikGfAibKYedGcD4PvvvzcjR440KKeMEBAC4UVA5B/eupPmAUAAxEhix7d+7OA3ZcoUc+aZZ5p27drZsCj1kEeMGGE3KVq2bFkA0JcKQkAIZIuAyD9b5JROCPyOAHvFzz33nBk1apSd2X/ggQfaUIaFGSw2btjQ4X2YyyTdhUDcEdCEv7g/ASp/VgiQCEHucKNH/Oqrr5oDDjjAHHPMMVYmlvxFwbABA5vuKJRLZRACcUZA5B/n2lfZs0KAxI/EcL/22mvm2WefNV27drXD/VEhfS84LDdslh22GgRepHQvBIKPQDS6JsHHWRpGFAESf5cuXexJfVElQhA+yta3b18zd+5cu4QR91Etb0QfVxVLCCQQUM8/AYUcQiB9BEB6L7/8shl9tWCuAAAgAElEQVQ2bJjZc889zfnnn2/CvJa/vJKT5KtWrWpwyQgBIRBuBET+4a4/aV8EBDCLn1v2Ijv0gp9//nlL/iD+fv36mYoVo/9XYu+fdhGgVxZCQAgUCIHov7EKBJzExgcB9Hq5ZS+I7/HHH7d79h900EHmjDPOiHSPH7WM8pPwacen9lVSIRBNBPTNP5r1qlIVAAEczXvnnXead955x2Cvfqzl51A/h8ULkG3JRboNn9GjR5vTTjvNLF++3OoV5XKXHHgpIAQKiIDIv4DgSnR0EPjpp5/MzTffbD799FNz+umn2+V87qx+9IhhokiG7PmjfN98842d5/Drr7/a8rLc9kY/QkAIhAYBDfuHpqqkaLEQ8BLa119/be6++267V/+VV16ZdDRvFMk+Fc5uI4AYxan8qXCRvxAIIwIi/zDWmnQuCgIgOAxz4xs/Tue74oorTMOGDW3ecSQ9lJnlhq0GQFEeQ2UiBAqCgMi/ILBKaJgRAKmtWbPGPPPMM3a7Xpxf379/f7PtttuGuVg56U6ix0FF+++/v9lyyy0TDYGcBCuxEBACJUFA5F8S2JVpkBH47rvvzL333mt++OEH85e//MVO7nOX+rH3G+Qy5Fs3lrlXr16mZ8+eSSsA8p2X5AkBIVB4BET+hcdYOQQMAfZiYZPUoOKGDRvsEr6hQ4eaatWqGXzf32WXXaz2iOfGDViRiqIOyk/skGHc8SgK6MpECBQIAZF/gYCV2OAiQBJDb37jxo2WxBYvXmzuv/9+M2vWLLP77rubs846y2yzzTbBLUSRNQPp43JHQLyNpyKrpOyEgBDIAQGRfw7gKWk4EeC6ddjo7b/++ut2t75KlSqZiy66yJI/SiZy+6N+2ctft26dwTI/jIy4DYE/YsolBIRAGBAQ+YehlqRj3hEAsU+dOtU8+eSTZsGCBebPf/6zOf74402NGjWS8mJvN8kzhjdsCA0ePNh+DpkyZYqpW7euhv5j+CyoyNFAQOQfjXpUKRwEQFRlmUWLFpmnnnrKTJ482dSuXdtcc801pl27domeLHq0NOzx8j7u9m+//WZ++eUXC4OwifvToPKHGQGRf5hrT7pnhMCKFSvs7nQffvih7bGefPLJBvvzY7ifRoRGJJJt4MLeP0LKa2Alp9adEBACQUNA5B+0GpE+eUGA5IRe/LJly8xrr71mxowZYyf4HXLIIeawww4z22+/vUgsA7TZMCK2SIp5EzzfIANRiioEhECJERD5l7gClH1hEABRYZ3+yJEj7QXC6tq1qyV9bFQD0oIhodHtElthNAufVGACnHB169bNPPzww2a77bazDSf3E0n4SiaNhUB8ERD5x7fuQ1tyl6DpJkGB1HH4zKhRowyG9ytWrGj22Wcfc/jhh5t69eolypyKtNzGQCKyHBYBYNyyZUt7wQNYEX9BJASEwP8j4P4n+F4iNrwPwntG5M9akR0aBNw/F5TGPY7bxQz0t99+23z55Ze2Z3rwwQeb3r172+F9xoMdhD9eaMB2SJ648QWGhhb9wlQe6SoECo0A/yP4f9CN/ws6HUH5z4j8C/0USH5eEeAfCULxJ8LmPBMnTjSvvPKK+fnnn+3BOyeddJLp0aOH2WqrrWzeTMM/Yl4ViokwvrCIJfGPSfFVTCGQNgLofGAZMTYQw6RimPHjx5tJkyaZfv362XMx0hZWwIhbbMK/WUYIBBgB7yO6du1ac+utt5qxY8eanXbayTYCWrVqZYf2d955ZzsBzUtSbHW7/gEucqBUc/H/6quvLO54qVWtWjVQPZlAgSZlYosAlsK++eab5pRTTrGjkfPmzTPnnnuuWbJkicG5IdWrVw8ENiL/QFRDvJVwycVFwu2pIw424/nggw8s+Sxfvtxgzfmxxx5rDjzwQHvkLomdvVRXlty5IwB8H3zwQXPJJZfYeRWYQyGsc8dVEqKFAD+H4b20xx572HlH2EsES4133HHHwPxnNOwfrecuEqVhYwA2hvJJ+AsXLrQ9zV133dXMnDnTjBgxwgwfPjzxZxIRRaL6VQghEGoE8F0fDQAcBY5jwTFahk+QtWrVClS5RP6Bqo54K0PSX716tZ2pj+9kc+fOtfvvN2/e3B62A+LHMrOrr77a+ruEDzdlxBvJwpdeOBceY+UQTgRA/DBNmzY1eJdtvfXWiUl/QSqRyD9ItRFjXdDDHz16tPniiy/Mt99+aw+PadCggZ0wg2N109lHXoQU4wdIRRcCAUEAnRDMS/rxxx/tUP+ECRPMnnvuaUcog/SOEvkH5IEJsxrpPtCI5/bU0av/+OOPzbRp08zXX39th8qaNGlijj76aLPbbrsZbMaD+G4aL05lhXnj6j43BID1OeecY6/cJCm1EIgOAnivue82uO+44w77P3n55Zft8uPWrVsbTPxDR8b7zkJ813jD3bB8ukX++UQzxrLwwPIPAJsPNB9k3OM4WPTscYHwv//+ezszH4fqYPZ4hw4dkjbiiTGcKroQEAIhQYDvOhx1jVVHbdq0Mdddd539xo9Jf0OHDjU77LCD6d+/v2+JmN59h/pGzLOnyD/PgMZVHB9g2nyQZ8+ebWbMmGHJHqSPzXiw1AUtYOy616lTJ7Ptttta2NhQcDH083PD5RYCQkAIlBIBvqMqV65sHnroIdOiRQv7vR/+l112mdlrr73MUUcdZScrcz4A9cX7EqOfWAlw6KGH0rsotpb6FQXmcGZCIk9He8TFw47jcrG9LmbjY/Odn376yS51adSokW0Vg+zbtm1r/wiQy3S0IYNu2n75X3XVVeaRRx6x39X8wuVXOARQLzB86RUuJ0kWAsFHgP8Hv/cV/fheQ2nc/w3CsSHZ+++/b8/MwAZlqbYezzcSWfX8uY6RyriFoZ/s4CLAB5Ia4h6G9ch72HgQYePiA8x4eA6WLl1qN66YNWuW+fTTTw2W4yEc61nxnQtD+rvvvrtv7x7xKIu2q4frR11hUz+EUy83XO78IkC8IRVLl2666Sb7sqpTp47Fv1gvq/yWStKEQH4Q4HuKtivV9XPdiMP/1dNPP21w0ujpp59uVwdgF0AuFyzkfysr8ncLp5evi0Y43O5DyAfQfRjh/uyzz8zNN99s98nHrnlwg8xdssdwFTbbQSMA37QwOx+nvmFmK9a0eh9cN69ckYIs7xBarjKV3h8BPC+sOyxdmj9/vsUeft469pcgXyEgBFIhgHcn9jLp1auXOe+88+zeJn//+98362ylSp+tf1bkz5cBbJdIslVC6UqDAF/obu7wQy8eJI4XPep3zpw59lhcHJJDwsVa+8aNG5vu3bubzp07257+lltu6YpKenghl88K7aTIGdxQVq5yMsgy1lFR58Qa2OOCgR/C1ACI9eOhwmeJAP9TSL7NNtvYLYFPPPFEg0+a+Fx6yy23FPQcgKzIH5O27rzzTjt5K8tyK1nAEKhQoYKpVKmSqVmzphkzZowlfqjIF/2aNWvsxjvYuAKzWbfffnu7Fn/y5MkGV7ENjuvFHto33HBDgpiKrUOc8mODC/M4cGDJXXfdZV9Y7gssTniorEIgnwjw/4XVAjizBEsFMQlwwIABBTsLIKsJf+gRYkYjXr4ymSOAFyZJNfPU/in8ZKJH5u21ITX88AIH0deoUcMO2aPliS0oK1asaIf6sRTPqyNmsyKO199fo/z7kmiQPxqgWDpYrVq1kumT/xJKohAQAlFHwO9d7fphyeCGDRtsB2vIkCF2dLUQmGRF/iAPDgnjRcyXciEUjKNMkitxde/pJi7uQwM3yR5uPECwV61aZfeXxkY62GgC3+nRcENc9PgxcQtr7LHBDk7Je+eddzbbyAVynnrqKbs8D24Y2tSlWDYwwKSzJ5980q4s0LBzsZD//3z8nsHiaqDchED4EeD/CO9RvIvxHsOs/xNOOMH29rFBEM4HKNT7Lathf8DOtdlwl4oEwl/92ZeADw4l4B4Xl9lhAx3MvMcWk/h+xPgg+P32289g6R22zAXxo9ePOsSFeH379jXjxo0zgwcPtuLhj5moxxxzjO35w7OUdQ4dMQIBHfAcFurPQWxlCwEhIAQKhQCIH+bVV1+1m53hnfzKK6/YJdGFyhNysyJ/98WPF7F7X0hloyibpOyWzQ9PNx6OskUPHhPxQPCLFy+2F9bYIwzpq1SpYtfVY+IeHiYQPQgfk/IQ7tYb7+mH4f3HH3/cnH322fZgHUzsw0YVjAe71IZ4BEGXUmNR7PyFfbERV35RRwCjqtgBEAeYgfjxzoUp5PstZ/IvpHJBrXC+/Fg5vAcWdHt1J04Mx707RE9/pKM/hutB8ujFg9ixzI5uDOnDgMzRm8cyPPToMWyPq3bt2kkPDvOnXuXdozfdtWtXs/feeweuZw3dqT9tlkt2/hHAswmcYeN5xHyQ/fff36CRCKM6yD/mkhgfBAYNGmSuuOIKO8T/0ksvbfbuLhQSWZF/oZQJi1z3ZccXI3XnS5L3tEnuCOcwD9yYRf/DDz/YnvuyZcsswS9YsMDgwkl3jI/hbRx0g5bhPvvsY3vzDRs2tETPPGHTzXyzsakr0rry4JaJJwJ8Jt58801z8cUXG2zbnM5Ji/FES6UWAukjgKXV6GSB+LH1ebGMyD8LpNkzZ1I2AGjTHzbiYoIdiB3f3tF7xz16UNgsBW4YECtm2mNnPFxdunSxG+WA8LG8DhtBuPl68yIxM04upI20bnq3PHILAc6x4DMnRISAEMgegf/+97929RXe/3iv8/+VvcT0Uor808MpEYuVQ/LFkjOsfACJo6eOtZlw41v8d999Z4fpOUQPIfgWjzXyWGKHw21A6himb9as2WY9KeaBdKncCcV+d/CFjPi5GKT/8ssvzbPPPmtuu+22XEQpbQQRyPX5iiAkKpIQyAoBvLOx6gqG7++sBGWYKBTkX9aLxg8sxGea/2vvTKDuHM44PhwNJVQQkpCECLFErKldCGKJ2IIWqaidc1Rpe6rRlracEj1FY2lJVY+l9uXYYqulgghiTaiKxE4qBG01aHp+0/5v5r7u/b4v3/fd777vvf85573zzswzzzzzn7nzzPbOVAqvhBH0otU738LzSZwelDrKnVE8U/Wsv7PZjtE2D/HYhc6358sss0xU8EzTs/7OCJ61eaZ16NmRltLLylNJ9pQ2fc/G7Qy30v/tb38bbrjhhnDyySfHXfVKV3ZnpGUe+UdA/wfVC7mp8101Ssk/SpbQCLQPAbWnstvHZdFj5V75q6FRw6MsZt34p+DxLhreaahEw0icA2J4Pv300zhiR6Fr1I5inzNnTlTydACIywif3hmbnBi9swbP+fVcXMNhOYze+/TpEzj2Fjrtqpe8qZ3KmfrrvbVw0dXSJv+33357/FTwnnvuCaNHj65lcuadYwRUH7HHjh0bDjjggNjBlX+ORbdoRqAQCNTjv5RL5S+lTakBSupOSzIFjE/cUOqyP/roo6i8Ga3rQbmj1NhFDx0KHRv+KGudcMfoHIXODnrW33lneh5lD42eVJZq7/BOR0fV8lItfj38wZXPTcaPHx+OP/74eNTkPvvsU5aPesjlNOuHgP6HdH45Bjr979VPKqdsBIxAexFot/KXEuvMRkA8NdqWjbJmPR2FLkXOO/4KYxSPItdoXkqXUbhG4lpr52x61txxM1LHZjTPQ8PGxgsUNiP+SvlTQyi7JfCzNJX4tRS/HmF0ipju59CJKVOmBD5FefbZZ8NGG21UD3GcZg4QSP8LRajDOYDMIhiBXCPQLuWfNgQoWSk43qXA5Q8tD6NuFDiftrHrXe+cY8yaOv5MwaePpubpBKjBkY1ClzLnQARG5qy1c1odD9PySy+9dHyk2BVXdmslk47Ys7TiITsbXiR3WmbIzTT/jjvuGGc4jj322HDBBRfEjX9W/kUqVctqBIyAEaiOQLuU/5VXXlk2yma6XZcRoLw1CseP0TgPHQAUNraMFCeKmxE4CpvNcYzKUdxMMbJxjncpcGz8sDHioXd1OqTQlJbttiEAntddd12YMGFCjMAmxS222CJcdNFF8apJlkRSzNvG1VSNgsBdd90VLr300nDhhRfGDnej5Mv5MALNhkC7lD/TwShmjbJZA0dxS4Gzfk641gexpbyZUicebpQ6caVMpLhVCKm//FqyFV+Kn/h6VzzxlNv2QgTAhul9NvqxARKDH181MDNDuXPuv3BeGNNvjY6A/jevvvpqrAfnnHNOzLLrQqOXvPPXqAi0S/kzMuRPn/7x1Tjgh1GYbPygUThuZgHSeHqXndJGphV4yF92GkdpV/ITve1yBLhCkpul1ltvvVIAszlrr712HO2NGTPGG/9KyDTHS/q/1X9TOW9paUw0to2AEcgfAot3RCQ1BLLFCzeNQmqLBlsPywB6F73o4KWw1K7mn9Kk75XoJaftcgS4AZDLgtgQmRo2Qe63335h6tSp4ZFHHolBdNx40o5VGsfvjYNAWsZ6z9qNk1vnxAg0BwLtVv5ZBSu48LcpDgI04jwo8vPOOy9svvnmsdNFDlSW2AcffHCkY7qXPR4KK05OLWlHEFB5czPk7rvvXrZc1xG+jmsEjEB9EGi38q+PuE61FgjQsHOM76OPPhpH988880xMRqM7ZgSuueaaMGzYsLgX4Mwzz4yfXNZCFvPMJwKqC7vttlu4/vrr45kX8sunxJbKCBiBlhBYbIH/wS3h0xRh2b0XZJoOAVVDI74sEGlYNZpsnM50jxs3LkycODGeQNiZfM3rywikTYTKXX5e8/8yXvYxAkVAoF0b/oqQMcvYdgTSBjxV5Ol7lltLYVlauxsHAZU7No86A42TQ+fECDQHAp72b45ydi6NQLsR0CgfBpzdwbXUOn2z3Uwd0QgYgboiYOVfV/iduBHIPwIa7SMpe0MGDRoU78jIv+SW0AgYgWoIWPlXQ8b+RsAIlBBQB4AvPTjwSftE5F8i9IsRMAKFQMDKvxDFZCGNQH0RSKf+2SMit+z6SufUjYARWFQErPwXFTHTG4EmR8AKv8krgLPfEAhY+TdEMToTRqC2CEjhDx8+PFx22WXx8q3apmjuRsAI1BIBf+pXS3TN2wg0CAJa219rrbXCwIED/Zlfg5Srs9G8CLRL+c+fPz/cdNNN8Sa/XXfdNaKnxoHrfmUYLWy11VZhwIABpTXC5557LvAQRhzOjd9///0VJdo33nhj3FSEA5rBgweHDTfcsLTJiBPn7r777nh/ABuPWIMcOXJkvFUQN2bKlCnhlVdeie/wWGmllcKIESOiWz/sXNaIBj9kXX311UtxuOEOWWW4iXDfffeVM8bltDPwIA14bbzxxvFSHOWPG/EeeOCBUhzuM9hzzz3jjYbyfPjhh8Nrr70WncTr3bt3YIQlHnxWdfXVV4s8prXtttuGvn37lvyQk5P5kAPDjYl77713yY2f8iu+m222Wdy5LSazZ88OyIKBhtsZ99prr3g7o/weeuih8Prrr5f4rrbaamH77bcvycp1zlz8lBrC+/TpE72QL8UVt3BV2eFHPUIGrS9LVvxkuHDoiiuuiLKAK2WDzDL33HNPaVc68SjbrbfeOgbj5rM16jGGNPHbaaedwsorr1zKXyordNxeCSapQYbUDB06NF6EhB98X3zxxfDEE0+USJCRuxLS8xUmTZpUukkRQpTs17/+9RgHuT7++ONw6623lnjwQn3u2bNnyQ9ZeeALlj169Ij/C+ILR/0/lV+Oc15zzTUjD2ioR/AgDg83ciIr7xjiIQfyyHAXxCabbBJpCJ87d2648847S5hCx5HAyCMDHi+99FKJhmu9d9lllxgMD4zKVnG23HLLUlsCjWRVOG3J6NGjo1M8KF82KOIGkyFDhsRHcWhLqCcyYEBbstxyy8krPPbYY6W2BE/qx84771wKBxv9t5Qu9YzrsIVbtmz4f6ZtCcz439CWyFRrS0gDvshKW8IV5zL8f/kfY6Dr1atXbEsUDgZc3pUatSWSXbLi5qEOSFbSxY/2iHZJRv9PudWWiJ7bXNO2BDraRdpHDDxXXXXVeIqo3Py/b7jhhlK9gpfaEugxyJq20WCxzz77lHCHRnU+RgghSFbJxn0m3FkinsiabUvuu+++8M4774hFWVuC5z/+8Y9w8803x3D4Uja04eCvdCSr3LQllJ/cRKbOExc/DFeojxo1Kr53+g8n/C2q+fDDDxcsu+yyC4YOHbrgP//5T9mD3uBZbLHF4jNx4sQY/sUXX0T71FNPLYVD17Nnz5h8yqdfv35lND/96U8XfP7556V07r///hiutJZYYokFM2bMWEAaSuewww6L6UuW7bbbriyc9BRfsl5++eWl+PCRrIsvvnik7dWrV5QBgYkPTY8ePcr4nH322SUe0Nxyyy2lcNJbcsklF7zyyiulvEDzjW98I9IgBzQjRowo8SCNTz/9tIwHNNdee20ZD2RVPghfddVVSzJK1m7dupWlc95555XxuPrqq0vpwOurX/3qgnfeeadEQ7732GOPEq6ks/fee5elM3fu3BIPwnluvfXWEg9kycpKeat8VX6Ki40sEyZMKOPxox/9qCyd7t27L5g3b16ZLNtvv30ZzcEHH1wWPmvWrBgu3EnrvvvuK0snK+vAgQO/9HdJZeX9oosuKkvn3HPPLcPsa1/72oL58+eXpbPZZpuVaJDnqKOOinUAPHimT59elhdoJk+eXFZPxo0bV+KBHIMHD45pqL6CfZpX3i+++OIyOU4//fSydFZYYYWyvCDLuuuuW8INHieeeGJJDtJ4/PHHy9JBlqeeeqosnRNOOKFEAw/yr3qKnZUVHr///e/L6on+n4Tx0JakPJCVuqU88z8mjvhjV2pLwDqlUVsiPsOGDSvLL7Skr3De07aE8Gw9UltCGAZZaUvgIT7jx48vS4f/UZoObcnMmTNLNPBQWwIdD21JmpdqbUlaRySreNCWwDvlo7ZENL/5zW/Kwq+66qpSPqBZeumlF7z99ttlNCNHjowyiofaEuHx/vvvl4VDBwbIInmydaB///5laSCz+Ms+//zzI414oKNS3NFt6DiVDXa2LRkzZkxJBsJnz54d00n53HvvvWWynHbaaWWyrLnmmmU84KOyR1behwwZEutHLX7adbwvvUeNdrt161bWc2Hkp54MNqMcRmUY3DocRD0b/OlZpm56fPR+SIfeGPHpjYkG+7PPPit1hHAjB3Fk+CQJGvWisKFJDenIKB3SUg+Q+MjAI96MLlIDDqQv2ZBT+YUPcVNZiQsmhCkdeCivhJMWfFIDrqLHn3Do8FP6KhP84IesioNNfqGVn2QVPWWj3rzoVH7KH3kRDfGyuBJWKb/QiQdlg3xKg/wga+omv6mRrPhB9+Mf/zhccsklcRZCdJSv8oafcCVt0hMP0cAnxRV3pXqU5hc+zFSkJq1H+FP+4AY/HuKAidKFJs0vNIRDBw1u8VA6hJEfGeiydQA5JSt00GTrfPb/CQ8eGZWf5AA71VdoGGX9+c9/DlztzCgLOtEgN25kFa7io7ooGuoAD+EYeCAH4bxjp/nFLUzEA8x455ERrnLDg3B4Ildan0mb/EoO8VF+JRvpCFf8eESjdFRfUx6kCS1+aTrEwY92LzXZsiGNlAfyIytxxVe4iifhpKVw4qsOSLYUV+IpHfGFB2nxyJAOBr48zJphy1B2avfwQwZww0gW/beUTooZNMgKH/GFTuUnP8Khk0nzix9xUlyJl/1/igdh0JNPZJUbG1mxZQhP8cjKKh7wkxEP+PCQF+igkZ/KRnHAVfmDDkyzNKLtqN0u5Z9mEAEEkvyxyYDc0PAuOgmNGzCUWfkrHuF6V1g1P/HJ0isediUZqoWLtlJ6aZy2vIuH7GpxlGYarvwQl4d8YneVycqUdUsOyVQpHD/5Q8e7jPxly781+5RTToln+7/77ruRlPiqR5X4t8avreFpOtXiQCMZWsIlGz+LTTYcnqIRX2jkn31XfNFKJvnLTuPLr5r9u9/9Lpx44onh5ZdfjtO0opNccmNX8kvDeV+UtLNxK7nTNNvKW/iIX/ofS/kpPGtXSyf1T9+Jn7rT92q8U5r0PUsvt2hkp/nI5heaakbxCVc80ac8FV9+aTyFyW6NRumk5SB+rcWFTu2A0svakl88CVc80sRIBr0rTgzMlJ/8OmIrX5V4pLJUCm+v38Iu/yJwqCaM/LM2rOWXTaZSQaW06bviVvITn0phLcVTGHYat9p7St/Wd1UcVeaUd8pD/tDz4NYjOuVT7lrbkknpZN3yl10pPJuHlEbvssWnrbbiySZetfe28myJLuVdjQ6aLF3W3VLc1sKyvFJ3+p7lUy2smr/ip/VRdZk4aR2FthKfSn7iWy1OGt6ed6Upe1F5ZP9jrfGpFp76p+/ZfGfDUnkVJjsbN6VN30Uvu6V4KU3Ko1qclD59V1z8KvkrXHxbo0nLIaVtiX9Kl6aXvotGtuTBTtOsFEd+aVz5ddSuBc+WZGqX8m+JocO+jACNJFNcTIWpwfwy1UIfppWphG2hXRjLb0ag9gjQQFEvMV3dWNU+d07BCDQPAlb+XVDWNJLshmZncbWeZSrGo48+Gnc1u3FNUfF7vRFA6VN/WSOWcQdVSNg2AsVCwMq/C8rrrbfeig0mn4qwgaMtHQCtPXWBeE7CCLSKAB1RniOPPDIcccQR8d2Kv1XYTGAEcouAlX8NikbTomLNd94cjMIu07aatnQQ2srLdEagowikddpKv6NoOr4RqD8CC7+ZqL8sDSkBDeWMGTPCRhtt1JD5c6aaAwE6o4z8qc/qmGKnnYLmQMK5NAKNgYCVfw3LUQ0lpw1y2pONESgqAlqGSvehUL9Td1HzZrmNQDMiYOVfw1KnYeQQCo7E1VGtNUzOrI1AzRDQqJ/jR1nC0vkKNUvQjI2AEagpAm1fhK6pGOXM9T18ue/CgxjSqUaNrrO0eXFzbjRngafr/an8WTnznp+svHY3PgLUSY3wOcOcDax0alP/xkfBOT1oHtYAABmUSURBVDQCjYVALkf+amiyShJ/Ogbyl53HIpFsHIfKxSfKUyVZRYvdEl2luPYzArVGgDpJ3VT91Jp/rdM1fyPQbAhIF3RFvnOp/JXxtJERKGqAZOddWU6fPj1ssMEGZUpd66fKk/KgRlb5t20E8oIAdVP1Ux1w+eVFRsthBIqGADogfbpS/lxO+wMGlyCcffbZ4cEHHwyHHHJI+Na3vhVxuf322+OZ7lzVSeOTZ0M+Jk+eHMaPH1+arUBeOjXvv/9+zAdXNnJoCvnlAgrOTbcxAnlCgHqs/xrXoY4bNy5en42MaVieZLYsRqBICOj/xf9JRu8Kk39n2blU/mSWW6S4vIU1cxS+lD/3YHP3MmuO3HYkgAQIypZbx1oy3Pc8ePDgqiRZsEmDR/6yxSArA/7QICN3lmfT4tO/Aw88MN5VzV3q3K990kknhfPPP18sbRuB3CCQ1neUP6dVpn65EdSCGIH/d0g7Cwjqedr+Sw+ozVd4ajMzxlHumiHj/YMPPog3B3700UeBfTMc9/7xxx9Hm4GfbvzDn3cMPFdaaaV4g2Zn5Sflk0vlLwHJPIrzmmuuKSn5HXfcMZxwwgnxpDwKIP3WGPcaa6wRdtllF7GoaPfp06dMmatARayClRs7TSf113slHs8880yUh3xgoKHA6XycccYZAcWPH50EzF577VUmV/T0jxGoMwL6P1CPadCws/W9ziI6eSPwJQTSeqo6nBIpPPVTvRa9bGiYmeUrl7///e/x+fDDD8M///nP8Mknn0Sb63hR7ChvrgDGja3NsfBKl7KVruTAZnDLVdky6gjI3Zl2rpU/Ge3fv3/cXcw74LAMQIeAY3IBU4Uju3fv3mVXjWbBUhwpc8WDburUqeEnP/lJNkp0kzYNHyP2b3/721+iIVxG79OmTQvrr79+WUN55plnxvumR40aVeL72GOPhe222y4st9xyYmHbCOQKAdVpCYWb/07WX+G2jUC9EFDdVNsuO5UHGnVkGY2/+eabgePXWY7FzYNS16NRungpDewVVlghsHy7/PLLB/RP9+7dowJHkTM7vdRSS0Wljs7i0jb8WOpldhubL8HwS3lKVqUnd2fauVf+q622WrwQh94TvSguvTn55JMjBlLgiwIIAGMoeIzcvA8dOjRMmjSpzA9/CoAnpY2R/x+GTC+88EI89xw68eZYXzoqxOOhIl1++eVxXZ/CxlDpmNlgHbVaGkrLthGoBwLp/2z27Nnh2WefDSNGjIiNWj3kcZrNhwBtI6ZSG5xFQ7T4M3KeM2dOeO2116Jinzt3bpyCnzdvXnQzHQ+N2l5slDXT7T179gxrrbVWVOQMzJZddtnSO6NzlDx+1Yx4Eo7cuOWXtcVDMwOEK57COtvOvfKnJ4UyZarltttuC4cffnhZBRCosu+44464lt5SJTn66KOjoheN7GrgEp7SqBBZn/nZz34W9yTQKLLc0Ldv31JB0yk47rjjIlvyQK+Syrf22muXkmJTI9NHu+++e1kaJQK/GIEcIED9p97fddddsfP6t7/9LbB8lv4vciCmRSgAAlJsiKp6xTv+qk+8p51OhWlgldIxKmcQ9fbbb8c2lnaWG1RR+tiiJS7KmnNXVlllldCvX784WteoHWXfo0ePqNSzMGZlzoZXcitdheGWX9YWjWyFy10LO/fKnwaGSnDdddeFQYMGxd5YFghVFPxRonvssUeWpMytipQWaBlBGxwUDtM4pMUyACf4nXvuueFXv/pVLGDWe6iM7EFQekzvMM3D8sJWW20Vd/vT69xwww1j3kTXhuRNYgSMgBEoPALZNlhtIO2r3rGZ9aU9ZWoehc7DYVMMuphRZVMdcZhqR7EzFc/GVKbiV1xxxTgdz0CSEXuWN26ZrDzyb0Q718qfQkHx01uj18ZUYyUDHeHYPG0pQGhER5z2GBT51ltvHaPyNQK79bnudJ111okKPr3Mh7ToyBx00EHx07+//OUv4ZJLLgnf/e53w6677hrlbo8MjmMEjIARKAoCandp19Vmp7Kj5FHwbKzjef3118Mrr7wS3njjjdheE5+1chQ8U/Obb755HBAy48r+MPxaMmlbn74TJ+tO+bQUltIV6T23yl+KmV4dCpMd/i0VAGHEwdZ7tQIV75b4tbUQlebPf/7zcP3114ezzjorXHbZZeGJJ56IJ/tJBqV54YUXBh4MywZs9jv++OPLZG5r2qYzAvVEoDP+P/WU32nXDwHaQ9bbGcGj7FHyzz//fJg1a1ZJKEbpjNpR6ByUxv4vTktlEJUatcGpX/Y91Qlqi2VDm4Zn4zaqO7fKn8JAOTKNfvrpp8fCqVYIaoRkqzBbo68W3pp/Nh0qEWtF3/nOd8Jpp50WjjnmmMCxviNHjoysoFccVVRsziPAf9iwYaVerehak8HhRqCrEFAjSd087LDDwtixY+PuZNJXfe4qWZxO/RFI64PeK9UF6guje2jYYIeiR7mzX4T2kR31TNez+ZnpedbimQUdMGBAVPK0qSyt8rRm2tJuikZ2lmc1/yxdo7hzpfxVkTjch00YrOmws7+lHZV5KQhk//73vx938/M5H5WcTkvWUMGUzxtvvDHstNNOpYa02SpfFhu7840A9ZalLhpr1WHX2XyXWWdLp3LP8tUUPuF8+85mu7/+9a/xkDMOOkP5s7+JT91o29n0jJLHZsqeHfbUrfYY18H2oBZCrpS/ssD6Dus8EyZMiOs5VCiePBcysrGhj47LoYceGjf6rbrqqspSmU3H4NJLLw0PPPBAnDHA3mGHHcpo7DACeUFA/zs1/KmtsLzIajk6HwGVN5wpbxS9DGv0HGzz8ssvx8+dOb2UL7NYrqWTyMCNw8zWW2+9eBU0a/V8IsdoXh0GeKVpiLft2iKw2IIcoo5IPJUaFvzZLJInk0JIheYUQj4jYe0fk5U3rfTZfOQxf1kZ8+DmXISJEyfGqcQ8yNPoMuj/mNZ1/p/yb/T8N1v+VM4qY+Wf81b4NJk9TSh6BjJ8Woc/yh5Fz/o838cz+GGUn07bt6e+VNIDksd2+xHI5cifwq5W4NX82w9Bx2OmMlHRr7766hYbxWxnIJUg5ZX6+90I1AsBKQI6rRyCxVcqPGzGsikeAipP2hq9V8sFZc5InjNLmMbn0zrc+DOC32KLLUpT92zEY/oeU60dq+ZfLX371w6BXCr/2mW3azgztcWfiseVvWswdyq1Q0D1mLpM489BP2zGlX/tUjbnWiJA+WUNfozsp0yZEpU9m/NYv0fZ8+08n9YNHDgwrtfTzjHYcT3IolgMt5V/jcpJSt9/jBoBbLZdhgB1WYqCdxSB3F0mhBPqVAQoP2YgsenIPf300/FTO43uKWNOvOOgHKby2ZiX/cROAmXbOrkVbjufCFj516hc9AeQXaNkzNYIdAkCKAoUAoZ36rXrdpdA3ymJZDtrHIeLwudOEh46AHxXz2mjXDI2ZMiQuGafLlFS3imfrFudiU4R2ExqjoCVfw0gdqNYA1DNsq4IaKS4+uqrhz333LO0tltXoZx4RCBVyCkkqXJmVz4b9KZPnx4363GoDsqaz+04OZVNerotNdt+pfyzYa25U3n8ni8ErPzzVR6WxgjkDgEpfmwOYeECK5v8IYAiZnZGCpkLb1D43MLIceLcR8/onuPHt99++7DNNtvEw3XICXFSJZ/mTvxSP78XHwEr/+KXoXNgBGqKQKpUpCBSv5ombuatIkCZSHlzah7r9o8//nh45JFH4id4HI+76aabxs16rOGnn96JucpVbtuNj4CVf+OXsXNoBDqEAIqBKeJ0VCmF0yHGjtwpCHDIzosvvhhH908++WRU+L169Yqje9bv2bCHwvcIvlPgbhgmVv4NU5TOiBGoDQLpKJ+T2xhd6jOv2qTY3FzTUbhG9OpsyWYKn1P17r///jBt2rQ4pc+5CzvvvHPYdttt45G5Oi7XSr+561O13Fv5V0PG/kbACJQQkBLiAKsf/vCHcR252vHVpUh+aTcCUvLqCIA/l+BwOc7dd98dv8PnsB0uv0HZ87AZUwqfhJmpSXfrt1sYR2xIBKz8G7JYnSkj0PkISAFxcYuVSufjK45S+HKzU5+rv1H6M2fOjNizdn/00UeHQYMGxS8v1FlQHGzKqJJ/SuP35kXAyr95y945NwJtRkAKScpE3/y3mYEJFxkBduvffPPNUfEzyu/du3c4/PDD4zf4bOKTUZnITSdN+zN4tzEClRCw8q+Eiv2MgBEoQwAlkiqZrLuM2I4WEVBHKiVKseUb/FtuuSWwee9f//pX2HLLLcOxxx4b77jP7tSnHCopeM/MpOj6vRICVv6VULGfETACZQhIwfCNP+v+rDXLr4zQjjYhkHYApMD5RO/WW28NTz31VFy7Z/MeV33rWF3FMe5tgthErSBg5d8KQA42Akbgf/eto3zYVMZj0zEEUOAa7b/00kvhtttuC1OnTg3LLbdc2G+//eK14Msuu2ypgyVapvOzo/+OSeLYzYqAlX+zlrzzbQQWEQEpLKIxrax15UVk0/TkUuRM7990003xMB5mUvbff/8wcuTIsOSSS0alDx1G9Lx7Or/pq0+nAWDl32lQmpERaFwEUgVELqWYGjfH7c9ZNWzk/8EHH8Q1fa5GZnQ/evTosPfee4du3brFRDWtn7XbL5FjGoEvI2Dl/2VM7GMEjEAVBJ5//vnw0EMPhbFjx4bu3buXpqWrkNs7GbnPnz8/PPjgg+FPf/pTnDUZPnx4GDNmTFh66aWNkxHocgSs/LsccidoBIqHgEahnBf/gx/8IOyzzz5R+WdnBIqXs9pInOICdtymd8kll4R33nknXptL54lP9zCi1cyAsK6NZOZqBP6HgJW/a4IRMAKLhADKSYpqkSI2EbEU+IcffhiuvPLKMHny5NCzZ894OuKQIUPK1u5FK7uJYHJW64iAlX8dwXfSRqBICEg5aaRaJNlrKWu2IyR8pkyZEi699NLAiYis67OZj3V9bdoTnrWUzbyNQDUErPyrIWN/I2AEyhBgdz9nx3uNugyWOAuirx8I4WS+iRMnxk/3BgwYEI466qjQv3//UiQr/RIUfqkjAlb+dQTfSRuBoiDAaBYFd9hhh8UHt0a4RclDreQEF2Hx9NNPR8XPjv6DDz447LrrrmGJJf7XzKL0obMxAnlAwMo/D6VgGYxAzhGQ4pKS8+h1YYGBCTfusYv/jjvuCKuttlo46aSTwhprrBGJhJkV/0LM/FZ/BKz8618GlsAI5BoBFD1T/hh1AlI718LXQDiUuBQ67OfMmRMmTJgQOJ6X448PPPDAeFBPOiNQAzHM0gh0CAEr/w7B58hGoPERSE/yS5VeM4/+pdinTZsWLr744vDvf/87nHDCCWGLLbYonX3gkX7j/zeKnMPFiyy8ZTcCRqD2CEjJo8yuuOKKMHDgwPDuu+829fo10/x33nln+PWvfx1P6fvFL35RpvhrXypOwQh0DAGP/DuGn2MbgaZBgE4An6299dZbcY1bnYKmAeD/Gf3ss89iJ2jSpElh6NCh4bjjjvMXEM1WCRogv1b+DVCIzoIRqDUC6XQ/aeFmOYDp70Y35JWHvPIZ3wUXXBDY1T9q1Khw0EEH+Za9Rq8ADZo/K/8GLVhnywjUGgGtezfDDAB5nDt3bjjrrLPizAff7g8bNqy0vl9rrM3fCHQ2Alb+nY2o+RmBBkRACn6zzTYLp5xySlznzs4GNGC2S1l67733wumnnx4++eSTcOKJJ4aNN964bMd/idAvRqAgCFj5F6SgLKYRqDcCKPtNN900Ps2k+GfPnh3OPvvsuKN/3LhxccNjM+W/3vXO6dcGASv/2uBqrkagYRFA8WnKvxEzSf5kZs2aFc4888zoPPXUU+MBPsyCaCZEdLaNQNEQaPzdOkUrEctrBHKIQKoQES/rzqHIHRbp1VdfDePHj498GPH37du3wzzNwAjkBQEr/7yUhOUwAjlGQKPdmTNnhhtuuCF+8pdjcTssGucYnHPOOfGLBhR/v379mqLD02HgzKAwCFj5F6aoLKgRqD8C99xzTzjkkEPCvHnzGk4ZMpvB89FHH4Vf/vKXpTV+3cjXDLMd9a9hlqCrELDy7yqknY4RKDACqeKTkmyUdW/yo7sLOMTojDPOiB2A733ve6WreMkr+xwaJc8FrooWvZMQsPLvJCDNxgg0MgKp0kvfGynP8+fPD+edd154/fXXw/HHHx/WWmutUqegkfLpvBgBELDydz0wAkagqRHQTAb3FjzzzDPhmGOOCRtttFEc5TPatzECjYiAa3YjlqrzZAQ6EYF0yp+T7Tjopnfv3g2z5s9Mxr333hvuvvvusNdee4Vtt922NL2f5r0TITUrI1B3BPydf92LwAIYgXwjgHLk0dW+vKMUi/qtf1ahT58+Pfzxj3+Ml/QccMABMV/5LhFLZwQ6joCVf8cxNAcj0PAIpAqTd3UAippx5Yfz+s8999ywyiqrxOn+JZZwk1jUMrXci4aAp/0XDS9TG4GmRUAKX3bRgfjiiy/iDX1c0cvO/mWWWaZhljKKXjaWv/YIWPnXHmOnYAQaBoE77rgjjBw5Mt5wpxmAomVOct98882BKf/DDz88HttbtHxYXiPQEQSs/DuCnuMagSZAQFPkZPWNN94IDzzwQPj0008LNfVPHtizIMX//PPPh5tuuilss8028ZE/sxo2RqAZELDyb4ZSdh6NQAcQSDf2SYkWSUkiM0Yyc5DPxIkTQ69evcKRRx7ZAWQc1QgUFwEr/+KWnSU3Al2CgHb5KzEpUbmLYEtmOgJXXXVV4Oz+I444InTr1q0I4ltGI9DpCFj5dzqkZmgEGhMBFCin3vE53FJLLRUzKaWa9xxr9M90P9/0jxo1KqyzzjpxNoA8FCUfecfZ8hUHAX/XUpyysqRGoC4ISDGiQHfcccf4SBD8FC6/vNrs6r/ssstCnz59wujRo6PcRZI/r7harmIi4JF/McvNUhuBuiGAsi+a0kTea6+9Nm5YHDt2bFhyySUjfkXpuNStsJ1wwyJg5d+wReuMGYHaICDFj10Eg5zvvfde4Dri4cOHhw033LAIYltGI1BTBKz8awqvmRuBxkBAip677mfNmhU4IKdI5g9/+EMUl+l+GY/6hYTtZkTAyr8ZS915NgKLiIBut7vmmmvijXeMpDF5VKB0VNJnxowZ4dlnnw3f/OY3w4orrlgmdx7lX8SiMbkRaBcC3vDXLtgcyQg0DwIoSJQpNiN+7r2XW3be0EAuDJ8pclXvCiusEIYNG5bLzkresLM8zYGAR/7NUc7OpRFoNwJSpFL02HryOnKWXFOnTg2vvvpqGDNmTOE+T2x3gTmiEWgDAlb+bQDJJEbACCxEAMUq5brQNz9v6qx8/vnn4brrrouf9m266aa5ljk/6FmSZkHA0/7NUtLOpxHoBAR222230L9//ziNrpmATmDbqSzomCDbtGnTwptvvhlOOumk8JWvfKVT0zAzI1B0BKz8i16Clt8IdBECKNS+ffvGRwo2rx0A5GJzIvJuvPHGXYSQkzECxUHA0/7FKStLagTqggCKNDXa+a8OQBqWl/fnnnsujvr33XffOOrP5iEvcloOI1AvBKz864W80zUCBUFASh6bhx30MrjzaG655Zaw8sorB9b6MXmVM4/YWabmQMDKvznK2bk0Ah1CQMrz6aefDuecc0745JNP4rp6h5h2UmRG9XromMycOTNMnz497LzzzvHWPmRPn05K1myMQKERsPIvdPFZeCNQewRQnJo2f+yxx8Ipp5wS5s2bl6vRtDonoDF58uSwzDLLhB122KH24DgFI1BQBKz8C1pwFtsIdBUCUvykpxE076l/V8lSLR1kQTZmJCZNmhRPIezevXuuZKwmu/2NQD0QsPKvB+pO0wgUCAEp1ryLjJxPPPFEPIVw5MiRUfGnMwJ5l9/yGYGuRMCf+nUl2k7LCBQQARSoRvlchdujR4+gHf95y87DDz8cVl999fjkTTbLYwTyhIBH/nkqDctiBHKIgEb+dAIOPfTQ+Aldr169cicpB/q88MILYZNNNomdE4/6c1dEFihHCFj556gwLIoRyCMClZQofpX86yn/k08+GWUaMWJE7mSrJy5O2whUQsDKvxIq9jMCRqAwCKgTgvIfNGhQWH755UvLFIXJhAU1Al2MgJV/FwPu5IxA0RDQKL+aXc/8sCTBt/1z5swJL7/8cthggw3iqD+VtZ7yOW0jkFcErPzzWjKWywgYgTYhgKJ/6qmnArf4bbXVVm2KYyIj0OwIWPk3ew1w/o1AgRFA8WNmzJgR+vXrF3r37l3g3Fh0I9B1CFj5dx3WTskIGIEaIMC0P8f5rrvuupG7PkusQVJmaQQaBgEr/4YpyubJCI27Pj9zQ9885V4tp6+99lr4+OOPw4ABA6qR2N8IGIEMAlb+GUDsLAYCTPcy4rPyL0Z51VJKru/94osvwvrrr1/LZMzbCDQUAlb+DVWcjZ+ZVNlrR3fq1/gIOIcpApT9W2+9FVZaaaV4hS9h2geQ0vndCBiBcgR8vG85HnYVAAEa/OHDh8fvuQsgrkWsMQKDBw/2Rr8aY2z2jYfAYgs8bGq8UnWOjEATIaAmzCP+Jip0Z7XDCHjk32EIzcAIGIF6ImClX0/0nXZREfCaf1FLznIbASNgBIyAEWgnAlb+7QTO0YyAETACRsAIFBUBK/+ilpzlNgJGwAgYASPQTgSs/NsJnKMZASNgBIyAESgqAlb+RS05y20EjIARMAJGoJ0I/BdXa4IGi3UscgAAAABJRU5ErkJggg=="></p>
<p>either graph passing through (or touching) A <em><strong>A1</strong></em></p>
<p>correct shape and vertical asymptote with correct equation for either graph <em><strong>A1</strong></em></p>
<p>correct horizontal asymptote with correct equation for either graph <em><strong>A1</strong></em></p>
<p>two completely correct sketches <em><strong>A1</strong></em></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="a\left( { - \frac{1}{2}} \right) + 1 = 0 \Rightarrow a = 2"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>a</mi> <mo>=</mo> <mn>2</mn> </math></span> <em><strong>A1</strong></em></p>
<p>from horizontal asymptote, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{a}{b}} \right)^2} = \frac{4}{9}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{a}{b} = \pm \frac{2}{3} \Rightarrow b = \pm 3"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> <mo>=</mo> <mo>±</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo stretchy="false">⇒</mo> <mi>b</mi> <mo>=</mo> <mo>±</mo> <mn>3</mn> </math></span> <em><strong>A1</strong></em></p>
<p>from vertical asymptote, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b\left( {\frac{4}{3}} \right) + c = 0"> <mi>b</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> = 3, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> = −4 or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span> = −3, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c"> <mi>c</mi> </math></span> = 4 <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="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="specification">
<p>Two airplanes, <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>, have position vectors with respect to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> given respectively by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mtext mathvariant="bold-italic">A</mtext></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>19</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> represents the time in minutes and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Entries in each column vector give the displacement east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the displacement north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the distance above sea level, all measured in kilometres.</p>
</div>
<div class="specification">
<p>The two airplanes’ lines of flight cross at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the three-figure bearing on which airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is travelling.</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>Show that airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</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 acute angle between the two airplanes’ lines of flight. Give your answer in degrees.</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>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the length of time between the first airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and the second airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> represent the distance between airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math> be the required angle (bearing)</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mn>90</mn><mo>°</mo><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for a labelled sketch.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ϕ</mi><mo>=</mo><mfrac><mrow><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></mrow><mrow><msqrt><mn>1</mn></msqrt><mo>×</mo><msqrt><mn>20</mn></msqrt></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>4472</mn><mo>…</mo><mo>,</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mtext>arccos</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4472</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>063</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>063</mn><mo>.</mo><mn>6</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn><mo>.</mo><mn>4</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><msup><mn>10</mn><mi>c</mi></msup></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced></math> be the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced></math> be the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></p>
<p>attempts to find the speed of one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>19</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>0</mn><mn>2</mn></msup><mo>+</mo><msup><mn>12</mn><mn>2</mn></msup></msqrt></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><mn>7</mn><mo>.</mo><mn>48</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>56</mn></msqrt></mrow></mfenced></math> (km min<sup>-1</sup>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><mn>4</mn><mo>.</mo><mn>89</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>24</mn></msqrt></mrow></mfenced></math> (km min<sup>-1</sup>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>></mo><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>speed</mtext><mo>=</mo><mfrac><mtext>distance</mtext><mtext>time</mtext></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>A</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced></mrow></mfenced><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>B</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced><mo>-</mo><msub><mi>r</mi><mi>B</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced></mrow></mfenced><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p>for example:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>A</mi></msub><mfenced><mn>1</mn></mfenced><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mfenced><mn>0</mn></mfenced></mrow></mfenced><mn>1</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>B</mi></msub><mfenced><mn>1</mn></mfenced><mo>-</mo><msub><mi>r</mi><mi>B</mi></msub><mfenced><mn>0</mn></mfenced></mrow></mfenced><mn>1</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><mn>1</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup></msqrt><mn>1</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mn>7</mn><mo>.</mo><mn>48</mn><mo>…</mo><mfenced><mrow><mn>2</mn><msqrt><mn>14</mn></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mn>4</mn><mo>.</mo><mn>89</mn><mo>…</mo><mfenced><msqrt><mn>24</mn></msqrt></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>></mo><msub><mtext>speed</mtext><mi>B</mi></msub></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</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>attempts to use the angle between two direction vectors formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mn>4</mn></mfenced><mo>+</mo><mfenced><mn>2</mn></mfenced><mfenced><mn>2</mn></mfenced><mo>+</mo><mfenced><mn>4</mn></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mrow><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>7637</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>7</mn><msqrt><mn>84</mn></msqrt></mfrac></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mtext>arccos</mtext><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7637</mn><mo>…</mo></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><mo>.</mo><mn>4399</mn><mo>…</mo></mrow></mfenced></math></p>
<p>attempts to find the acute angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo><mo>-</mo><mi>θ</mi></math> using their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>40</mn><mo>.</mo><mn>2</mn><mo>°</mo></math> <em><strong>A1</strong></em></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>for example, sets <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced></math> and forms at least two equations <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn><mo>-</mo><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>12</mn><mo>-</mo><mn>2</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for equations involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> only.</p>
<p><br><strong>EITHER</strong></p>
<p>attempts to solve the system of equations for one of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to solve the system of equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> value into the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>7</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>9</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>OP</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd></mtr></mtable></mfenced></math>. Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> km east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> km north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> km above sea level.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find the value of <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></math> <em><strong>(M1)</strong></em></p>
<p><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><mn>2</mn><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> minutes (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>18</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>11</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msqrt><msup><mfenced><mrow><mn>10</mn><mi>t</mi><mo>-</mo><mn>18</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>+</mo><msup><mfenced><mrow><mn>11</mn><mo>-</mo><mn>6</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>18</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>11</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msqrt><msup><mfenced><mrow><mn>18</mn><mo>-</mo><mn>10</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>11</mn><mo>+</mo><mn>6</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0M0A0</strong></em> for expressions using two different time parameters.</p>
<p><br><strong>THEN</strong></p>
<p>either attempts to find the local minimum point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> or attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>'</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> (or equivalent) <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>8088</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>123</mn><mn>68</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>01459</mn><mo>…</mo></math></p>
<p>minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>01</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msqrt><mn>1190</mn></msqrt><mn>34</mn></mfrac></mrow></mfenced></math> (km) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for attempts at the shortest distance between two lines.</p>
<p> </p>
<p><em><strong>[5 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;">
<p>General comment about this question: many candidates were not exposed to this setting of vectors question and were rather lost.</p>
<p>Part (a) Probably the least answered question on the whole paper. Many candidates left it blank, others tried using 3D vectors. Out of those who calculated the angle correctly, only a small percentage were able to provide the correct true bearing as a 3-digit figure.</p>
<p>Part (b) Well done by many candidates who used the direction vectors to calculate and compare the speeds. A number of candidates tried to use the average rate of change but mostly unsuccessfully.</p>
<p>Part (c) Most candidates used the correct vectors and the formula to obtain the obtuse angle. Then only some read the question properly to give the acute angle in degrees, as requested.</p>
<p>Part (d) Well done by many candidates who used two different parameters. They were able to solve and obtain two values for time, the difference in minutes and the correct point of intersection. A number of candidates only had one parameter, thus scoring no marks in part (d) (i). The frequent error in part (d)(ii) was providing incorrect units.</p>
<p>Part (e) Many correct answers were seen with an efficient way of setting the question and using their GDC to obtain the answer, graphically or numerically. Some gave time only instead of actually giving the minimal distance. A number of candidates tried to find the distance between two skew lines ignoring the fact that the lines intersect.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean 0.2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Lucca eats at least one banana in a particular day.</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 expected number of weeks in the year in which Lucca eats no bananas.</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>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X"> <mi>X</mi> </math></span> be the number of bananas eaten in one day</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X \sim {\text{Po}}(0.2)"> <mi>X</mi> <mo>∼</mo> <mrow> <mtext>Po</mtext> </mrow> <mo stretchy="false">(</mo> <mn>0.2</mn> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>⩾</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0.181{\text{ }}( = 1 - {{\text{e}}^{ - 0.2}})"> <mo>=</mo> <mn>0.181</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>0.2</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Y"> <mi>Y</mi> </math></span> be the number of bananas eaten in one week</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{Y}} \sim {\text{Po}}(1.4)"> <mrow> <mtext>Y</mtext> </mrow> <mo>∼</mo> <mrow> <mtext>Po</mtext> </mrow> <mo stretchy="false">(</mo> <mn>1.4</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Y = 0) = 0.246596 \ldots {\text{ }}( = {{\text{e}}^{ - 1.4}})"> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.246596</mn> <mo>…</mo> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>1.4</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Z"> <mi>Z</mi> </math></span> be the number of days in one week at least one banana is eaten</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Z \sim {\text{B}}(7,{\text{ }}0.181 \ldots )"> <mi>Z</mi> <mo>∼</mo> <mrow> <mtext>B</mtext> </mrow> <mo stretchy="false">(</mo> <mn>7</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0.181</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}(Z = 0) = 0.246596 \ldots "> <mrow> <mtext>P</mtext> </mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.246596</mn> <mo>…</mo> </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="52 \times 0.246596 \ldots "> <mn>52</mn> <mo>×</mo> <mn>0.246596</mn> <mo>…</mo> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 12.8{\text{ }}( = 52{{\text{e}}^{ - 1.4}})"> <mo>=</mo> <mn>12.8</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>52</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>1.4</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> given by</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnalign="left"><mtr><mtd><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</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>recognition of the need to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></math> (or equivalent) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>u</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>attempt to use correct limits for their integrand and set equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mi>k</mi><mrow><mn>16</mn><mo>+</mo><mi>k</mi></mrow></msubsup><mo>=</mo><mn>1</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mn>0</mn><mn>4</mn></msubsup><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mfenced><mrow><mn>16</mn><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mi>k</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mfenced><mrow><mo>⇒</mo><mfrac><mn>1</mn><msqrt><mi>k</mi></msqrt></mfrac><mo>-</mo><mfrac><mn>1</mn><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></mfrac><mo>=</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>645038</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>645</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;">
[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 polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4} + p{x^3} + q{x^2} + rx + 6">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>q</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>r</mi>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span> is exactly divisible by each of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 1} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 2} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 3} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Find the values of <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">
<mi>q</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</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>substitute each of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 1,2 and 3 into the quartic and equate to zero <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p + q + r = - 7">
<mi>p</mi>
<mo>+</mo>
<mi>q</mi>
<mo>+</mo>
<mi>r</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4p + 2q + r = - 11">
<mn>4</mn>
<mi>p</mi>
<mo>+</mo>
<mn>2</mn>
<mi>q</mi>
<mo>+</mo>
<mi>r</mi>
<mo>=</mo>
<mo>−</mo>
<mn>11</mn>
</math></span> or equivalent <em><strong> (A2)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9p + 3q + r = - 29">
<mn>9</mn>
<mi>p</mi>
<mo>+</mo>
<mn>3</mn>
<mi>q</mi>
<mo>+</mo>
<mi>r</mi>
<mo>=</mo>
<mo>−</mo>
<mn>29</mn>
</math></span></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong> </em>for all three equations correct, <em><strong>A1</strong> </em>for two correct.</p>
<p>attempting to solve the system of equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = −7, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = 17, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> = −17 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Only award <em><strong>M1</strong></em> when some numerical values are found when solving algebraically or using GDC.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to find fourth factor <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {x - 1} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to expand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x - 1} \right)^2}\left( {x - 2} \right)\left( {x - 3} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</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="{x^4} - 7{x^3} + 17{x^2} - 17x + 6">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>17</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>17</mn>
<mi>x</mi>
<mo>+</mo>
<mn>6</mn>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> = −7, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = 17, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> = −17) <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A2</strong> </em>for all three values correct, <em><strong>A1</strong> </em>for two correct.</p>
<p><strong>Note:</strong> Accept long / synthetic division.</p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{sec}}\,x + 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sec</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span>, stating its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> ≥ 3 <em><strong>A1</strong></em></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="x = {\text{sec}}\,y + 2"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mn>2</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Exchange of variables can take place at any point.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,y = \frac{1}{{x - 2}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {\text{arccos}}\left( {\frac{1}{{x - 2}}} \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≥ 3 <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Allow follow through from (a) for last <em><strong>A1</strong></em> mark which is independent of earlier marks in (b).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{{x^2}}}{{x - 3}}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = m\left( {x + 3} \right)">
<mi>y</mi>
<mo>=</mo>
<mi>m</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{R}">
<mi>m</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>Find the set of values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> such that the two graphs have no intersection points.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><em><strong>METHOD 1</strong></em></p>
<p>sketching the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{{x^2}}}{{x - 3}}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x + 3 + \frac{9}{{x - 3}}">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo>+</mo>
<mfrac>
<mn>9</mn>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>) <em><strong>M1</strong></em></p>
<p>the (oblique) asymptote has a gradient equal to 1 </p>
<p>and so the maximum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> is 1 <em><strong>R1</strong></em></p>
<p>consideration of a straight line steeper than the horizontal line joining (−3, 0) and (0, 0) <em><strong>M1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> > 0 <em><strong>R1</strong></em></p>
<p>hence 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> ≤ 1 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>attempting to eliminate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> to form a quadratic equation in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<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="{x^2} = m\left( {{x^2} - 9} \right)">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>m</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {m - 1} \right){x^2} - 9m = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mi>m</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>EITHER</strong></em></p>
<p>attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 4\left( {m - 1} \right)\left( { - 9m} \right) < 0">
<mo>−</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>9</mn>
<mi>m</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>0</mn>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> <em><strong>M1</strong></em> </p>
<p> </p>
<p><em><strong>OR</strong></em></p>
<p>attempting to solve <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> < 0 <em>ie</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{9m}}{{m - 1}} < 0\,\left( {m \ne 1} \right)">
<mfrac>
<mrow>
<mn>9</mn>
<mi>m</mi>
</mrow>
<mrow>
<mi>m</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo><</mo>
<mn>0</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mo>≠</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> <em><strong>M1</strong></em></p>
<p> </p>
<p><em><strong>THEN</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 0 < m < 1">
<mo stretchy="false">⇒</mo>
<mn>0</mn>
<mo><</mo>
<mi>m</mi>
<mo><</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>a valid reason to explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 1">
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> gives no solutions <em>eg</em> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 1">
<mi>m</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>,</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {m - 1} \right){x^2} - 9m = 0 \Rightarrow - 9 = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mi>m</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> and so 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> ≤ 1 <em><strong>R1</strong></em></p>
<p> </p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>k</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mi>x</mi><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for the product of the roots, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> such that the equation has one positive and one negative real root.</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>product of roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>9</mn></mrow><mi>k</mi></mfrac></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>recognition that the product of the roots will be negative <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>9</mn></mrow><mi>k</mi></mfrac><mo><</mo><mn>0</mn></math></p>
<p>critical values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac></math> seen <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><mo><</mo><mi>k</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the minimum point on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> showing clearly the minimum point and any asymptotic behaviour.</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>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>This region is now rotated 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 about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule or product rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}{\text{ }}\left( { = \frac{1}{{2\sqrt x \sin x}} - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}} \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{2\sqrt x \sin x}}"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> or equivalent and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}"> <mo>−</mo> <mfrac> <mrow> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> or equivalent.</p>
<p> </p>
<p>setting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} - \sqrt x \cos x = 0"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> </mrow> </mfrac> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} = \sqrt x \cos x"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> </mrow> </mfrac> <mo>=</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> or equivalent <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.17"> <mi>x</mi> <mo>=</mo> <mn>1.17</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x \leqslant 1.17"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo>⩽</mo> <mn>1.17</mn> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x"> <mn>0</mn> <mo><</mo> <mi>x</mi> </math></span> and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \leqslant 1.17"> <mi>x</mi> <mo>⩽</mo> <mn>1.17</mn> </math></span>. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < 1.17"> <mi>x</mi> <mo><</mo> <mn>1.17</mn> </math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.19.25.png" alt="N17/5/MATHL/HP2/ENG/TZ0/10.b/M"></p>
<p>concave up curve over correct domain with one minimum point above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> asymptotically <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi "> <mi>x</mi> <mo>=</mo> <mi>π</mi> </math></span> asymptotically <strong><em>A1</em></strong></p>
<p> </p>
<p>Note: For the final <strong><em>A1 </em></strong>an asymptote must be seen, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> must be seen on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis or in an equation.</p>
<p> </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="f’(x){\text{ }}\left( { = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}} \right) = 1"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.96"> <mi>x</mi> <mo>=</mo> <mn>1.96</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(1.96 \ldots )"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1.96</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1.51"> <mo>=</mo> <mn>1.51</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi \int_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{{x{\text{d}}x}}{{{{\sin }^2}x}}} "> <mi>V</mi> <mo>=</mo> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mrow> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>M1 </em></strong>is for an integral of the correct squared function (with or without limits and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2.68{\text{ }}( = 0.852\pi )"> <mo>=</mo> <mn>2.68</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>0.852</mn> <mi>π</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br>