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<h2>HL Paper 3</h2><div class="specification">
<p><strong>This question is about modelling the spread of a computer virus to predict the number of computers in a city which will be infected by the virus.</strong></p>
<p><br>A systems analyst defines the following variables in a model:</p>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the number of days since the first computer was infected by the virus.</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> is the total number of computers that have been infected up to and including day <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</li>
</ul>
<p>The following data were collected:</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>A model for the early stage of the spread of the computer virus suggests that</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>β</mi><mi>N</mi><mi>Q</mi><mfenced><mi>t</mi></mfenced></math></p>
<p style="text-align: left;">where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> is the total number of computers in a city and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> is a measure of how easily the virus is spreading between computers. Both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> are assumed to be constant.</p>
</div>
<div class="specification">
<p>The data above are taken from city X which is estimated to have <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>6</mn></math> million computers.<br>The analyst looks at data for another city, Y. These data indicate a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mn>9</mn><mo>.</mo><mn>64</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>8</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>An estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>′</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>t</mi><mo>≥</mo><mn>5</mn></math>, can be found by using the formula:</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>≈</mo><mfrac><mrow><mi>Q</mi><mfenced><mrow><mi>t</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mo>-</mo><mi>Q</mi><mfenced><mrow><mi>t</mi><mo>-</mo><mn>5</mn></mrow></mfenced></mrow><mn>10</mn></mfrac></math>.</p>
<p>The following table shows estimates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>'</mo><mo>(</mo><mi>t</mi><mo>)</mo></math> for city X at different values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>An improved model for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, which is valid for large values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, is the logistic differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>k</mi><mi>Q</mi><mfenced><mi>t</mi></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mi>Q</mi><mfenced><mi>t</mi></mfenced></mrow><mi>L</mi></mfrac></mrow></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> are constants.</p>
<p>Based on this differential equation, the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>Q</mi><mo>'</mo><mfenced><mi>t</mi></mfenced></mrow><mrow><mi>Q</mi><mfenced><mi>t</mi></mfenced></mrow></mfrac></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> is predicted to be a straight line.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, Pearson’s product-moment correlation coefficient.</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 it would not be appropriate to conduct a hypothesis test on the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> found in (a)(ii).</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>Find the general solution of the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>β</mi><mi>N</mi><mi>Q</mi><mfenced><mi>t</mi></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the data in the table write down the equation for an appropriate non-linear regression model.</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>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mn>2</mn></msup></math> for this model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence comment on the suitability of the model from (b)(ii) in comparison with the linear model found in part (a).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering large values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> write down one criticism of the model found in (b)(ii).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.v.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answer from part (b)(ii) to estimate the time taken for the number of infected computers to double.</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 in which city, X or Y, the computer virus is spreading more easily. Justify your answer using your results from part (b).</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>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>. Give your answers correct to one decimal place.</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>Use linear regression to estimate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The solution to the differential equation is given by</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mi>L</mi><mrow><mn>1</mn><mo>+</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>k</mi><mi>t</mi></mrow></msup></mrow></mfrac></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is a constant.</p>
<p>Using your answer to part (f)(i), estimate the percentage of computers in city X that are expected to have been infected by the virus over a long period of time.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3090</mn><mi>t</mi><mo>-</mo><mn>54000</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>3094</mn><mo>.</mo><mn>27</mn><mo>…</mo><mi>t</mi><mo>-</mo><mn>54042</mn><mo>.</mo><mn>3</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award at most <em><strong>A1A0</strong></em> if answer is not an equation. Award <em><strong>A1A0</strong></em> for an answer including either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>755</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>754741</mn><mo>…</mo></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>t</mi></math> is not a random variable <strong>OR</strong> it is not a (bivariate) normal distribution</p>
<p><strong>OR</strong> data is not a sample from a population</p>
<p><strong>OR</strong> data appears nonlinear</p>
<p><strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> only measures linear correlation <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not accept “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> is not large enough”.</p>
<p> </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>attempt 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><mi>Q</mi></mfrac><mo>d</mo><mi>Q</mi><mo>=</mo><mo>∫</mo><mi>β</mi><mi>N</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mi>Q</mi></mfenced><mo>=</mo><mi>β</mi><mi>N</mi><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for LHS, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mi>N</mi><mi>t</mi></math>, and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>.</p>
<p>Award full marks for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mi>β</mi><mi>N</mi><mi>t</mi><mo>+</mo><mi>c</mi></mrow></msup></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mi>β</mi><mi>N</mi><mi>t</mi></mrow></msup></math>.</p>
<p>Award <em><strong>M1A1A1A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mi>β</mi><mi>N</mi><mi>t</mi></mrow></msup></math></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at exponential regression (<em><strong>M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>15</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>292</mn><mi>t</mi></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mi>Q</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>14864</mn><mo>…</mo><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>292055</mn><mo>…</mo><mi>t</mi></mrow></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p>attempt at exponential regression (<em><strong>M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>15</mn><mo>×</mo><mn>1</mn><mo>.</mo><msup><mn>34</mn><mi>t</mi></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>14864</mn><mo>…</mo><mo>×</mo><mn>1</mn><mo>.</mo><mn>33917</mn><msup><mo>…</mo><mi>t</mi></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone answers involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>. Condone absence of “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo></math>” Award <em><strong>M1A0</strong></em> for an incorrect answer in correct format.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>999</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>999431</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>comparing something to do with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mn>2</mn></msup></math> and something to do with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Examples of where the <em><strong>M1</strong> </em>should be awarded:</p>
<p style="padding-left:90px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mn>2</mn></msup><mo>></mo><mi>r</mi></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>></mo><mi>r</mi></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>999</mn><mo>></mo><mn>0</mn><mo>.</mo><mn>755</mn></math><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>999</mn><mo>></mo><mn>0</mn><mo>.</mo><msup><mn>755</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>563</mn></mrow></mfenced></math><br> The “correlation coefficient” in the exponential model is larger. <br>Model B has a larger <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mn>2</mn></msup></math></p>
<p style="text-align:left;padding-left:60px;">Examples of where the <em><strong>M1</strong> </em>should <strong>not</strong> be awarded:</p>
<p style="text-align:left;padding-left:90px;">The exponential model shows better correlation (since not clear how it is being measured)<br>Model 2 has a better fit<br>Model 2 is more correlated</p>
<p> </p>
<p>an unambiguous comparison between <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>r</mi><mn>2</mn></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> leading to the conclusion that the model in part (b) is more suitable / better <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone candidates claiming that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is the “correlation coefficient” for the non-linear model.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>it suggests that there will be more infected computers than the entire population <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any response that recognizes unlimited growth. </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.v.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>15</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>292</mn><mi>t</mi></mrow></msup><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>15</mn><mo>×</mo><mn>1</mn><mo>.</mo><msup><mn>34</mn><mi>t</mi></msup><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mn>2</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>292</mn></mrow></mfrac></math> <strong>OR</strong> using the model to find two specific times with values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mfenced><mi>t</mi></mfenced></math> which double <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>37</mn></math> (days) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not <em><strong>FT </strong></em>from a model which is not exponential. Award <em><strong>M0A0</strong></em> for an answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>13</mn></math> which comes from using <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>,</mo><mo> </mo><mn>20</mn><mo>)</mo></math> from the data or any other answer which finds a doubling time from figures given in the table.</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>an attempt to calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> for city X <em><strong>(M1)</strong></em></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>292055</mn><mo>…</mo></mrow><mrow><mn>2</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mfrac><mrow><mi>ln</mi><mo> </mo><mn>1</mn><mo>.</mo><mn>33917</mn><mo>…</mo></mrow><mrow><mn>2</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>12328</mn><mo>…</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>7</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>this is larger than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>64</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>8</mn></mrow></msup></math> so the virus spreads more easily in city X <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> It is possible to award <em><strong>M1A0R1</strong></em>. <br>Condone “so the virus spreads faster in city X” for the final <em><strong>R1</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>38</mn><mo>.</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>3086</mn><mo>.</mo><mn>1</mn></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong> </em>if values are correct but not to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> dp.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>Q</mi><mo>'</mo></mrow><mi>Q</mi></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>42228</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>5561</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>6</mn></mrow></msup><mi>Q</mi></math> <em><strong>(A1)(A1)</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for each coefficient seen – not necessarily in the equation. Do not penalize seeing in the context of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<p><br>identifying that the constant is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> <strong>OR</strong> that the gradient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi>k</mi><mi>L</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>422</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>422228</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><mi>L</mi></mfrac><mo>=</mo><mn>2</mn><mo>.</mo><mn>5561</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>6</mn></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mn>165000</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mn>165205</mn></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>164843</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> sf value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, or any other value from plausible pre-rounding.<br>Allow follow-through <strong>within</strong> the question part, from the equation of their line to the final two <em><strong>A1</strong></em> marks.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the eventual number of infected <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>165205</mn><mo>…</mo></mrow><mn>2600000</mn></mfrac><mo>=</mo><mn>6</mn><mo>.</mo><mn>35</mn><mo>%</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>35403</mn><mo>…</mo><mo>%</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept any final answer consistent with their answer to part (f)(i) unless their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is less than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>120146</mn></math> in which case award at most <em><strong>M1A0</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>A significant minority were unable to attempt 1(a) which suggests poor preparation for the use of the GDC in this statistics-heavy course. Large numbers of candidates appeared to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>,</mo><mo> </mo><mi>Q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> interchangeably. Accurate use of notation is an important skill which needs to be developed.</p>
<p>1(a)(iii) was a question at the heart of the Applications and interpretations course. In modern statistics many of the calculations are done by a computer so the skill of the modern statistician lies in knowing which tests are appropriate and how to interpret the results. Very few candidates seemed familiar with the assumptions required for the use of the standard test on the correlation coefficient. Indeed, many candidates answered this by claiming that the value was either too large or too small to do a hypothesis test, indicating a major misunderstanding of the purpose of hypothesis tests.</p>
<p>1(b)(i) was done very poorly. It seems that perhaps adding parameters to the equation confused many candidates – if the equation had been <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>'</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>5</mn><mi>Q</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> many more would have successfully attempted this. However, the presence of parameters is a fundamental part of mathematical modelling so candidates should practise working with expressions involving them.</p>
<p>1(b)(ii) and (iii) were done relatively well, with many candidates using the data to recognize an exponential model was a good idea. Part (iv) was often communicated poorly. Many candidates might have done the right thing in their heads but just writing that the correlation was better did not show which figures were being compared. Many candidates who did write down the numbers made it clear that they were comparing an <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mn>2</mn></msup></math> value with an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> value.</p>
<p>1(c) was not meant to be such a hard question. There is a standard formula for half-life which candidates were expected to adapt. However, large numbers of candidates conflated the data and the model, finding the time for one of the data points (which did not lie on the model curve) to double. Candidates also thought that the value of t found was equivalent to the doubling time, often giving answers of around 40 days which should have been obviously wrong.</p>
<p>1(d) was quite tough. Several candidates realized that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> was the required quantity to be compared but very few could calculate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> for city X using the given information.</p>
<p>1(e) was meant to be relatively straightforward but many candidates were unable to interpret the notation given to do the quite straightforward calculation.</p>
<p>1(f) was meant to be a more unusual problem-solving question getting candidates to think about ways of linearizing a non-linear problem. This proved too much for nearly all candidates.</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">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.v.</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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><em>In this question you will explore possible models for the spread of an infectious disease</em></p>
<p>An infectious disease has begun spreading in a country. The National Disease Control Centre (NDCC) has compiled the following data after receiving alerts from hospitals.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">A graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> against <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> is shown below.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">The NDCC want to find a model to predict the total number of people infected, so they can plan for medicine and hospital facilities. After looking at the data, they think an exponential function in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = a{b^d}">
<mi>n</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>b</mi>
<mi>d</mi>
</msup>
</mrow>
</math></span> could be used as a model.</p>
</div>
<div class="specification">
<p>Use your answer to part (a) to predict</p>
</div>
<div class="specification">
<p>The NDCC want to verify the accuracy of these predictions. They decide to perform a <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\chi ^2}">
<mrow>
<msup>
<mi>χ<!-- χ --></mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> goodness of fit test.</p>
</div>
<div class="specification">
<p>The predictions given by the model for the first five days are shown in the table.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>In fact, the first day when the total number of people infected is greater than 1000 is day 14, when a total of 1015 people are infected.</p>
</div>
<div class="specification">
<p>Based on this new data, the NDCC decide to try a logistic model in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = \frac{L}{{1 + c{e^{ - kd}}}}">
<mi>n</mi>
<mo>=</mo>
<mfrac>
<mi>L</mi>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>c</mi>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mo>−<!-- − --></mo>
<mi>k</mi>
<mi>d</mi>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Use the data from days 1–5, together with day 14, to find the value of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an exponential regression to find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, correct to 4 decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the number of new people infected on day 6.</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>the day when the total number of people infected will be greater than 1000.</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>Use your answer to part (a) to show that the model predicts 16.7 people will be infected on the first day.</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 number of degrees of freedom is 2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Perform a <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\chi ^2}">
<mrow>
<msup>
<mi>χ</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> goodness of fit test at the 5% significance level. You should clearly state your hypotheses, the p-value, and your conclusion.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give two reasons why the prediction in part (b)(ii) might be lower than 14.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence predict the total number of people infected by this disease after several months.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the logistic model to find the day when the rate of increase of people infected is greatest.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</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="a = 9.7782{\text{,}}\,\,b = 1.7125">
<mi>a</mi>
<mo>=</mo>
<mn>9.7782</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mn>1.7125</mn>
</math></span> <em><strong>M1A1A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\left( 6 \right) = 247">
<mi>n</mi>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>247</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>number of new people infected = 247 – 140 = 107 <em><strong>M1A1</strong></em></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>use of graph or table <em><strong>M1</strong></em></p>
<p>day 9 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>9.7782(1.7125)<sup>1</sup> <em><strong>M1</strong></em></p>
<p>= 16.7 people <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2 parameters (<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>) were estimated from the data. <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\upsilon = 5 - 1 - 2">
<mi>υ</mi>
<mo>=</mo>
<mn>5</mn>
<mo>−</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>= 2 <em><strong>AG</strong></em></p>
<p><em><strong>[2 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="{H_0}\,{\text{:}}">
<mrow>
<msub>
<mi>H</mi>
<mn>0</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
</math></span> data is modeled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\left( d \right) = 9.7782{\left( {1.7125} \right)^d}">
<mi>n</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>9.7782</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.7125</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{H_1}\,{\text{:}}">
<mrow>
<msub>
<mi>H</mi>
<mn>1</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
</math></span> data is not modeled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\left( d \right) = 9.7782{\left( {1.7125} \right)^d}">
<mi>n</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>9.7782</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.7125</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>p-value = 0.893 <em><strong>A2</strong></em></p>
<p>Since 0.893 > 0.05 <em><strong>R1</strong></em></p>
<p>Insufficient evidence to reject <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{H_0}">
<mrow>
<msub>
<mi>H</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>. So data is modeled by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n\left( d \right) = 9.7782{\left( {1.7125} \right)^d}">
<mi>n</mi>
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>9.7782</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.7125</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>vaccine or medicine might slow down rate of infection <em><strong>R1</strong></em></p>
<p>People become more aware of disease and take precautions to avoid infection <em><strong>R1</strong></em></p>
<p><em> Accept other valid reasons </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>1060 <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>108 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>0.560 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>As <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \to \infty ">
<mi>d</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \to 1060">
<mi>n</mi>
<mo stretchy="false">→</mo>
<mn>1060</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sketch of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}n}}{{{\text{d}}d}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>n</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>d</mi>
</mrow>
</mfrac>
</math></span> or solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}n}}{{{\text{d}}{d^2}}} = 0">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>n</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<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="d = 8.36">
<mi>d</mi>
<mo>=</mo>
<mn>8.36</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>Day 8 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>An estate manager is responsible for stocking a small lake with fish. He begins by introducing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math> fish into the lake and monitors their population growth to determine the likely carrying capacity of the lake.</p>
<p>After one year an accurate assessment of the number of fish in the lake is taken and it is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> be the number of fish <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> years after the fish have been introduced to the lake.</p>
<p>Initially it is assumed that the rate of increase of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> will be constant.</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>8</mn></math> the estate manager again decides to estimate the number of fish in the lake. To do this he first catches <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math> fish and marks them, so they can be recognized if caught again. These fish are then released back into the lake. A few days later he catches another <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn></math> fish, releasing each fish after it has been checked, and finds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn></math> of them are marked.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of marked fish caught in the second sample, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> is considered to be distributed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mi>n</mi><mo>,</mo><mo> </mo><mi>p</mi></mrow></mfenced></math>. Assume the number of fish in the lake is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math>.</p>
</div>
<div class="specification">
<p>The estate manager decides that he needs bounds for the total number of fish in the lake.</p>
</div>
<div class="specification">
<p>The estate manager feels confident that the proportion of marked fish in the lake will be within <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> standard deviations of the proportion of marked fish in the sample and decides these will form the upper and lower bounds of his estimate.</p>
</div>
<div class="specification">
<p>The estate manager now believes the population of fish will follow the logistic model <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mi>L</mi><mrow><mn>1</mn><mo>+</mo><mi>C</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>k</mi><mi>t</mi></mrow></msup></mrow></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the carrying capacity and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The estate manager would like to know if the population of fish in the lake will eventually reach <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5000</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this model to predict the number of fish in the lake when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>8</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming the proportion of marked fish in the second sample is equal to the proportion of marked fish in the lake, show that the estate manager will estimate there are now <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math> fish in the lake.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></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>State an assumption that is being made for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> to be considered as following a binomial distribution.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that an estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mo>(</mo><mi>X</mi><mo>)</mo></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>38</mn><mo>.</mo><mn>25</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the variance of the proportion of marked fish in the sample, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mfrac><mi>X</mi><mn>300</mn></mfrac></mfenced></math>, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>000425</mn></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>Taking the value for the variance given in (d) (ii) as a good approximation for the true variance, find the upper and lower bounds for the proportion of marked fish in the lake.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find upper and lower bounds for the number of fish in the lake when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>8</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given this result, comment on the validity of the linear model used in part (a).</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>Assuming a carrying capacity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5000</mn></math> use the given values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mn>0</mn></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mn>1</mn></mfenced></math> to calculate the parameters <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use these parameters to calculate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mn>8</mn></mfenced></math> predicted by this model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment on the likelihood of the fish population reaching <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5000</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mn>8</mn></mfenced><mo>=</mo><mn>1000</mn><mo>+</mo><mn>200</mn><mo>×</mo><mn>8</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2600</mn></math> <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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>45</mn><mn>300</mn></mfrac><mo>=</mo><mfrac><mn>300</mn><mi>N</mi></mfrac></math><strong> <em>M1A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>2000</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>300</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>=</mo><mfrac><mn>300</mn><mn>2000</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>15</mn></math> <strong> <em>A1A1</em></strong></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>Any valid reason for example: <em><strong>R1</strong></em></p>
<p>Marked fish are randomly distributed, so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> constant.</p>
<p>Each fish caught is independent of previous fish caught</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><mi>n</mi><mi>p</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>p</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>300</mn><mo>×</mo><mfrac><mn>300</mn><mn>2000</mn></mfrac><mo>×</mo><mfrac><mn>1700</mn><mn>2000</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>38</mn><mo>.</mo><mn>25</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Var</mtext><mfenced><mfrac><mi>X</mi><mn>300</mn></mfrac></mfenced><mi mathvariant="normal">=</mi><mfrac><mrow><mtext>Var</mtext><mo>(</mo><mi>X</mi><mo>)</mo></mrow><msup><mn>300</mn><mn>2</mn></msup></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>000425</mn></math> <em><strong>AG</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>15</mn><mo>±</mo><mn>1</mn><mo>.</mo><mn>5</mn><msqrt><mn>0</mn><mo>.</mo><mn>000425</mn></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>181</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>119</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 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"><mfrac><mn>300</mn><mi>N</mi></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>181</mn><mo>…</mo><mo>,</mo><mo> </mo><mo> </mo><mfrac><mn>300</mn><mi>N</mi></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>119</mn><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p>Lower bound <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1658</mn></math> upper bound <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2519</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Linear model prediction falls outside this range so unlikely to be a good model <em><strong>R1A1</strong></em></p>
<p> </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn><mo>=</mo><mfrac><mn>5000</mn><mrow><mn>1</mn><mo>+</mo><mi>C</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn><mo>=</mo><mfrac><mn>5000</mn><mrow><mn>1</mn><mo>+</mo><mn>4</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>k</mi></mrow></msup></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mo>-</mo><mi>k</mi></mrow></msup><mo>=</mo><mfrac><mn>3800</mn><mrow><mn>4</mn><mo>×</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>7916</mn><mo>…</mo></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>2336</mn><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mn>8</mn></mfenced><mo>=</mo><mfrac><mn>5000</mn><mrow><mn>1</mn><mo>+</mo><mn>4</mn><msup><mi>e</mi><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>2336</mn><mo>×</mo><mn>8</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>3090</mn></math> <em><strong>M1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any answer that rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3000</mn></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This is much higher than the calculated upper bound for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>(</mo><mn>8</mn><mo>)</mo></math> so the rate of growth of the fish is unlikely to be sufficient to reach a carrying capacity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5000</mn></math>. <strong><em>M</em></strong><em><strong>1R</strong></em><em><strong>1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.</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>
<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>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> : <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}">
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> defined by</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {x + y,\,\,x - y} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {xy,\,\,x + y} \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</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="\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State with a reason whether or not <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> commute.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = f\left( {g\left( {\left( {x{\text{,}}\,\,y} \right)} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = f\left( {\left( {xy,\,\,x + y} \right)} \right)">
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</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=" = \left( {xy + x + y,\,\,xy - x - y} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mi>y</mi>
<mo>−</mo>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong><em>[3 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="\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = g\left( {f\left( {\left( {x{\text{,}}\,\,y} \right)} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = g\left( {\left( {x + y,\,\,x - y} \right)} \right)">
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\left( {x + y} \right)\left( {x - y} \right),\,\,x + y + x - y} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {{x^2} - {y^2}{\text{,}}\,\,2x} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></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>no because <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f \circ g \ne g \circ f">
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
<mo>≠</mo>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</math></span> <strong><em>R1</em></strong></p>
<p><strong>Note:</strong> Accept counter example.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {a{\text{,}}\,\,b} \right) \Rightarrow \left( {x + y,\,\,x - y} \right) = \left( {a{\text{,}}\,\,b} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</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="\left\{ {\begin{array}{*{20}{c}} {x = \frac{{a + b}}{2}} \\ {y = \frac{{a - b}}{2}} \end{array}} \right.">
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {\frac{{x + y}}{2},\,\,\frac{{x - y}}{2}} \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>A suitable site for the landing of a spacecraft on the planet Mars is identified at a point, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">A</mtext></math>. The shortest time from sunrise to sunset at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">A</mtext></math> must be found.</strong></p>
<p>Radians should be used throughout this question. All values given in the question should be treated as exact.</p>
<p>Mars completes a full orbit of the Sun in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>669</mn></math> Martian days, which is one Martian year.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>On day <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo> </mo></math>, the length of time, in hours, from the start of the Martian day until sunrise at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> can be modelled by a function, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, where</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi>b</mi><mi>t</mi></mrow></mfenced><mo>+</mo><mi>c</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is shown for one Martian year.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Mars completes a full rotation on its axis in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn></math> hours and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math> minutes.</p>
</div>
<div class="specification">
<p>The time of sunrise on Mars depends on the angle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math>, at which it tilts towards the Sun. During a Martian year, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math> varies from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>0</mn><mo>.</mo><mn>440</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>440</mn></math> radians.</p>
<p>The angle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>, through which Mars rotates on its axis from the start of a Martian day to the moment of sunrise, at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ω</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>839</mn><mo> </mo><mi>tan</mi><mo> </mo><mi>δ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>ω</mi><mo>≤</mo><mi>π</mi></math>.</p>
</div>
<div class="specification">
<p>Use your answers to parts (b) and (c) to find</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> be the length of time, in hours, from the start of the Martian day until <strong>sunset</strong> at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> on day <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> can be modelled by the function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn><mo>)</mo><mo>+</mo><mn>18</mn><mo>.</mo><mn>65</mn></math>.</p>
<p>The length of time between sunrise and sunset at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>, can be modelled by the function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn><mo>)</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>)</mo><mo>+</mo><mi>d</mi></math>.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn><mo>)</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>)</mo></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>d</mi></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Im</mtext><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>−</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> are complex functions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≈</mo><mn>0</mn><mo>.</mo><mn>00939</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle through which Mars rotates on its axis each hour.</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>Show that the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>98</mn></math>, correct to three significant figures.</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>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn></math>, correct to two significant figures.</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</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>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">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub></math> in exponential form, with a constant modulus.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find an equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo> </mo><mi>sin</mi><mo>(</mo><mi>q</mi><mi>t</mi><mo>+</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find, in hours, the shortest time from sunrise to sunset at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> that is predicted by this model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognition that period <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>669</mn></math> <em><strong>(M1)</strong></em></p>
<p><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>669</mn></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>00939190</mn><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for a correct expression leading to the given value or for a correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> to 4 sf or greater accuracy.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≈</mo><mn>0</mn><mo>.</mo><mn>00939</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>length of day<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>24</mn><mfrac><mn>2</mn><mn>3</mn></mfrac></math> hours <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"><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>666</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mover><mn>6</mn><mo>¯</mo></mover></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>667</mn></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mrow><mn>24</mn><mstyle displaystyle="true"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mfrac><mn>360</mn><mrow><mn>24</mn><mstyle displaystyle="true"><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow></mfrac></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>255</mn></math> radians <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>254723</mn><mo>…</mo><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>37</mn></mfrac><mo>,</mo><mo> </mo><mn>14</mn><mo>.</mo><mn>5945</mn><mo>…</mo><mo>°</mo></mrow></mfenced></math> <em><strong>A1</strong></em><br><br></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>substitution of either value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math> into equation <em><strong>(M1)</strong></em></p>
<p>correct use of arccos to find a value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Both <em><strong>(M1)</strong></em> lines may be seen in either part (c)(i) or part (c)(ii).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ω</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>839</mn><mo> </mo><mi>tan</mi><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>440</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>97684</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>1</mn><mo>.</mo><mn>98</mn></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> For substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>98</mn></math> award <em><strong>M0A0</strong></em>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>440</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>16</mn><mo> </mo><mo> </mo><mo>(</mo><mn>1</mn><mo>.</mo><mn>16474</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mtext>max</mtext></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>97684</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>25472</mn><mo>…</mo></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>7</mn><mo>.</mo><mn>76</mn><mo> </mo><mtext>hours</mtext><mo> </mo><mo> </mo><mo>(</mo><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>70</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>98</mn></math>.<br><br></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mtext>min</mtext></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>16474</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>25472</mn><mo>…</mo></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>57</mn><mo> </mo><mtext>hours</mtext><mo> </mo><mo> </mo><mo>(</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo><mo>)</mo><mo> </mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>55</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>56</mn></math> from use of rounded values.<br><br></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</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>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>-</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo></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>1</mn><mo>.</mo><mn>59408</mn><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for substituting their values into a correct expression. Award <em><strong>A1</strong></em> for a correct value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> from their expression which has at least 3 significant figures and rounds correctly to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>6</mn></math>.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>1</mn><mo>.</mo><mn>6</mn></math> (correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> sf) <em><strong>AG</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><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mrow><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>+</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo></mrow><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>12</mn><mo>.</mo><mn>333</mn><mo>…</mo></mrow><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>57258</mn><mo>…</mo><mo>+</mo><mn>1</mn><mo>.</mo><mn>59408</mn><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>76075</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>59408</mn><mo>…</mo></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>.</mo><mn>17</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>16666</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>16</mn></math> from use of rounded values. Follow through on their answers to part (d) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>6</mn></math>.<br><br><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>18</mn><mo>.</mo><mn>65</mn><mo>-</mo><mn>6</mn><mo>.</mo><mn>16666</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn><mo>.</mo><mn>5</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>12</mn><mo>.</mo><mn>4833</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Follow through for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn><mo>.</mo><mn>65</mn></math> minus their answer to part (f).<br><br><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at least one expression in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><msup><mtext>e</mtext><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced><mtext>i</mtext></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn></mrow></mfenced><mtext>i</mtext></mrow></msup><mi mathvariant="normal">,</mi><mo> </mo><mo> </mo><msub><mi mathvariant="normal">z</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi></mrow></mfenced><mtext>i</mtext></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>83</mn></mrow></mfenced><mtext>i</mtext></mrow></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mfenced><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi></mrow></mfenced><mtext>i</mtext></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mo>.</mo><mn>83</mn><mtext>i</mtext></mrow></msup><mo>-</mo><mn>1</mn><mo>.</mo><mn>6</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mfenced><mrow><mn>3</mn><mo>.</mo><mn>06249</mn><mo>…</mo><msup><mtext>e</mtext><mrow><mn>2</mn><mo>.</mo><mn>99086</mn><mo>…</mo><mtext>i</mtext></mrow></msup></mrow></mfenced></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>06249</mn><mo>.</mo><mo>.</mo><mo>.</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>150729</mn><mo>.</mo><mo>.</mo><mo>.</mo></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>99086</mn><mo>.</mo><mo>.</mo><mo>.</mo></math> <em><strong>(M1)</strong></em><em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> variables (or equivalent) must be seen.<br><br><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>06</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>99</mn><mo>)</mo><mo>+</mo><mn>12</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>06248</mn><mo>…</mo><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>99086</mn><mo>…</mo><mo>)</mo><mo>+</mo><mn>12</mn><mo>.</mo><mn>4833</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>Note:</strong> Accept equivalent forms, e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>06</mn><mo> </mo><mi>sin</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>00939</mn><mi>t</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>151</mn><mo>)</mo><mo>+</mo><mn>12</mn><mo>.</mo><mn>5</mn></math>.<br>Follow through on their answer to part (g) replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>.</mo><mn>5</mn></math>.</p>
<p> <em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>shortest time between sunrise and sunset</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>.</mo><mn>4833</mn><mo>…</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>06249</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>42</mn></math> hours <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>9</mn><mo>.</mo><mn>420843</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>.</mo><mn>44</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> sf values.<br><em><strong><br></strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">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>
<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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.iii.</div>
</div>
<br><hr><br><div class="specification">
<p><em>This question explores methods to determine the area bounded by an unknown curve.</em></p>
<p>The curve <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> is shown in the graph, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x \leqslant 4.4">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>4.4</mn>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The curve <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> passes through the following points.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">It is required to find the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4.4">
<mi>x</mi>
<mo>=</mo>
<mn>4.4</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>One possible model for the curve <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> is a cubic function.</p>
</div>
<div class="specification">
<p>A second possible model for the curve <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> is an exponential function, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p{{\text{e}}^{qx}}">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>q</mi>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p{\text{,}}\,\,q \in \mathbb{R}">
<mi>p</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</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>Use the trapezoidal rule to find an estimate for the area.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.</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>Use all the coordinates in the table to find the equation of the least squares cubic regression curve.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coefficient of determination.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for the area enclosed by the cubic function, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4.4">
<mi>x</mi>
<mo>=</mo>
<mn>4.4</mn>
</math></span>.</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 the value of this area.</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y = qx + {\text{ln}}\,p">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mi>q</mi>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>p</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence explain how a straight line graph could be drawn using the coordinates in the table.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By finding the equation of a suitable regression line, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 1.83">
<mi>p</mi>
<mo>=</mo>
<mn>1.83</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 0.986">
<mi>q</mi>
<mo>=</mo>
<mn>0.986</mn>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the area enclosed by the exponential function, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 4.4">
<mi>x</mi>
<mo>=</mo>
<mn>4.4</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.iv.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{1.1}}{2}\left( {2 + 2\left( {5 + 15 + 47} \right) + 148} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>1.1</mn>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>5</mn>
<mo>+</mo>
<mn>15</mn>
<mo>+</mo>
<mn>47</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>148</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Area = 156 units<sup>2</sup> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The graph is concave up, <em><strong>R1</strong></em></p>
<p>so the trapezoidal rule will give an overestimate. <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 3.88{x^3} - 12.8{x^2} + 14.1x + 1.54">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3.88</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12.8</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>14.1</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1.54</mn>
</math></span> <em><strong>M1A2</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{R^2} = 0.999">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0.999</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int\limits_0^{4.4} {\left( {3.88{x^3} - 12.8{x^2} + 14.1x + 1.54} \right)} dx">
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>4.4</mn>
</mrow>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3.88</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>12.8</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>14.1</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1.54</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>d</mi>
<mi>x</mi>
</math></span> <em><strong>A1A1</strong></em></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>Area = 145 units<sup>2</sup> (Condone 143–145 units<sup>2</sup>, using rounded values.) <em><strong>A2</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y = {\text{ln}}\left( {p{{\text{e}}^{qx}}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>p</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>q</mi>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</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{ln}}\,y = {\text{ln}}\,p + {\text{ln}}\left( {{{\text{e}}^{qx}}} \right)">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>p</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>q</mi>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y = qx + {\text{ln}}\,p">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mi>q</mi>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>p</mi>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Plot <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</math></span> against <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>. <em><strong>R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Regression line is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,y = 0.986x + 0.602">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mn>0.986</mn>
<mi>x</mi>
<mo>+</mo>
<mn>0.602</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p>So <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = ">
<mi>q</mi>
<mo>=</mo>
</math></span> gradient = 0.986 <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = {e^{0.602}} = 1.83">
<mi>p</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>0.602</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>1.83</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int\limits_0^{4.4} {1.83{e^{0.986x}}dx = 140} ">
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>4.4</mn>
</mrow>
</munderover>
<mrow>
<mn>1.83</mn>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>0.986</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mn>140</mn>
</mrow>
</math></span> units<sup>2</sup> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.iv.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<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">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.iv.</div>
</div>
<br><hr><br>