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<h2>HL Paper 3</h2><div class="specification">
<p><strong>This question explores models for the height of water in a cylindrical container as water drains out.</strong></p>
<p><br>The diagram shows a cylindrical water container of height <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>2</mn></math> metres and base radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> metre. At the base of the container is a small circular valve, which enables water to drain out.</p>
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"></p>
<p style="text-align: left;">Eva closes the valve and fills the container with water.</p>
<p style="text-align: left;">At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, Eva opens the valve. She records the height, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres, of water remaining in the container every <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> minutes.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Eva first tries to model the height using a linear function, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mi>t</mi><mo>+</mo><mi>b</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Eva uses the equation of the regression line of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, to predict the time it will take for all the water to drain out of the container.</p>
</div>
<div class="specification">
<p>Eva thinks she can improve her model by using a quadratic function, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>p</mi><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>t</mi><mo>+</mo><mi>r</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><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Eva uses this equation to predict the time it will take for all the water to drain out of the container and obtains an answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> minutes.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> be the volume, in cubic metres, of water in the container at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> minutes.<br>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> be the radius, in metres, of the circular valve.</p>
<p>Eva does some research and discovers a formula for the rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>π</mi><msup><mi>R</mi><mn>2</mn></msup><msqrt><mn>70</mn><mo> </mo><mn>560</mn><mi>h</mi></msqrt></math></p>
</div>
<div class="specification">
<p>Eva measures the radius of the valve to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>023</mn></math> metres. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> be the time, in minutes, it takes for all the water to drain out of the container.</p>
</div>
<div class="specification">
<p>Eva wants to use the container as a timer. She adjusts the initial height of water in the container so that all the water will drain out of the container in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> minutes.</p>
</div>
<div class="specification">
<p>Eva has another water container that is identical to the first one. She places one water container above the other one, so that all the water from the highest container will drain into the lowest container. Eva completely fills the highest container, but only fills the lowest container to a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> metre, as shown in the diagram.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> Eva opens both valves. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> be the height of water, in metres, in the lowest container at time <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>Find the equation of the regression line of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></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>Interpret the meaning of parameter <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> in the context of the model.</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>Suggest why Eva’s use of the linear regression equation in this way could be unreliable.</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 equation of the least squares quadratic regression curve.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down a suitable domain for Eva’s function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>p</mi><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>t</mi><mo>+</mo><mi>r</mi></math>.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><msup><mi>R</mi><mn>2</mn></msup><msqrt><mn>70</mn><mo> </mo><mn>560</mn><mi>h</mi></msqrt></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><msup><mi>R</mi><mn>2</mn></msup><msqrt><mn>70</mn><mo> </mo><mn>560</mn><mi>h</mi></msqrt></math>, show that the general solution is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>17</mn><mo> </mo><mn>640</mn><msup><mfenced><mrow><mi>c</mi><mo>-</mo><msup><mi>R</mi><mn>2</mn></msup><mi>t</mi></mrow></mfenced><mn>2</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the general solution from part (d) and the initial condition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>3</mn><mo>.</mo><mn>2</mn></math> to predict the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find this new height.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>H</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>≈</mo><mn>0</mn><mo>.</mo><mn>2514</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>009873</mn><mi>t</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1405</mn><msqrt><mi>H</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>T</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> minutes to estimate the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>Alessia is an ecologist working for Mediterranean fishing authorities. She is interested in whether the mackerel population density is likely to fall below <math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="bold">5000</mn></math> mackerel per <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext mathvariant="bold">km</mtext><mn mathvariant="bold">3</mn></msup></math>, as this is the minimum value required for sustainable fishing. She believes that the primary factor affecting the mackerel population is the interaction of mackerel with sharks, their main predator.</strong></p>
<p>The population densities of mackerel (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> thousands per <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>km</mtext><mn>3</mn></msup></math>) and sharks (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> per <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>km</mtext><mn>3</mn></msup></math>) in the Mediterranean Sea are modelled by the coupled differential equations:</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>M</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>α</mi><mi>M</mi><mo>-</mo><mi>β</mi><mi>M</mi><mi>S</mi></math></p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>S</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>γ</mi><mi>M</mi><mi>S</mi><mo>-</mo><mi>δ</mi><mi>S</mi></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is measured in years, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math> are parameters.</p>
<p>This model assumes that no other factors affect the mackerel or shark population densities.</p>
<p>The term <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mi>M</mi></math> models the population growth rate of the mackerel in the absence of sharks.<br>The term <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mi>M</mi><mi>S</mi></math> models the death rate of the mackerel due to being eaten by sharks.</p>
</div>
<div class="specification">
<p>Suggest similar interpretations for the following terms.</p>
</div>
<div class="specification">
<p>An equilibrium point is a set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo> </mo></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>M</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>S</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>.</p>
<p>Given that both species are present at the equilibrium point,</p>
</div>
<div class="specification">
<p>The equilibrium point found in part (b) gives the average values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi></math> over time.<br><br>Use the model to predict how the following events would affect the average value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. Justify your answers.</p>
</div>
<div class="specification">
<p>To estimate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math>, Alessia considers a situation where there are no sharks and the initial mackerel population density is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mn>0</mn></msub></math>.</p>
</div>
<div class="specification">
<p>Based on additional observations, it is believed that</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>549</mn></math>,</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>236</mn></math>,</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>244</mn></math>,</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>39</mn></math>.</p>
<p>Alessia decides to use Euler’s method to estimate future mackerel and shark population densities. The initial population densities are estimated to be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mn>0</mn></msub><mo>=</mo><mn>5</mn><mo>.</mo><mn>7</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn></math>. She uses a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> years.</p>
</div>
<div class="specification">
<p>Alessia will use her model to estimate whether the mackerel population density is likely to fall below the minimum value required for sustainable fishing, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5000</mn></math> per <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>km</mtext><mtext>3</mtext></msup></math>, during the first nine years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mi>M</mi><mi>S</mi></math></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mi>S</mi></math></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>show that, at the equilibrium point, the value of the mackerel population density is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>δ</mi><mi>γ</mi></mfrac></math>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of the shark population density at the equilibrium point.</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>Toxic sewage is added to the Mediterranean Sea. Alessia claims this reduces the shark population growth rate and hence the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> is halved. No other parameter changes.</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>Global warming increases the temperature of the Mediterranean Sea. Alessia claims that this promotes the mackerel population growth rate and hence the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> is doubled. No other parameter changes.</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>Write down the differential equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> that models this situation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the expression for the mackerel population density after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> years is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><msub><mi>M</mi><mn>0</mn></msub><msup><mtext>e</mtext><mrow><mi>α</mi><mi>t</mi></mrow></msup></math></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Alessia estimates that the mackerel population density increases by a factor of three every two years. Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>549</mn></math> to three significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>M</mi><mi>n</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>n</mi></msub></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method to find an estimate for the mackerel population density after one year.</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>Use Euler’s method to sketch the trajectory of the phase portrait, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>≤</mo><mi>M</mi><mo>≤</mo><mn>7</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo>≤</mo><mi>S</mi><mo>≤</mo><mn>3</mn></math>, over the first nine years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your phase portrait, or otherwise, determine whether the mackerel population density would be sufficient to support sustainable fishing during the first nine years.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State <strong>two</strong> reasons why Alessia’s conclusion, found in part (f)(ii), might not be valid.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.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 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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>
<br><hr><br><div class="specification">
<p>This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method.</p>
<p>It is desired to solve the coupled system of differential equations</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot x = x + 2y - 50">
<mrow>
<mover>
<mi>x</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mn>50</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot y = 2x + y - 40">
<mrow>
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mn>40</mn>
</math></span></p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> represent the population of two types of symbiotic coral and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is time measured in decades.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equilibrium point for this system.</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>If initially <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 100">
<mi>x</mi>
<mo>=</mo>
<mn>100</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 50">
<mi>y</mi>
<mo>=</mo>
<mn>50</mn>
</math></span> use Euler’s method with an time increment of 0.1 to find an approximation for the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 1">
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Extend this method to conjecture the limit of the ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{y}{x}">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \to \infty ">
<mi>t</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</math></span>.</p>
<p> </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>Show how using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X = x - 10{\text{,}}\,\,Y = y - 20">
<mi>X</mi>
<mo>=</mo>
<mi>x</mi>
<mo>−</mo>
<mn>10</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>Y</mi>
<mo>=</mo>
<mi>y</mi>
<mo>−</mo>
<mn>20</mn>
</math></span> transforms the system of differential equations into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<mover>
<mi>X</mi>
<mo>˙</mo>
</mover>
</mrow>
<mo>=</mo>
<mi>X</mi>
<mo>+</mo>
<mn>2</mn>
<mi>Y</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mover>
<mi>Y</mi>
<mo>˙</mo>
</mover>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>X</mi>
<mo>+</mo>
<mi>Y</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve this system of equations by the eigenvalue method and hence find the general solution for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> of the original system.</p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the particular solution to the original system, given the initial conditions of part (b).</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>Hence find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 1">
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, giving the answers to 4 significant figures.</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 part (f) to find limit of the ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{y}{x}">
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \to \infty ">
<mi>t</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.</p>
<div class="marks">[1]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>If instead the initial conditions were given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 20">
<mi>x</mi>
<mo>=</mo>
<mn>20</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 10">
<mi>y</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>, find the particular solution for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> of the original system, in this case.</p>
<div class="marks">[2]</div>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.</p>
<div class="marks">[2]</div>
<div class="question_part_label">k.</div>
</div>
<br><hr><br><div class="specification">
<p>This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.</p>
<p>It is desired to solve the coupled system of differential equations</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot x = 3x + y">
<mrow>
<mover>
<mi>x</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot y = - x + y.">
<mrow>
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>.</mo>
</math></span></p>
</div>
<div class="specification">
<p>The general solution to the coupled system of differential equations is hence given by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = A\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right){e^{2t}} + B\left( {\begin{array}{*{20}{c}} t \\ { - t + 1} \end{array}} \right){e^{2t}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>As <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \to \infty ">
<mi>t</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</math></span> the trajectory approaches an asymptote.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 1}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right){e^{2t}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> is a solution to the above system.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} t \\ { - t + 1} \end{array}} \right){e^{2t}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> is also a solution.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>If initially at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0{\text{,}}\,\,x = 20{\text{,}}\,\,y = 10">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>20</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mn>10</mn>
</math></span> find the particular solution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 2">
<mi>t</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</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 equation of this asymptote.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><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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TrCA8RH+Fw7bsCpshrEzZuOUfLrGYip6Y8gRHltUkwDQsMRqdyz1oDKt36P3JpY5GQ/htHm/amJkxxzFutuzXmcOt2XOWGo3YPsd9TIWj4Hw5RlZk0VeGvl66iJW4nspDD0UxZ3Fc+8O4Zz2PfOXnFHK9Z67ZJETLbzZ6GVfx6BkKH9YPjmM+ysaoZqXDAGmzfthSFTHsHjE8TeUzUPBZ9nYarF7/EbHIxxqmZUHT6JC8qyHkPuLIEJUWMwSLp/SYfj9SqoB6ja3nD/OzByUiDqj+twSVnUM1Wh7FSD8rMNxs3M3Gk9NQm3K8uMWsUwnUW92JEOMBfNYpiC7sIk1OG47rKyRCKNmypB/pr1+PK5hZgTZLnT6AECHkBazlxgz3LMyxJ3kCe349WjsdCsmAJ5l+o03+pRvuNPOIHB+ElZJsLkMUURSM37mxd/IE3c4U5JRU6cGJb0NGSVnMTJgrdwNCEPK6YOFNd3Jr8U8skBPRJm/szFuHqRJi1K8nPx8pdzsXSOuu312DBcaRD3e+MQFXaHskTRWIUdm8qkXSvKMqOM49bGpCCvTG8stDubY7ax9jLWcXIj1xwx6FCUrcHorCSE3277ThnQWP4xNp0Q34qffIbMMCnOwzAmdSPK9C3KNjdXR7nSpUzFm1ilISTkTpuKth6V+/9qLAQTn0JCSB/xDamHTrzfe2AA+srbGJmLQHtHFX0DMLC3+H/5WdT2sFNXclxaUpH1cIjFG9wbg/r39cwb3i35Y1DELxGnqsbOd/+M09JgbYMeXxw+juvKFkaMm9xp5ZQhduMChLc7E2BieYbUvtbK1xA149fI3FaGqsxELNTolB1MT9ELQfEvQpM9DTX5/4EZCeWIzbQ9u+Ig31p1+GJ3NVTRs/DwM5twsroUBcmDsW/tc8gq8uJY+qkRv3YLsqO/Q37STCQcjUJmuzMdzvPLRO7/tJGYGW5xdcMX2nFrJXKjpiEpcytOVL2AOR1+CFE621aClsXP4WHLKz6i1q8rsbs5CNFxj+GZTaWoLt2K5BGHsTYpB0W1YpHRyRyzH2tv0VGcXM+1jklXQv6Ig7HZeDa8v7LM0lV8/cXnaFZFIe7hxdh0Uvpg6gKM2LcaSVl7uuTDpR7OfeUyrXJPYtDuwrpt1WKlNxc5aQ84OFownj6uQrDXHlV0CWnHnPcVEtYnO9gxk8QvKA5ZBS8g8sgKzBw7BrGr8qHZc1hsgPdi4t2WhxU9mdJpzXj6R5+J8w9fgirdCRwsyMXiaGBP1oeo6mkHYvqv8JejvTH/td8hccRepM9IR/7pzn2GsXdYBCZKl9L8gjB12VIkqmqxZ1elF1/BaIH+5Kc46v84Xsv+DUbsWYQZ87cZD8A6zZsuK3aSfzgWV+lw4uD/x2uLY8QG9CY+qGr/OVFD7V7kHZ2B9U/9ooNLgwMQFjUeQ8Tw+A2ZhmXPPwZVczF2lX6nrHc1x7w71s7j5D7pSkjOwclY4eDsq6x3GKImDhG36YUhUxfg+cRgNO85gNILN79j9Mg7ZLzEqtyxJBUqr74hFnBBiMtZat65+N0eCLX4//VLjWI9rDCN41NNt39UcbURl6RTNOpAtDuD6rMaULn+VVTOXWI91mKgGmGB9Sg/d7FtfErrRZwrr0dgmBrSxZKeSWxgkxdg88lvodOdxt7l9wE1wNj5szFJ6rwYN9F3KN31MT5KfwDB8iWdYESka8TlGqRHBGNC7jH0V9+FQGhxrvaa8SGi1tqzKMedCFMPUJaYBCBk6mw8FBUC1J+F7lIPqvaayrA+eQEO3LcQzyak4KUPdyI7shKZz+ajUips3Mk32/G5XkcaI7YRyXM/xX1LFyIh+UV8uD9HPABbjWffqkAT/MWwdCK/7F1W9Kl27Id+IVMQ/9D/QYi9S4tSjmWewtwse+Px7HE0vk/RUY558yVcu3HqZK51qBUXSg/go48W4b5g4+VZdcQi7EM99qXf72Bqlf4InTxB+fnm80wZpIxdUeEiKv76d+NYAGkerlVpSN8zGMkFGrwR31YR+426HwkTVGj+shrfKQd7bZdwH8CdZ0+2G09g/JCHChMS7lcGsfq6FtRqNmLXxOXImhpk/cb6qzFxVjDqi4/hnCl+546huD7YPAi6xzPUomTtq9g2Ig2/Nx3pMW6i4YjffEwshqWCWLpVozQvXlwej7zSalQtDkefu8MxS1WN4mPnlcs813DuWAXqOzxD2guDg0dBNWESwgb1nOwzDv5G2+XEfmFIWr4QE06cgU4q9pzlm531xvG516Ea3N9qiIv3uIozn+zGCfOlM7GYGf0Ili+fhBPHz8vzm/l3Ir/sXlb0wXZsPGFiMybPoIPmlU8xcfUSTLX6EEWb9rEUHyaPWwvA4P7/YjdWHeWY117CdRCnzuRax/wxJH6DRZ8p3ko3IFosI6PzPoOuagnC/fvi7on3QlVfgWPnTIWl8QNbUAWif9+bX7R4qAwyjl0p2pKGYdt/gwip+h2biF0DU6Ap3YFs22LFbwQiEyYB2rM4b3tqXyzqGoeEyKek25g+5DEJCZEjPPUibyHpk8kb8HrDbCwxx05alofcEmn48UBMSVuGOO0ObDtUC4NYWBfl74A2bhnSpvTc83pG0sSYGuTOj0dS6b0oyLccl8a4uUQ+eIuF9v3tOKS/hibtXuS/X4O4nFRMsXN5x6AvxtqXv8QjS6J9+EDsOi40XDXvUCXGHUk9juzabRyEbdCj4s+fQjv2HqjlnHOWb4GYNPdRjK16A2sKjouFkJi7h7bjfW2skyEv3Zmy00Mpdv2pVD5olyaY/fPhGowNG2o86HI5vzq6rOhj7Vg+MF2PikfmIXaUMtZMWrZ6CxqeTG0rYKRl2RtQYp6lQhQwAXPnj0XVmlwUSMMHxG0ObXMnx7zhEu5lNFyxOYXmLE6d7MvMLjTgik1p4pgfAibNxvyx5VizZrs8ZMGgP4JtrjyXu5QpWMxuyTxUiht1pULByjRhVfF5ZU4bB9rNf+OAabLcHjGp8vfCmcJVQoz4/NLfYHWzmFdKnryx9E3jJNbyBI6Fwhm7cyzdWp6LmyD8UPGq8HMpTqHzhHWFdiYWlTFu1jqYw+pGnVC6fp4QKufeQ8JKZQJYmXneTOMtNOVVofBHTh7qrq6Nmyk2ba/Vev6tG8KVM/uEdSmTlfVSPr1rMymys3yzXC/eYlYpk1V3Lem5uo7Yhx1Y3/aapPwpKLVuj47yy8Th5L6eacc3J24XheKVkUpsxJttf3XllFC4Upq0V1lvvnUwcblVLN3MsS6YSNmS9LzusZxUWbpZzqHnYpxcyTWJ5aTK8s12Xj8LHcy9aZwn2NQfOHguF0m/pyO3Sf8odZ9Mur4snXbsUuIRbdmGDCSt/Yv8qVvZ2N8ge0EMIn8ZhRDxKFc6utt96nZMmxpiHjxpqNVg4Yy3MejNP7a/VGkiXQ7OWYYV3z0KzWtPujhu4ce7JXHzQYybexg39zBu7mHc3MO4uYdxc4+juHnm/KtfXwT+aw6Oi39UdWkRXn9tCaJrtiIz/QnMkD4VmfE2dtcGYZZFoSfxC4rHG8WZCNn/DorafeedpBX6A9tRFpqJ4k2Jt6zQIyIiIuquPHNmzwcxbu5h3NzDuLmHcXMP4+Yexs09jJt7HMWNp76IiIiIfBiLPSIiIiIfxmKPiIiIyIex2CMiIiLyYSz2iIiIiHwYiz0iIiIiH2Z36hUiIiIi8i4dTb3CefZuEiluH3ywXblHrnrssUeYb25gO3UP4+Yexs09jJt7uD91j6P9KS/jEhEREfkwFntEREREPozFHhEREZEPY7FHRERE5MNY7BERERH5MBZ7RERERD6MxR4RERGRD2OxR0REROTDWOwRERER+TAvKPZuoOlcKY5ov4fVV30Q0c1n0KPy4/eRmzoDqZp/KAtNLqEkI0qe3d76FoWMkkvKNtKvqMTHO9chdcwiaPStylLnpMdpclMwRvqdE9ah0vWHelZjCTLG2MZEvI3JREmjQdlIIcVXI8VG2iYGuZVNygqFs/UmnXlOX6TXIFV5zWMyStCoLPYJDtugi+2rSYuDprakHo3YDA20TTZ50XQamozZxryJzYRGaxlF19q6fS3QV2rEvz9CfsyE3Ep4S1P2ZR4u9qSk2IN8U8LJSfkeKvUtynqRoMOfsxYgq/hbmFNVagzbfocMOw3B6B9iEv8O2yr1bY8hIidaoS9ajfi0Zcjdd1FZ1sagPYC/Dn4FpdXfyl/JI92qD67GBNV0zAwPVLb6B4peSEbac+uwr1lZ5JTYD5SsRlzEv2PjpQfwiqYc1VVLEO6vrO7WrkH70VEMLhD/ZiUmOt3XOJj9IFQJ0xEe0NbFGvQHkRn3IOI3/jeiXikS4/gXLA7vp6x1vr6N68/ps4bEY3P1QWRPGI2EmT9DgLLY+zlug661rwZUvrUBn0x8Ecd1Z1FakAhsW4Q5y4pQa9ohGnTQLEvEiv9OwdFqcZvne2HjnFXQ1Br3va61dTsMtSjJnIuI+LdxKSoLmtKzqFocDq9oyj7Oc72CfFQhJsWTO9AY/5bcaVWXvoGIimzET/tPc9IJl07i09O/xOLZ9+B/yY87jvz5D2PlqZ8j+ZdD5W3aG445z0RBu/JhLNToWPARucRf3IdugK50A6KVJW0MuOo3GQsXPYAh5l7jGr45sh9aqwJjOOI3V6I0L16574wBTZUbkZxUhGHZO/HhSymYHT7Ei8aXtMDvvnlYNNnibzbU4MhOvXUR0lSG9ckLsX3YC9j/4YtInh1uEUeRs/VWXHxOH2f45jPs1EY6Lj68jqM2KHGhfTUew+GBC5E1NUjMj14YMnUBnk8MRvOeAyi9IJ1jE9tclQYb94Rg+dJYBPmJ20x5BI+H7MWK1/+GRpfbui2xyFyfhqTtQ5G9/z28lByH8CG9lHXkaR12J11KKtieS0T6tu8Ql/P/zJ2W35CfY/ov7gCa9+KdfefElLuBhqqjuLjgPzDzzv8tbiEdsbyANf7L8M6L8QjpZ/3nG/QleF2jFR8n/a4ZyPpDKmrTn8X6ygbjBkTkJj/0GzXKugC5GQVGUwXeWvk6auJWIjspDPbOY3VvARgVMsiqI21fhEj91u+RWxOLnOzHMNqm33K+3pYrz+nrXCk+eqiAKUhPHG2RH/0ROnmC8rPkKs58shsnAn+B8SP7GBf5DcX4aWJBuLsSX7e609alg7YCrMw9L+7TVyJpdE855PAeHmglxoItc18tVGIHnzlHbeePaEbVzs/wjeEy/v75T/D0v90DqdQzaDXIyr2IhMejxKMR45ZGLdCXbcT8aW9BFTbc/Pv8Qh7CksSLyM3SQMvTe+QqeZhABmLloQUxSE2NgXpOPnPIxo8vMAxoLP8Ym04AI37yGTLDjOOCxqRuRJnlUA6vYqcIaazCjk1l4osEyjKVcVBjUpBXpgwzcbbeqR5Y+MjFRw0iB36FNbGjjTGLXY0Sr82brtSKKw2XgQmTEDZIuqB6GbrjdYA6ELeb06UPgkaGAPVnobvUfoSd87Zej/Idf8IJDMZPyjIRJvedEUjN+xv07De7hVveMxhqS7BJ6tgQ3L5oM/w3qr80jlNQjQvG4Nv6IOTXyYgaIF3ANXZoVWKPGDLU4vjfNEZg7mrsa/4Ea/KPWgzW7YehIWIPWrUfR765piwjckDKp6yn8OQnw/Fi6VlUH/x3XNhXjQkJ92NUD9mPuuZmFBhX8fUXn6NZFYW4hxdj00lpfNECjNi3GklZe9rGF3kTO2dAWr+uxO7mIETHPYZnNpWiunQrkkccxtqkHBTVtjhd71RPvYRbNQCDB96LZ/acxIn9OYiu2Yjn/usrfhjAlpwfrVicFY8Qy6Y6qL9FseeIC229VYcvdldDFT0LDz+zCSerS1GQPBj71j6HrCIOpeoObvHuS0yafduxRxpYam+gZ1MdtFp5JUJC7kS/28RiLfinxsGdcsKWi6tGIXiwxTgAvyBMSZwL6SS1KvFdfP7SVIsOrxcGB48Sf9uXOHzc3mBXIkvXoC34TySVzsR7r83H5CG94Dc4GONUwZg2fqgnToN3XzezwOgdhqiJ0lAOe+OLvEvHZ0AGICxqvHxpzG/INCx7/jGomouxq/Q7F9d3rMdewk1ciuWJk8WYSZcdwxDWW1lNFlpQW1SAowmZeCq8v7KskzrR1nuHRWCiNE5P3C9PXbYUiapa7NlViQvKevKcW7v/MhVskpC7MNRqbIoBjZV/xU651nsMSxJCrP84QzMu68SVvfsjoK/lGtMZv2A7yeiHvgH90Vs6xXzuIo/4yLHGo8hfU4PE559AuJybSk7iXky8u69xG5J1XYFhO77Im7h6ttMPAaHhiFTutedsvSVXn9OH2Ck+DOeOobg+GLMmqvnJTzNpHN0mZFZOR5bVeNgBUIfdCZSfRa15p3gNtee0QOBdUA+0jqDbbT1gFCZH9pQDkO7PY72DfJnW8tkNWuxc9wGaEYS4nFRMcbXjMp/xc/KRcCKHTIWdRR41VeDtlz8AetKO1CU3q8Doi7sn3gtVfQWOnTMNs1DGF6kC0d/qoM4LdHAGxP/ucMxSVaP42Hnz5SzDlQZcELca3P9fnK53qBNnXXyFXHy0/BvmTjL19w2o2vURqsY+arGMDLW78cquMVidNcP6wxb22p3hPI4VV0M1Kxx3W9V6LrZ1fzUmzgpGffExnDMn8VU0XLgO1eD+4jOSp93a3tTvpwged4dyx5J0qvkNrKlq7vhDG34qDFCrgOsNaLzaNgJAbvjS4+wmowFXGxsgphvUA1Seq2zJC4i50lAvHmx8j4amFjRp9yPvzXfx95reuHfgFZztcQO/r+NCw1Vz8WHF5QLjMhquODqf7oeASbMxf2w51qzZjtNNBhj0R7Dt/ZrOHfB1Ex2eAQmYgLnzx6JqTS4KTjeKG9bi0LYd0MYtQ9qUgc7XO9B1Z1i7K6n4KAEej8EE6ey7PHnwUjy5aTCy1ycrZ+R9hYM2KOu4fUlzNq5+vRFPLplmLvSkZdm5h9AotbspqciJq8H7245Ab2iEtug9vK+NRU7aA9Zt2uW2HohJcx/F2CpxP15wHE3ShyYPbbf/O8kjbnHLGIgpacsQJ9ZszRWHlcmTxUTTrMa89L0YkbwVxW/E23zSVuE3ApEJk8QHfoPq70w7XuNRh3wJd3ogzlbZfnqtBd9VfyPuwCchIXIEiz1ywB+D7vsVkkfsRfp9U/DcLuBf4x+EfzOgP30NqkE9Yb6oVug1i6COWIR90ifiM2cgOFUDvbLWxHGB8Q9oUsMRka4Rf/4EmTMiLL4FoAmVuTFQqy1m/u83Gc/mb8Fv8QfMHKtGcMRruLHgbayNt/cp/e7M0RmQ/gh/9k3s+O0PWDNzLNTB0/DyjRQUrZ2j9HXO1tuJm+xmnWH1Jr1wZ2QyErTLECx94nPsk/gvzMZ7xXlI9onpPpy1QUftS2IQ618NViUvxOZty+U2ZfzCgmFi23oH6ocijIWXnxpzsl5BXO0KRASPxYyNwIKiFxEfZN3PddzWbXPSD/3CFyB/xzxgTQzGqu9CxMstdn8necZtgkj5WSYlhTRbdteRkvEg3lqTgdx9tfISVfQS5Cx8AnOcTKZq0OYjfsZmjCvQ4KWp0hHvNWjzUzAjU4voZa9htdUkkBLpK1/ikfRlKg5qkq0/iXSTSXH74IPtyj1y1WOPPdLF+eabur6d+ibGzT2Mm3sYN/dwf+oeR/tTDxwOSp+cGocHH4xFdPJW+etYTm5egngXZs33C4lH1uI7sG3dLmXOsz4ISX5PfHGl2JxuW+iJxWHtYby/8472HzknIiIi6iFucQlknPw4NWwS4jPewb783yAieDYy8jUoMX0Js/wl0Ifaf2mzTLrcsR55QZvxTNZBB5M1Kqey561F6/LV7n/knIiIiMjL3eJizx/9Amdi9fFvoasuh+b1tVgc/R22ZS5C0oyxUMdmIH93LYJmTWn3VWhmfmrEv/Ehfh9yGH8oshyrYOk8DuT/HaG//xCbkr3xK5iIiIiIbg4PjNnzTRxj4B6O2XMP26l7GDf3MG7uYdzcw/2pe7rZmD0iIiIiulVY7BERERH5MBZ7RERERD6MxR4RERGRD2OxR0REROTD7H4al4iIiIi8S0efxm1X7BERERGR7+BlXCIiIiIfxmKPiIiIyIex2CMiIiLyYSz2iIiIiHwYiz0iIiIinwX8D2zlde5Mr1CaAAAAAElFTkSuQmCC"></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>
<br><hr><br><div class="specification">
<p>Consider the system of paired differential equations</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot x = ax + by">
<mrow>
<mover>
<mi>x</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot y = cx + dy">
<mrow>
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
<mo>+</mo>
<mi>d</mi>
<mi>y</mi>
</math></span>.</p>
<p>This system is going to be solved by using the eigenvalue method.</p>
<p> </p>
</div>
<div class="specification">
<p>If the system has a pair of purely imaginary eigenvalues</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that if the system has two distinct real eigenvalues then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a - d} \right)^2} + 4bc > 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>c</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find two conditions that must be satisfied by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> must have opposite signs.</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>In the case when there is a pair of purely imaginary eigenvalues you are told that the solution will form an ellipse. You are also told that the initial conditions are such that the ellipse is large enough that it will cross both the positive and negative <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> axes and the positive and negative <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> axes.</p>
<p>By considering the differential equations at these four crossing point investigate if the trajectory is in a clockwise or anticlockwise direction round the ellipse. Give your decision in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>. Using part (b) (ii) show that your conclusions are consistent.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Illustrate graphically the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo><</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down a lower bound for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</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>Find an upper bound for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of brown squirrels, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> , in an area of woodland can be modelled by the following differential equation.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{x}{{1000}}\left( {2000 - x} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2000</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span></p>
</div>
<div class="specification">
<p>One year conservationists notice that some black squirrels are moving into the woodland. The two species of squirrel are in competition for the same food supplies. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> be the number of black squirrels in the woodland.</p>
<p>Conservationists wish to predict the likely future populations of the two species of squirrels. Research from other areas indicates that when the two populations come into contact the growth can be modelled by the following differential equations, in which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in tens of years.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = \frac{x}{{1000}}\left( {2000 - x - 2y} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2000</mn>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> ≥ 0</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = \frac{y}{{1000}}\left( {3000 - 3x - y} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>y</mi>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>3000</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> ≥ 0</p>
<p style="text-align: left;">An equilibrium point for the populations occurs when both <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>When the two populations are small the model can be reduced to the linear system</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 2x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = 3y">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>3</mn>
<mi>y</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>For larger populations, the conservationists decide to use Euler’s method to find the long‑term outcomes for the populations. They will use Euler’s method with a step length of 2 years (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0.2">
<mi>t</mi>
<mo>=</mo>
<mn>0.2</mn>
</math></span>).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equilibrium population of brown squirrels suggested by this model.</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>Explain why the population of squirrels is increasing for values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> less than this value.</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>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 800"> <mi>x</mi> <mo>=</mo> <mn>800</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 600"> <mi>y</mi> <mo>=</mo> <mn>600</mn> </math></span> is an equilibrium point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the other three equilibrium points.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using separation of variables, show that the general solution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 2x"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = A{{\text{e}}^{2t}}"> <mi>x</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the general solution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = 3y"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <mi>y</mi> </math></span>.</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>If both populations contain 10 squirrels at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </math></span> use the solutions to parts (c) (i) and (ii) to estimate the number of black and brown squirrels when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0.2"> <mi>t</mi> <mo>=</mo> <mn>0.2</mn> </math></span>. Give your answers to the nearest whole numbers.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the expressions for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x_{n + 1}}"> <mrow> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y_{n + 1}}"> <mrow> <msub> <mi>y</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></span> that the conservationists will use.</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>Given that the initial populations are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 100"> <mi>x</mi> <mo>=</mo> <mn>100</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 100"> <mi>y</mi> <mo>=</mo> <mn>100</mn> </math></span>, find the populations of each species of squirrel when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 1"> <mi>t</mi> <mo>=</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use further iterations of Euler’s method to find the long-term population for each species of squirrel from these initial values.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the same method to find the long-term populations of squirrels when the initial populations are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 400"> <mi>x</mi> <mo>=</mo> <mn>400</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 100"> <mi>y</mi> <mo>=</mo> <mn>100</mn> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method with step length 0.2 to sketch, on the same axes, the approximate trajectories for the populations with the following initial populations.</p>
<p>(i) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1000"> <mi>x</mi> <mo>=</mo> <mn>1000</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1500"> <mi>y</mi> <mo>=</mo> <mn>1500</mn> </math></span></p>
<p>(ii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1500"> <mi>x</mi> <mo>=</mo> <mn>1500</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1000"> <mi>y</mi> <mo>=</mo> <mn>1000</mn> </math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
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<p>Given that the equilibrium point at (800, 600) is a saddle point, sketch the phase portrait for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≥ 0 , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> ≥ 0 on the same axes used in part (e).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
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