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<h2>HL Paper 2</h2><div class="specification">
<p>An ice-skater is skating such that her position vector when viewed from above at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds can be modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>with respect to a rectangular coordinate system from a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, where the non-zero constants <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> can be determined. All distances are in metres.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the displacement of the ice-skater is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> and the velocity of the ice‑skater is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the velocity vector at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the magnitude of the velocity of the ice-skater at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the magnitude of the velocity of the ice-skater when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, the ice-skater is skating parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis for the first time.</p>
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OP</mtext></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>use of product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mover><mi>x</mi><mo>˙</mo></mover></mtd></mtr><mtr><mtd><mover><mi>y</mi><mo>˙</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi><mo>+</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced open="|" close="|"><mi mathvariant="bold-italic">v</mi></mfenced><mn>2</mn></msup><mo>=</mo><msup><mover><mi>x</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>+</mo><msup><mover><mi>y</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>=</mo><msup><mfenced open="[" close="]"><mrow><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced open="[" close="]"><mrow><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi><mo>+</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> It is more likely that an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">v</mi></mfenced></math> is seen.<br> <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mover><mi>x</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>+</mo><msup><mover><mi>y</mi><mo>˙</mo></mover><mn>2</mn></msup></msqrt></math> is not sufficient to award the <em><strong>M1</strong></em>, their part (a) must be substituted.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>+</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>t</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mn>2</mn><mi>b</mi><mi>t</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>use of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>t</mi><mo>=</mo><mn>1</mn></math> within a factorized expression that leads to the final answer <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mn>2</mn><mi>b</mi><mi>t</mi></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>magnitude of velocity is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>=</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>t</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math> result from part (b) is an alternative approach.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn><mo>×</mo><mn>2</mn></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup></mrow></mfenced></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>51</mn><mo> </mo><mo> </mo><mo>(</mo><mn>1</mn><mo>.</mo><mn>50504</mn><mo>…</mo><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mfenced><mrow><mi>b</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>t</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>t</mi><mo>=</mo><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>53</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>53086</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>correct substitution of their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> to find <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> <em><strong>(M1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>697</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>696591</mn><mo>…</mo></mrow></mfenced></math> <strong>and </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>488</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>487614</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>use of Pythagoras / distance formula <em><strong>(M1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OP</mtext><mo>=</mo><mn>0</mn><mo>.</mo><mn>850</mn><mo> </mo><mtext>m</mtext><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>850297</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[6 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of coupled differential equations.</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mi>x</mi></math></p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi></math></p>
</div>
<div class="specification">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues and corresponding eigenvectors of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math>.</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>Hence, write down the general solution of the system.</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>Determine, with justification, whether the equilibrium point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> is stable or unstable.</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>(i) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p>(ii) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a phase portrait for the general solution to the system of coupled differential equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>6</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>6</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>6</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>6</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mo>-</mo><mn>4</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <strong> <em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>4</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>4</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn><mi>x</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn><mi>y</mi></mtd></mtr></mtable></mfenced></math> <strong> <em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> This <em><strong>M1</strong></em> can be awarded for attempting to find either eigenvector.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>y</mi></math></p>
<p>possible eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple)<strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn><mi>x</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>y</mi></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn></math></p>
<p>possible eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> (or any real multiple)<strong> <em>A1</em></strong></p>
<p><em><strong><br>[6 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"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong> <em>(M1)A1</em></strong></p>
<p><strong><br>Note:</strong> Award <em><strong>M1A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math>, <em><strong>M1A0</strong></em> if LHS is missing or incorrect.</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>two (distinct) real negative eigenvalues <strong><em>R1</em></strong></p>
<p>(or equivalent (<em>eg</em> both <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mn>4</mn><mi>t</mi></mrow></msup><mo>→</mo><mn>0</mn><mo>,</mo><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>→</mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>→</mo><mo>∞</mo></math>))</p>
<p>⇒ stable equilibrium point <strong><em>A1</em></strong></p>
<p><strong><br>Note:</strong> Do not award <em><strong>R0A1</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi></mrow><mrow><mo>-</mo><mn>4</mn><mi>x</mi></mrow></mfrac></math> <strong><em>(M1)</em></strong></p>
<p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <strong><em>A1</em></strong></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><img src="data:image/png;base64,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"> <strong><em>A1</em></strong><strong><em>A1</em></strong><strong><em>A1</em></strong><strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for a phase plane, with correct axes (condone omission of labels) and at least three non-overlapping trajectories. Award <em><strong>A1</strong></em> for all trajectories leading to a stable node at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. Award <em><strong>A1</strong></em> for showing gradient is negative at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>4</mn></math>. Award <em><strong>A1</strong></em> for both eigenvectors on diagram.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A ball is attached to the end of a string and spun horizontally. Its position relative to a given point, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, is given by the equation</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> where all displacements are in metres.</p>
</div>
<div class="specification">
<p>The string breaks when the magnitude of the ball’s acceleration exceeds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the ball is moving in a circle with its centre at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and state the radius of the circle.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the velocity of the ball at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</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>Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.</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>Find an expression for the acceleration of the ball at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at the instant the string breaks.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>How many complete revolutions has the ball completed from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> to the instant at which the string breaks?</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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"><mfenced open="|" close="|"><mi mathvariant="bold-italic">r</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn></math> <strong>R1</strong></p>
<p> </p>
<p><strong>Note:</strong> use of the identity needs to be explicitly stated.</p>
<p> </p>
<p>Hence moves in a circle as displacement from a fixed point is constant. <strong>R1</strong></p>
<p>Radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">v</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> <strong>M1A</strong><strong>1</strong></p>
<p> </p>
<p><strong>Note:</strong> <strong>M1</strong> is for an attempt to differentiate each term</p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">v</mi><mo mathvariant="bold">∙</mo><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced><mo>∙</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> <strong>M1</strong></p>
<p> </p>
<p><strong>Note:</strong> <strong>M1</strong> is for an attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">v</mi><mo mathvariant="bold">∙</mo><mi mathvariant="bold-italic">r</mi></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>×</mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>×</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>A</strong><strong>1</strong></p>
<p>Hence velocity and position vector are perpendicular. <strong>AG</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></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"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> <strong>M1A1A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></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"><msup><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>06</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>400</mn></math> <strong>(M1)(A1)</strong></p>
<p> </p>
<p><strong>Note:</strong> <strong>M1</strong> is for an attempt to equate the magnitude of the acceleration to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>18</mn><mo>.</mo><mn>3</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>18</mn><mo>.</mo><mn>256</mn><mo>…</mo></mrow></mfenced><mo> </mo><mfenced><mtext>s</mtext></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Angle turned through is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>18</mn><mo>.</mo><msup><mn>256</mn><mn>2</mn></msup><mo>=</mo></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>33</mn><mo>.</mo><mn>329</mn><mo>…</mo></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>33</mn><mo>.</mo><mn>329</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>33</mn><mo>.</mo><mn>329</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><mo>=</mo><mn>5</mn><mo>.</mo><mn>30</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> complete revolutions <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.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">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A change in grazing habits has resulted in two species of herbivore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math>, competing for food on the same grasslands. At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> environmentalists begin to record the sizes of both populations. Let the size of the population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, and the size of the population <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>. The following model is proposed for predicting the change in the sizes of the two populations:</p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mi>y</mi></math></p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi>y</mi></math></p>
<p style="padding-left: 60px;">for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>For this system of coupled differential equations find</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> has a population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math>.</p>
</div>
<div class="specification">
<p>It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> has an initial population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2900</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the eigenvalues.</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>the eigenvectors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the general solution of the system of equations.</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>Sketch the phase portrait for this system, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
<p>On your sketch show</p>
<ul>
<li>the equation of the line defined by the eigenvector in the first quadrant</li>
<li>at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as <em>t</em> increases</li>
</ul>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a condition on the size of the initial population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> if it is to avoid its population reducing to zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> at this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>. Give your answer to the nearest <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> herbivores.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This 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"><mfenced open="|" close="|"><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>4</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong>(M1)(A1)</strong></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Attempt to solve either</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>or </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>1</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></p>
<p>or equivalent <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong>A1</strong><strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> accept equivalent forms</p>
<p> </p>
<p><strong>[3 marks]</strong></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"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <strong>A1</strong><strong>A1</strong><strong>A1</strong></p>
<p> </p>
<p><strong>Note: A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> correctly labelled, <strong>A1</strong> for at least two trajectories above <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> and <strong>A1</strong> for at least two trajectories below <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math>, including arrows.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2000</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></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"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext mathvariant="italic">e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext mathvariant="italic">e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo>,</mo><mo> </mo><mn>2900</mn><mo>=</mo><mo>-</mo><mn>2</mn><mi>A</mi><mo>+</mo><mi>B</mi></math> <strong>M1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for the substitution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2900</mn></math></p>
<p> </p>
<p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>-</mo><mn>300</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mn>2300</mn></math> <strong>A1</strong><strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mo>-</mo><mn>300</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2300</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>79</mn><mo> </mo><mfenced><mrow><mn>6</mn><mo>.</mo><mn>7896</mn><mo>…</mo></mrow></mfenced></math> (years) <strong>A1</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></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"><mi>y</mi><mo>=</mo><mn>600</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>5</mn><mo>×</mo><mn>6</mn><mo>.</mo><mn>79</mn></mrow></msup><mo>+</mo><mn>2300</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>2</mn><mo>×</mo><mn>6</mn><mo>.</mo><mn>79</mn></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26827</mn><mo>.</mo><mn>9</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26830</mn></math> (to the nearest <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> animals) <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> moves along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><msup><mtext> m s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>t</mi><mo>−</mo><mn>24</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is at instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the magnitude of the particle’s acceleration at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> seconds.</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>Find the greatest speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> in the interval <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>6</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The particle starts from the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>. Find an expression for the displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> in the interval <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>4</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>0</mn></math><strong> <em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>t</mi><mo>=</mo><mn>6</mn></math><strong> <em>A1</em></strong></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>use of power rule<strong> <em>(M1)</em></strong></p>
<p><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><mn>4</mn><mi>t</mi><mo>+</mo><mn>16</mn></math><strong> <em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>t</mi><mo>=</mo><mn>6</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mo>-</mo><mn>8</mn></math><strong> <em>(A1)</em></strong></p>
<p>magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using a sketch graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> <strong> <em>(M1)</em></strong></p>
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD ONE</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>∫</mo><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p>attempt at integration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>2</mn><msup><mi>t</mi><mn>3</mn></msup></mrow><mn>3</mn></mfrac><mo>+</mo><mn>8</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>24</mn><mi>t</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> (use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math>) <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><msup><mi>t</mi><mn>3</mn></msup></mrow><mn>3</mn></mfrac><mo>+</mo><mn>8</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>24</mn><mi>t</mi></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD TWO</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>t</mi></msubsup><mi>v</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p>attempt at integration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mrow><mn>2</mn><msup><mi>t</mi><mn>3</mn></msup></mrow><mn>3</mn></mfrac><mo>+</mo><mn>8</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>24</mn><mi>t</mi></mrow></mfenced><mn>0</mn><mi>t</mi></msubsup></math><strong> <em>A1</em></strong></p>
<p>attempt to substituted limits into their integral <strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><msup><mi>t</mi><mn>3</mn></msup></mrow><mn>3</mn></mfrac><mo>+</mo><mn>8</mn><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>24</mn><mi>t</mi></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[4 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"><msubsup><mo>∫</mo><mn>0</mn><mn>4</mn></msubsup><mfenced open="|" close="|"><mi>v</mi></mfenced><mo> </mo><mo>d</mo><mi>t</mi></math> <strong> <em>(M1)(A1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for using the absolute value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>, or separating into two integrals, <em><strong>A1</strong></em> for the correct expression.<br><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>32</mn><mo> </mo><mtext>m</mtext></math><strong> <em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, of the spring is measured, in centimetres, from the equilibrium position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>. The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> can be modelled by the following second order differential equation, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, measured in seconds, after the initial displacement.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mn>3</mn><mover><mi>x</mi><mo>˙</mo></mover><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The differential equation can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mover><mi>x</mi><mo>˙</mo></mover></mtd></mtr><mtr><mtd><mover><mi>y</mi><mo>˙</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">A</mi><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>2</mn></math> matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mover><mi>x</mi><mo>˙</mo></mover></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvectors of matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the shock absorber is displaced <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>cm</mtext></math> and its velocity is zero, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mover><mi>x</mi><mo>˙</mo></mover><mo>⇒</mo><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mover><mi>x</mi><mo>¨</mo></mover></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>+</mo><mn>3</mn><mfenced><mi>y</mi></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong><em>R1</em></strong></p>
<p><br><strong>Note:</strong> If no explicit reference is made to <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mover><mi>x</mi><mo>¨</mo></mover></math>, or equivalent, award <em><strong>A0R1</strong></em> if second line is seen. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> used instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, award <em><strong>A0R0</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi></math> <strong><em>AG</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mo>-</mo><mi>λ</mi></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mfenced><mrow><mi>λ</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mo>=</mo><mn>0</mn></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mo>;</mo><mo> </mo><mi>λ</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <strong><em>A1</em></strong></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"><mfenced><mtable><mtr><mtd><mn>2</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>5</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>5</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></mtd><mtd><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong></em> for a valid attempt to find either eigenvector. Accept equivalent forms of the eigenvectors. <br>Do not award <em><strong>FT</strong></em> for eigenvectors that do not satisfy both rows of the matrix.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <strong><em>M1A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo> </mo><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>8</mn><mo>,</mo><mo> </mo><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi><mo>=</mo><mn>0</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>A</mi><mo>-</mo><mn>2</mn><mi>B</mi><mo>=</mo><mn>8</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mn>0</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo> </mo><mo>;</mo><mo> </mo><mi>B</mi><mo>=</mo><mo>-</mo><mn>5</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>10</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup></math> <strong><em>A1</em></strong></p>
<p><strong><br>Note:</strong> Do not award the final <em><strong>A1</strong></em> if the answer is given in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math>.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this problem, although simple errors often complicated things. In part (a) an explicit statement of the relationship between the second derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and the first derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> was often issing. Then in part (b) there seemed to be confusion about the matrix, with the correct values often placed in the wrong row or column of the matrix. Despite these errors, candidates made good attempts at finding eigenvalues and eigenvectors. It is to be noted that an error in solving the quadratic equation to find the eigenvectors means that follow-through marks are unlikely to be awarded since the eigenvectors are not reasonable answers and will not be consistent with the eigenvalues. Candidates need to take real care at this point of a question in part (c)(i). A significant number of candidates did not write down the final answer correctly, leaving their final answer in vector form, rather than “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo></math> ….” as asked for in the question.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The cross-sectional view of a tunnel is shown on the axes below. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mtext>AB</mtext><mo>]</mo></math> represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>8</mn></math>, relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>, and point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>8</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Find the height of the tunnel when</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</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>Hence find the maximum height of the tunnel.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>6</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.</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>Write down the integral which can be used to find the cross-sectional area of the tunnel.</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 find the cross-sectional area of the tunnel.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>evidence of power rule (at least one correct term seen) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>.</mo><mn>6</mn><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><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"><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>.</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>33</mn><mo> </mo><mfenced><mrow><mn>5</mn><mo>.</mo><mn>33333</mn><mo>…</mo><mo>,</mo><mo> </mo><mfrac><mn>16</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>5</mn><mo>.</mo><mn>33333</mn><msup><mo>…</mo><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>×</mo><mn>5</mn><mo>.</mo><mn>33333</mn><msup><mo>…</mo><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for substituting their zero for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo> </mo><mfenced><mrow><mn>5</mn><mo>.</mo><mn>333</mn><mo>…</mo></mrow></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>59</mn><mo> </mo><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><mo> </mo><mfenced><mrow><mn>7</mn><mo>.</mo><mn>58519</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M0A0M0A0</strong></em> for an unsupported <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>59</mn></math>. <br>Award at most <em><strong>M0A0M1A0</strong></em> if only the last two lines in the solution are seen. <br>Award at most <em><strong>M1A0M1A1</strong></em> if their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>33</mn></math> is not seen.</p>
<p><strong><br></strong><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>One correct substitution seen <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>4</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[1 mark]</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"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>2</mn><mfenced><mrow><mfenced><mrow><mn>2</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mn>6</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo>.</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>2</mn></math> seen. Award <em><strong>M1</strong></em> for correct substitution into the trapezoidal rule (the zero can be omitted in working).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>29</mn><mo>.</mo><mn>6</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msubsup><mo>∫</mo><mn>2</mn><mn>8</mn></msubsup><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>d</mo><mi>x</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msubsup><mo>∫</mo><mn>2</mn><mn>8</mn></msubsup><mi>y</mi><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for a correct integral, <em><strong>A</strong><strong>1</strong></em> for correct limits in the correct location. Award at most <em><strong>A0A1</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>d</mtext><mi>x</mi></math> is omitted.</p>
<p><strong><br></strong><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"><mi>A</mi><mo>=</mo><mn>32</mn><mo>.</mo><mn>4</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>A2</strong></em></p>
<p><strong><br>Note:</strong> As per the marking instructions, <em><strong>FT</strong></em> from their integral in part (d)(i). Award at most <em><strong>A1FTA0</strong></em> if their area is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>48</mn></math>, this is outside the constraints of the question (a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn></math> rectangle).</p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">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>
<br><hr><br><div class="specification">
<p>A particle moves such that its displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres, from a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds is given by the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The equation for the motion of the particle is amended to</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the particle is stationary at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> to show that this equation can be written as</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></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>Find the eigenvalues for the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the long-term velocity of the particle.</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>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> to write the differential equation as a system of coupled, first order differential equations.</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>Use Euler’s method with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> to find the displacement of the particle when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the long-term velocity of the particle.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><strong><br>Note:</strong> Award <strong>M1</strong> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mtable><mtr><mtd><mo>-</mo><mi>λ</mi><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><em><strong>Note:</strong></em> Award <em><strong>M1</strong> </em>for an attempt to find eigenvalues. Any indication that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det</mtext><mfenced><mrow><mi mathvariant="bold-italic">M</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> has been used is sufficient for the <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>λ</mi><mfenced><mrow><mo>-</mo><mn>5</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>λ</mi><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(on a phase portrait the particle approaches <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> increases so long term velocity (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>) is)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>Note:</strong></em> Only award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> if both eigenvalues in part (a)(ii) are negative. If at least one is positive accept an answer of ‘<em>no limit</em>’ or ‘<em>infinity</em>’, or in the case of one positive and one negative also accept ‘<em>no limit or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="italic">0</mn></math> (depending on initial conditions)</em>’.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>5</mn><mi>y</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> in any recurrence formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><msub><mi>y</mi><mi>n</mi></msub></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mfenced><mrow><mn>3</mn><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mn>4</mn><mo>-</mo><mn>5</mn><msub><mi>y</mi><mi>n</mi></msub><mo>-</mo><mn>6</mn><msub><mi>x</mi><mi>n</mi></msub></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>(when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>,) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>64402</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>644</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the velocity</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>It was clear that second order differential equations had not been covered by many schools. Fortunately, many were able to successfully answer part (ii) as this was independent of the other two parts. For part (iii) it was expected that candidates would know that two negative eigenvalues mean the system tends to the origin and so the long-term velocity is 0. Some candidates tried to solve the system. It should be noted that when the command term is ‘state’ then no further working out is expected to be seen.</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;">
<p>Forming a coupled system from a second order differential equation and solving it using Euler’s method is a technique included in the course guide. Candidates who had learned this technique were successful in this question.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The 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 10">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</math></span>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></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> passes through the following points.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfUAAAAxCAYAAADDRjAHAAANmElEQVR4Ae2dwUsXzxvHxx9dxUxPYR7SSyej1KDyUJBKHTPM8CAYhBYdorLM6FCGRnTokBoGHQKTOkWGevCQEWQWdYguGR7Ck2XWH+CP9/h9bD67s+5+9PPZmd15Bj7szuzsZ+d57TPz7Mw8O1uwsrKyIjgwASbABJgAE2ACiSfwv8RLwAIwASbABJgAE2ACkgAbdVYEJsAEmAATYAIpIbBFJ0dBQYEumdOYABNgAkyACTABgwTCZsy1Rh3lDTvRoEyxXxoPOcwjEzszyeSBGDPxM/GmMCMvEX+cGTETP4HVFOhGWODh9zBCfJwJMAEmwASYQEIIsFFPyI3iYjIBJsAEmAATCCPARj2MEB9nAkyACTABJpAQAmzUE3KjuJhMgAkwASbABMIIsFEPI2TJ8VevXonKykrpjIXtnTt3LCmZmWJ8//5dMqipqZFM4EBy8uRJgXQOmQQaGxslo8xUjhGBN2/eiM7OTrFt2zZx7do1SnZ6i/aF2hvULdQztEGuhM+fPwvUG+iGLvz8+VPqDNhQ24NzrAhYUc4bVp3fvanuxk3zGBsbw6p/KyMjI/ImDA4Oyji2poJpJsXFxSv4TU9PSwTYokwNDQ2mkMjrG7t4wIVJV0zfLyqeLeWg8vT390s9wnZubo6SjW5NM2publ6pqKhYq1uLi4srHR0dUr+pvsUNKC4mqqy4ZpC81dXVK/ghP/QG+2CGeD5DFA54VcsXopzoOynFCaZ5QGFQ0dQABcLPVDDNBNenhxxiAINuslwmr00M1O2nT5+kweru7jbKRS2TTYzIoIOTTcE0I1wfbLwhKN2bLx/xuJignYUhp06CzqhTJ0vVG7RFKKO3Tco1iygcrBx+p+FCbBEwpErDHK4NO0P22dlZsWfPnoyRnebmZjE3N+fscDPWDcBwOwc9AQwPnj59Wjx58kQUFRXpMzmcinrV1dUlrly5IqqqqhwmkZ3oadelp0+fioMHD64L5eXLl/K4qjdHjhyRaa9fv1733DgOWmnUx8fHRUdHh5iYmJAMdu7cKeMVFRXi8uXLcXCx5hoLCwuyLOXl5RllospFxzMOOhr59u2baGhocFT6TLHv3bsn6uvrxdGjRzMPcEwSuHv3rty2t7czEQ+B7u5u0dfXtzafTPPH1dXVoqmpyZPbvSgeCGGL1FBSUiKjOGY6WGnUAeXixYuSDZ6c4KwwOTkp0Gi7GsrKylwVPZLcQ0ND4tevX6K/vz9S/jRngkMT6suFCxekmFNTU2kWd0OyoV2BkXr06NGaQxgc5VwbCdTB6+3tlZ2ouro6OUJaWloqwGt0dFSQ8dKd51IanAhtDdYadfTO0evq6ekRbW1tTht0W5XHlnLB6/Tq1aviwYMHzg+loqfQ2toqhoeHtQ0wprTQ83I9LC0tyekrjIC9e/dOLgONoXgMybvk5a3TAxjwgYEBMT09Lblgmg9TXXgIsqEnqiszp/0jYK1RRxFhzKFQ6HW4Hr58+eI6Aq38aGQOHTokDTrPsQvx/PlzAYO1e/fuNT8UmsaCXwr2ube1qkow4tAZ4oGpveLiYkFzplqFS3kiHvhaWlpkT53mltHBunnzptQrmrZIOYZQ8WZmZkLzmMpgrVHH0yKcDjB3gYbK1VBYWChFX15e1iLYtWuXNt2FRDRAcBikxtkFmcNkhGGCE6H6Iz8DSgv7D5eP19bWOt0b/fr1q7z95LNDukAPPtxTF2Lr1q3yAYfYqNu9e/eqUSP7Vhp1MugYToWzD4bEXA3wsETv4dmzZxkIEMdwGFW2jIOORM6ePSsXxVCdJ7F4SNCCEY5gYTEjEEBnwVuncBp6YDY0zBFEyEuW7du3y/+dn5/P+H+asoFBcz1gZBBBnaah/QMHDpjHo3uPLsq7cLrzcpFG7/vRQhD0vmC+3/9br+wmeaBceGdUfQeSGOneoVxPjlweM80EDLyLPWDhB6SZ4mKaSdD9pff3870wRtD11XRbGNG7xrSAE9hggRUsaERtj1ruOPdNMyJ9oTZXZaO+m51mJqQfuvf1wQN6QovPII79OBa+iqIbGKbzhSgn+k7KQQIqE66NHyoYBTTUSNMBpjz53JriocoE2aFIKAt4QOlMBtNMSCdQDu+Pjfo/zaCVwEhv/h0xs2dab1SpYdBVPcLCI6aMllou04xgpNDe2MQmLiZYqAkGGtejH+JIVwP0hB5+kA/1DNzyHaJwKEAhvOMFcKjRJHuzORNnHv5bzUyYiZ9AeArrDTMKJ+DPwXqzyiQKByvn1P23lFOYABNgAkyACTCBMAJs1MMI8XEmwASYABNgAgkhEDj8npDyczGZABNgAkyACThDIGxqfEsQibATg85LY3qUeYw0yr2eTMzET4eZ+Jl4U5iRl4g/zoyYiZ/Aagp0Iyzw8HsYIT7OBJgAE2ACTCAhBNioJ+RGcTGZABNgAkyACYQRYKMeRoiPMwEmwASYABNICAE26gm5UVzMzRHAMo78Wc3NMeSzmYCOANctHRVzaWzUzbGPfGV8WrSzszPju89Y45zWY478RwnPCKMMR5Gwtd2Rx/s7duxYwqVfv/hR2bikS6gf+I5ETU3Nmj5gn9bpDiJKjPB9degRtqh/aa5vUfXHW68QT3vdgj7ga34ku/X6oFvWLspSdLrz0ppmkgeWI8T1sVQhrUmNJRyRZmrZ3P9WIYz1dhMHyB22DKwpNriuiRCVDeUzqUtxMsKyr6q+oP7QEqBUl7z3ixjhXFr2E8vJ4n/Upau95+UyHicjlJtkVlkFyYM8JtqduJmQ/MQGy8SSPqD9wZLdcaz1TuWgbRQO3FMPely3JP3v37+yJKOjowLfNUagr5JNTU1ZUsr8FgM9pOPHj4v+/v78XiiB/54NG9d06ffv377vgp84cULe5YWFBe3dJka3b99e+wLimTNnZN40fnY0G/3RAkt54sTEhJSwt7d3TR/wnXl87pmO2YbAOqOOoVUa5lArUWVlpUzHMJhLAQqENQPIoLskO8l6/fp1+d30/fv3UxJv/yOQDRvXdGl8fFzg881q+Pjxo8BnV8FCF3SMMPyK0NbWpjsl0WnZ6E+iBc1D4fFJbBuDdUYdlWpkZESyUp+mHz9+LNNOnTplI8dYy0QPO4cPH471uiYuhvnP9+/fCzwpc8gkkAs2rugSOguYF0XvfXJyMhNkQAxsMCePUaLBwUF5fkDWRCbnQn8SKXgWhW5qahIw3qpPBXSpr6/P98CYxd/mNat1Rh3S1tbWSqF//PixJvzbt29FQ0ND4BP2WkYHdrq7u0V1dbVob29PtbRoVM+fPy+Gh4elnNCBqAFTE+QghVGetHm+b4aNytAFXcLIX11dnTTmGPEqKipSEWj3oS/o0be0tMhGfceOHdp8SU3cjP6kvW6p9xT6Mjs7Kx/uSktL5WgxdKmjo8PahzwrjboKlfYfPnwoenp6KOrsFj0H9DRg6EpKSlLNobm5WVy6dElUVVX55ESjG+TFjAqHyojh18XFRVFfXy+6urpkxfT9UUITNspGFdcVXcL0FX4YiofM+/btC/Vkh98Kzpmbm5PGHR7eQ0NDKr5E729Uf1yoW+qNhc8BWGGUB20JdGJsbEwMDAwIvIFkZSCvOnUbxcNOzZ+PfdXLcmRkJDbPU50sNvBAucbGxqTXJTwyTYd8M4GHKa6x3i/MC15lFIe3ar6ZkDy5YGNKl+JiRKy8W7QlKAM82rMJFRUVsu5lc85G8+abUS70R5UtTXVLlQv78HrH/SDPdzpO6UFvUVC+XG+j6Ia1PXUMfc3Pz8sHIfTQvQ4vVj4h5bFQcNZpbW0VL1680PZc83hpI39NDkvUy8KWvN+np6flE3OQs5OuwJjSmZmZ0R1KXNpm2bimS+oNLisrk9Hl5WU1OXQfUzhLS0uh+ZKQYbP645UxTXXLK9uHDx9kkndUlKZwVL8v77mm4tYadVQizPtgmPXWrVum+FhxXXrtBA82qiHDnDGHTALQFx0XGHQMw7seXNKlxsZGny/Fnz9/pAqUl5drVQFOUFhcBJzUgDg6Gi4HF+sWvXUEW6QGeigsLCxUk63Yt9aoAybeA4RTBuYzXA547QSOcSoH9LbgwOFioIYZstMrkGhwKIALzX+iMYbnKnpZ586doyyp3YaxcU2X4KUMHUFAw3zjxo2MuqTTH+gKOJFhhy5Bp+7fv59avSHBwvTHtbpFb1uhDSHDDp3BnDr8C3T+PsTS2FY35h9l3F53Xi7TaNW0XP7nRv/LJA/M2eD6Qb+NyrTZ8+JmQnOhuC7m8MivgOYHaZUrpGPlL8yBEjOsDpbN/PtG2cTNhMoZlY0NuhQnI9xz6AL0hfQGcXV+1Ks/OIb5dlp5DudhHz4IcYU4GUGmqPrjYt0CH+gIrU6Ie4O2hdqbuHSCrhNFNwqQ2ftEgVdANMnebHmNY+gMiz2ovdO8XnCdP7eBxzrFM3KImfixMxM/E28KM/IS8ceZETPxE1hNiaIbVg6/Y6gDw+82GPQguJzOBJgAE2ACTMA2AtYYdTjGYa4CPXTMXbju7W6bonB5mAATYAJMwH4CgcPv9hedS8gEmAATYAJMwC0CYVPjW3Q4wk7SncNpTIAJMAEmwASYgFkC1gy/m8XAV2cCTIAJMAEmkHwCbNSTfw9ZAibABJgAE2ACksD/AWXNp4vvSNSCAAAAAElFTkSuQmCC"></p>
<p>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><em>-</em>axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span><em>-</em>axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 10">
<mi>x</mi>
<mo>=</mo>
<mn>10</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="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.</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 regression 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 = 10">
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>.</p>
<div class="marks">[1]</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>
<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{2}{2}\left( {2 + 2\left( {4.5 + 4.2 + 3.3 + 4.5} \right) + 8} \right)">
<mfrac>
<mn>2</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>4.5</mn>
<mo>+</mo>
<mn>4.2</mn>
<mo>+</mo>
<mn>3.3</mn>
<mo>+</mo>
<mn>4.5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>8</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Area = 43 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0.0389{x^3} - 0.534{x^2} + 2.06x + 2.06">
<mi>y</mi>
<mo>=</mo>
<mn>0.0389</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>0.534</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2.06</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2.06</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} = 0.991">
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0.991</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^{10} {y\,dx} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>10</mn>
</mrow>
</munderover>
<mrow>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>42.5 <em><strong>A2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.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>
<br><hr><br><div class="specification">
<p>At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.</p>
<p>The path of the ball is modelled by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><msub><mi>u</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mi>y</mi></msub><mo>-</mo><mn>5</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is the horizontal displacement from the archer and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the vertical displacement from the ground, both measured in metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, in seconds, since the ball was launched.</p>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub></math> is the horizontal component of the initial velocity</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub></math> is the vertical component of the initial velocity.</li>
</ul>
<p>In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="specification">
<p>An archer releases an arrow from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>. The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and an angle of elevation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>°</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the initial speed of the ball.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle of elevation of the ball as it is launched.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum height reached by the ball.</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>Assuming that the ground is horizontal and the ball is not hit by the arrow, find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> coordinate of the point where the ball lands.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the path of the ball, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the two positions where the path of the arrow intersects the path of the ball.</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>Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mn>10</mn><mn>2</mn></msup><mo>+</mo><msup><mn>8</mn><mn>2</mn></msup></msqrt></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>8</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>12</mn><mo>.</mo><mn>8062</mn><mo>…</mo><mo>,</mo><mo> </mo><msqrt><mn>164</mn></msqrt></mrow></mfenced><mo> </mo><mfenced><mrow><mtext>m</mtext><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>10</mn><mn>8</mn></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>896</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>3</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>896055</mn><mo>…</mo></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>3401</mn><mo>…</mo><mo>°</mo></math>) <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>897</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>4</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arcsin</mtext><mfenced><mfrac><mn>10</mn><mrow><mn>12</mn><mo>.</mo><mn>8</mn></mrow></mfrac></mfenced></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>t</mi><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mi>t</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>might be implied by a correct graph or use of the correct equation.</p>
<p> </p>
<p><strong>METHOD 1 – graphical Method</strong></p>
<p>sketch graph <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>might be implied by correct graph or correct maximum (eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>).</p>
<p><br>max occurs when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em><br><br></p>
<p><strong>METHOD 2 – calculus</strong><br><br>differentiating and equating to zero <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>10</mn><mo>-</mo><mn>10</mn><mi>t</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mo>=</mo><mn>1</mn><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3 – symmetry</strong></p>
<p>line of symmetry is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mo>=</mo><mn>1</mn><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><mo>+</mo><mn>8</mn><mo>×</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mo> </mo><mn>21</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em><br><br></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong> </em>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> is also seen.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></mfenced><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mo>×</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>21</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>13</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>5</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mn>5</mn><mrow><mfenced><mrow><mn>13</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>13</mn><mo>-</mo><mn>21</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>21</mn></mrow></mfenced></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>25</mn><mi>a</mi><mo>+</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi></math><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>169</mn><mi>a</mi><mo>+</mo><mn>13</mn><mi>b</mi><mo>+</mo><mi>c</mi></math><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>441</mn><mi>a</mi><mo>+</mo><mn>21</mn><mi>b</mi><mo>+</mo><mi>c</mi></math> <em><strong>M1A1</strong></em></p>
<p>solving simultaneously, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mo>,</mo><mo> </mo><mi>c</mi><mo>=</mo><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math>)</p>
<p> </p>
<p><strong>METHOD 4</strong><br><br>use quadratic regression on <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>13</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>21</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Question asks for expression; condone omission of "<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo></math>".</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>trajectory of arrow is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>10</mn><mo>+</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p>intersecting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>10</mn><mo>+</mo><mn>2</mn></math> and their answer to (d) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>8</mn><mo>.</mo><mn>66</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>53</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mfenced><mrow><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>52647</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>15</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>66</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mfenced><mrow><mn>15</mn><mo>.</mo><mn>0859</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>66006</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mtext>target</mtext></msub><mo>=</mo><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>,</mo><mo> </mo><mo> </mo><msub><mi>t</mi><mtext>target</mtext></msub><mo>=</mo><mfrac><mrow><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>457132</mn><mo>…</mo><mo> </mo><mtext>s</mtext></math> <em><strong>(A1)</strong></em></p>
<p>attempt to find the distance from point of release to intersection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>8</mn><mo>.</mo><mn>65705</mn><msup><mo>…</mo><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>3</mn><mo>.</mo><mn>52647</mn><mo>…</mo><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>8</mn><mo>.</mo><mn>79060</mn><mo>…</mo><mo> </mo><mtext>m</mtext></mrow></mfenced></math></p>
<p>time for arrow to get there is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>8</mn><mo>.</mo><mn>79060</mn><mo>…</mo></mrow><mn>60</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>146510</mn><mo>…</mo><mtext>s</mtext></math> <em><strong>(A1)</strong></em></p>
<p>so the arrow should be released when</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>311</mn><mo> </mo><mfenced><mtext>s</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>310622</mn><mo>…</mo><mo> </mo><mfenced><mtext>s</mtext></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question was found to be the most difficult on the paper. There were a good number of good solutions to parts (a) and part (b), frequently with answers just written down with no working. Part (c) caused some difficulties with confusing variables. The most significant difficulties started with part (d) and became greater to the end of the question. Few candidates were able to work through the final two parts.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>A biologist introduces <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> rabbits to an island and records the size of their population <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>)</mo></math> over a period of time. The population growth of the rabbits can be approximately modelled by the following differential equation, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is time measured in years.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></math></p>
</div>
<div class="specification">
<p>A population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> foxes is introduced to the island when the population of rabbits has reached <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math>. The subsequent population growth of rabbits and foxes, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the population of foxes at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, can be approximately modelled by the coupled equations:</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mfenced><mrow><mn>2</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>01</mn><mi>y</mi></mrow></mfenced></math></p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0002</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfenced></math></p>
</div>
<div class="specification">
<p>Use Euler’s method with a step size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>25</mn></math>, to find</p>
</div>
<div class="specification">
<p>The graph of the population sizes, according to this model, for the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> years after the foxes were introduced is shown below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>Describe the changes in the populations of rabbits and foxes for these <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> years at</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the population of rabbits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> year after they were introduced.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) the population of rabbits 1 year after the foxes were introduced.</p>
<p>(ii) the population of foxes 1 year after the foxes were introduced.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the non-zero equilibrium point for the populations of rabbits and foxes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mo>∫</mo><mn>2</mn><mo>d</mo><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>100</mn><mo>⇒</mo><mi>A</mi><mo>=</mo><mn>100</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>100</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>739</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>738</mn></math> for the final <em><strong>A1</strong></em>.</p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>25</mn></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> This may be inferred from a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> column, where this is seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>25</mn><msub><mi>x</mi><mi>n</mi></msub><mo> </mo><mfenced><mrow><mn>2</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>01</mn><msub><mi>y</mi><mi>n</mi></msub></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>25</mn><msub><mi>y</mi><mi>n</mi></msub><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0002</mn><msub><mi>x</mi><mi>n</mi></msub><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p style="padding-left:120px;"><img src="data:image/png;base64,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"> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for whole line correct when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>75</mn></math>. The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> column may be omitted and implied by the correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> values. The formulas are implied by the correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> columns.</p>
<p><br>(i) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2840</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2836</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2837</mn></math>) <em><strong>A1</strong></em></p>
<p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>58</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>59</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>both populations are increasing <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>rabbits are decreasing and foxes are increasing <em><strong>A1A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>setting at least one <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>DE</mtext></math> to zero <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4000</mn><mo>,</mo><mo> </mo><mo> </mo><mi>y</mi><mo>=</mo><mn>200</mn></math> <em><strong>A1A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Jorge is carefully observing the rise in sales of a new app he has created.</p>
<p>The number of sales in the first four months is shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Jorge believes that the increase is exponential and proposes to model the number of sales <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> in month <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> with the equation</p>
<p style="text-align: left; padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math></p>
</div>
<div class="specification">
<p>Jorge plans to adapt Euler’s method to find an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p>With a step length of one month the solution to the differential equation can be approximated using Euler’s method where</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math></p>
</div>
<div class="specification">
<p>Jorge decides to take the mean of these values as the approximation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> for his model. He also decides the graph of the model should pass through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>52</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The sum of the square residuals for these points for the least squares regression model is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>555</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Jorge’s model satisfies the differential equation</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>N</mi></math></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find three approximations for the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation for Jorge’s model.</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 sum of the square residuals for Jorge’s model using the values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment how well Jorge’s model fits the data.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give two possible sources of error in the construction of his model.</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 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"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup></math> <strong>(M1)A1</strong></p>
<p> </p>
<p><strong>Note: M1</strong> is for an attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>r</mi><mi>N</mi></math> <strong>AG</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept solution of the differential equation by separating variables</p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>⇒</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>≈</mo><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>≈</mo><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></math> <strong>M1A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math> <strong>AG</strong></p>
<p> </p>
<p><strong>Note:</strong> Do not penalize the use of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> sign.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Correct method <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>52</mn><mo>-</mo><mn>40</mn></mrow><mn>40</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>70</mn><mo>-</mo><mn>52</mn></mrow><mn>52</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>346</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>98</mn><mo>-</mo><mn>70</mn></mrow><mn>70</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <strong>A2</strong></p>
<p> </p>
<p><strong>Note: A1</strong> for a single error <strong>A0</strong> for two or more errors.</p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>349</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>34871</mn><mo>…</mo></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>68</mn><mn>195</mn></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>34871</mn><mo>…</mo><mo>×</mo><mn>2</mn></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>8887</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>9</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>349</mn><mi>t</mi></mrow></msup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></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"><msup><mfenced><mrow><mn>36</mn><mo>.</mo><mn>6904</mn><mo>…</mo><mo>-</mo><mn>40</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>+</mo><msup><mfenced><mrow><mn>73</mn><mo>.</mo><mn>6951</mn><mo>…</mo><mo>-</mo><mn>70</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>104</mn><mo>.</mo><mn>4435</mn><mo>…</mo><mo>-</mo><mn>98</mn></mrow></mfenced><mn>2</mn></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>66</mn><mo>.</mo><mn>1</mn><mo> </mo><mfenced><mrow><mn>66</mn><mo>.</mo><mn>126</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The sum of the square residuals is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> times as large as the minimum possible, so Jorge’s model is unlikely to fit the data exactly <strong>R1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>For example</p>
<p>Selecting a single point for the curve to pass through</p>
<p>Approximating the gradient of the curve by the gradient of a chord <strong>R1R1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></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;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.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>A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math></p>
<p>This equation links a variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with the temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The Arrhenius equation predicts that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> is a straight line.</p>
</div>
<div class="specification">
<p>Write down</p>
</div>
<div class="specification">
<p>The following data are found for a particular reaction, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is measured in Kelvin and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>mol</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>:</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASsAAAERCAYAAAAjRYF/AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAACwCSURBVHhe7d0PUBRnnjfw78tRajliqesq0TAhkvBHdEUi5rKkEjeswBlNlCSbvInLa4RVVy02JjG5DIHi2KybjRESNut5xggvas4kJ8N5pWc0Oia1JveicJTRU5kCkRwwiocEGALG6eftnumBYWgcghpp5vupmnLm6WYYpn/97e5nnvH5X0IGIqIhLkD9l4hoSGNYEZEuMKyISBcYVkSkC3062I3GO9V7RES3T13df6v3XDTDynslosFgLdFgadUOLwOJSBcYVkSkCwwrItIFhhUR6QLDioh0gWFFRLrAsCIiXWBYEZEuMKyISBcYVgMmofVoNqKMdzpH12rfIvBY0Tl5TaL+XIWtYj+KMpch8+hltY0GgmE1YM2oOPSfiF9XiMP/VYtTxcthUJpj3sDh2v9GbZkZf/j1A5gZ+hO+qdQP5YD3Bn6xeAWyd3yPe6eOUdtpILhfDZTUBvztW3hn3XzcO6ZFDq4jsMvNhpmhmCy/iwHBcfjfTy3H38VOcK1P1EcAxsY+ghTlKBczH/Fho1zNNCAMq4EKuBvzFkXBeSy8Vof/3Fcr3wlFyvwZGKu0yQJjHsKDY/mWUn/kM6uKIyixGxCT8gDuri7CY91dCGtRarumrkdauGcNgnT+JCzN8h3DI5jPMykaMKUrQTkj/6mzuyDw3sXIWfdLJK57D6VleVgcHKiuR1oYVj9YJ6qPHUKlfM+Q8ghieSZFAyX9D2q/blIPcoE4t6MYXz34JrauW4zY4BHqStQf7mk/lHQBx0pOyHd6XwIS+SJVf4WSSjtwfxBq3/415mdWYWrIRO6EA8T36Ydqb4TVqnSt38VPc+gHuIZLp0/IZ+QGTJ/6U8DRIbdV4vjZFtdi8olh9YO4O0jlu/w0h36QFpw9rnQezMEzy34ln5XHy7HVhK9r/wfXbKdRabvqWo36xbD6IdrPoGTngV5DFogGpPUUDpXUyoUThtDJozBm6jTcK1dSZfZjWPjPHbiHfVY+cXcbkGuwla6FcXoSsg82OFvsO5ZixmNFsHK4Og2AdLEWX8tHOfeHMgFhj2D5glBM//Wf8Off3OcaEkPXxQkj6JZhLdFgadUOz6yISBcYVkSkCwwrItIFhhUR6QLDioh0gWFFRLrAsCIiXWBYEZEuMKyISBcYVkSkC5pftyEiut28v27D7wbSLcNaosHidwOJSLcYVkSkCwwrItIFhhUR6QLDioh0gWFFRLrAsCLyGxLaKwqQHJOHCh3OVM+wIvITku0LvL95Hy6oj/WGYUXkD6Q67Pu4CYlLf4mRapPeMKw8yRu0dNVc5+hZ1y0CjxWdk0+eVe1WHM5PQ5S6LDlzFyq8JqeUbF+iIF19juRM7Kiw9fw86ZOvuriZ5Bo7WvoBMpPnIr30G7XR01XYjm9BelT/NdjXVTTsOwI8+Tgixv6N2qZDytdtPIWETFXv+RuHaCt/VyQlvSvK2xxqmwfHBWFeGSci04rFWWV52ylRmBYnQjzXbysTeUkzRVLWZ6LR0SUaLb8XSZEZwlzf5VruZ4ZHLfmoi5uqSVhM8c73LSRkpkgz16ntbuprCVkosiz1wuGoF5ashSJypVnUO19amyjPS1R/3uO2fJP4h1lebWlm0eh8zqFJeY3eGFZuzjBKFCvNF+SS6MtRVSgWhSSKvPI2tcXdFi4WFZ6Vf+Y7UVX4rAiJzBKWb9VncJwVhYvCRaTJIr51tfiVYVFLPuriVnDVlUZYadSTa914YbI0qS3X9335JjFr1iZR/r3aMERp1Q4vA50ktH6xDa/tP439GU9iRf4nOGptVZcpruHS6ROoVB+5BYQ9gJQYwGptRLt0AcdKTgDxsYgcq76tAXchPmUO7CVHUNHKi0H98VUXWjphLVqF9KLTaFdbXFpxrmgtUgZw+RgQNA6T1PuepOqvUFI5EvFxYRirtrlqsAklh07Jv2F4Y1g5BWDsvFycqatBWWkOHrxcjNSEX3oUXABGj5sAA+y43NrpbPFkv9iCjvZGObTsmBBtxES1HQhE0Ljx8grVqL3oq1+Bhh5fdaFlFO5NzcbSmg14oXs9JahMWF+zBJtTIwa500lor6+BFXcg2ijXlFvAaIybNBL2r2txcQDHw8DYF1FZ+SJiA9UGHWFY9TICwbELsOwPu3AoNxbHsl/H+xUtcrtctLGPIMVQix1vFqDUeXS9Clu5BX+VA8oweRxGd7Tgot35JBou40qbDge2kKq/uuhHwBTMy9noCqztB3C44BVnUG3NSUDwoPc4CR0tzfLhsh91zWgb5ifvDCtNYxGRug6vxvwX/u/nNfJFoNL0EF7b+x5+jY+QkTAdxuQs/HPpPhy0hyJl/ozu03JtEzE+SIeHMvKiURf9UQIr+yVEF6Tj+e3hyMm+kaAaAOMEBA3zvZlh1Z+AnyB05k/VB4oAjLl3Mf5w4JzzPwWrO7Aes3EFmP4rPDFnAjApCg/GGNB1uRUd6k8o/RcN562AIQyhk0eobaRrfeqiH1IDjuZuwumMbShcXoWc3MOw3dCZTyAmRc9BjHdXxLUmnD/RDMPMUExmWPkp6X9Q+/VU/J+Hp8ll4k3pg3gdv93xU6zbkIrYMfLbqNWZLtXjpKUWhpRHEOvudCd9u25dqJSgylmPndNMeGd5MhIy3sLGaWasyLmxwNLqTJfOn4SleSBn9/rHPUgh2VDxb6X4N/cAzvZzKM36e+z+xcv4Tew45youEtqtR1GU+Rzm/+l7/HbPP+J33ctHISzxKSzAAewsOYN2uZyse3dhtzUZf1zz82FfSMPSgOvCUyesxbmuoFoWjTHONvnycdkGZ2CtLvb9aaDU1oJL6MKllo7e6wbcjUQ5/FCyGyXn5LiSX8/eoj2wLngZax7q+Vhn2FKHMHTzy3FW6uA65W933pJMotBSJXpGVKnjU5zLFwpT4T5R3qg10NMh2qrMwpQUrj5PljBX+eMIKxfd19IA6uLm0hjU2WdM1LeiypwlkpzLw0WSySyqbvlg1R+f8rd744QRdMuwlmiwtGqHl4FEpAsMKyLSBYYVEekCw4qIdIFhRUS6wLAiIl1gWBGRLjCsiEgXGFZEpAuaI9iJiG437xHs/LoN3TKsJRosft2GiHSLYUVEusCwIiJdYFgRkS4wrIhIFxhWRKQLDCsi0gWGFZHfkNBeUYDkmDxU6HDOXYYVkZ+QbF/g/c37cEF9rDcMKyJ/INVh38dNSFz6S4xUm/SGYdWHMjfgFyjJT0OU8U4YjWtRanOfM1+F7fgWpEcp7RFIztyFCttVdZmLZPsSBelznV8XMCZnYod7zjnSuevVxU3UbsXR0g+QmTwX6aXfqI2efNdgX1fRsO8I8OTjiBj7N2qbDinfDfSkNV+X/3DPyabMx7ZN7C1vFD0zsjlEW/m78rKFIstSLxzqnHKRK82i3r1SW5nIS5opkrI+E42OLtFo+b1IiswQ5nqtOQaHv+FTS9eri5upSVhM8a75AkNmijRzndru5qsGNeYdVG7LN4l/mOXVlmYWjc7nHJqU1+iNYdWtS9SbM0RkSJxIKzzVdyJLx1lRuChcRJoscumqTVWFYlFIvDBZmuRH34mqwmdFSGSWsHyrlrLGz/iT4VFLPuriFnDVlUZY+axB35yT9faZOHXo0aodXgaqpIb9yH1tD7DAhNxU97TfPaTqr1BSORLxcWHdU8EHhD2AlJgmlBw6hVbpAo6VnADiYxE5Vn1bA+5CfMoc2EuOoKKVF4N65Ksu+uqEtWgV0otOo11tcWnFuaK1SCnyPX18QNA4TFLve/JZg2rbcMWwcupE9cFPsN8+BfFhp5AdrfQH3Imo9C047uwPkNBeXwMr7kC0cbzrRxQBozFu0kjYv67FxdZGWK12TIg2YqK6GAhE0Dh5fXs1ai/66legocdXXWgZhXtTs7G0ZgNe6A4sJahMWF+zBJtTIwa50w2gBgdwPAyMfRGVlS8iNlBt0BGGlcJ9VjR9Ph5OSMfWMzUo2/M64o+9gSdWfAirJKGjpRl2dfU+6prR1taCi/2ucBlX2nQ4sMXf+awLdT1vAVMwL2ejK7C2H8DhglecQbU1JwHBg97jBlCDw/zknWGlkOy4UmeH4b75eDw2WH5TRiA47hmsXhEHVB7CsepOdcV+GCcg6Lrv5ESMD9Lhoczf3UhdKIGV/RKiC9Lx/PZw5GTfSFANgM8a1D+GVb/GYMrdU+V/lbMiYFL0HMTIx7XLrR4Feq0J5080wzAzFJPviMKDMQZ0XW5Fh7pYuYxoOG8FDGEInTxCbSN986yL65wtSw04mrsJpzO2oXB5FXJyD8N2Q2c+gb5rkGHlBwKNmP1oqHZHuOF+zL5ntGZHpnT+JCzNoUiZPwNjtTrTpXqctNTCkPIIYt2d7qQfA6gLTUpQ5azHzmkmvLM8GQkZb2HjNDNW5NxYYPmsQbVtuOIe5DQRD615GQvwEd582+IsKMn2OTZvqUTiq09jjhI0AXcjUS48lOxGyTm5VNrPYW/RHlgXvIw1Dyld6qMQlviU/BwHsLPkDNrlcrLu3YXd1mT8cc3Ph30hDU8DqIs+OmEtznUF1TL3p4djEbFsgzOwVhf7/jRQamvBJXThUktH73V91uAwpw5h6Oa/46wcoq3KLExJ4c73ICRkoTAVl4nGXqP/3IMDleXKAEGzqGrzXMHrOZKyhLnKH0dYuQyPWhpIXdwsGoM6+4yJ8lWDw4NW7XB2G7plWEs0WFq1w8tAItIFhhUR6QLDioh0gWFFRLrAsCIiXWBYEZEuMKyISBcYVkSkCwwrItIFzRHsRES3m/cIdn7dhm4Z1hINFr9uQ0S6xbAiIl1gWBGRLjCsiEgXGFZEpAsMKyLSBYYVkd9oQUX+EsTkV0CPs1gyrIj8wlXYjhZj86fV6mP9YVgR+QGp4Qg+/mYOlibdobboD8OqDwnt1i9Qkp+GKOOdMBrXotTmPml2LSstykRyr/Yeku1LFKTPdY7ANSZnYkeFzefUS3Q7tcJ6tBRFmYtgTC+FTW319KNu03YrjpZ+gMzkuUgv/UZt9CSfIR3fgvQopTYjkJy5CxW2q+qyfkh12GcGnnzuZ/qeEk75uo0nrSlw/Id7miNliqNtYm95o+g9E5dFmCLd0yStEebGXnMkCdFWJvKSZoqkrM9Eo6NLNFp+L5IiM4S5vktdwb8M/VpyyJs0S0Q6t6d8SzOLRnVJtx91mzYJiylera+ZIs1cp7a7OURb+btyfS4UWZZ64XDUC0vWQhG50izqnYWqMZWXcluSJmb0atN67qFFeZ3eGFbdukS9OUMu3DiRVnhK3uz9+U5UFT4rv0/eYaW2R2YJy7dqxDnOisJF4SLSZJFj0P/oppbU7dQ3rG7PNnVUFYpFWoGi8btd68YLk6VJbbkeV5jNyisXXofZIUerdngZqJIa9iP3tT3AAhNyU90z6WoJRNC48ep9D9IFHCs5AcTHItI9U6/WlPI09ASMxrhJI9UHHga1TTthLVqF9KLTaFdbXFpxrmgtUop8z8gcEDQOk9T7nqTqr1BSORLxcWHdl3NaU8oPVwwrp05UH/wE++1TEB92CtnRSn/AnYhK34LjvvoD3NobYbXaMSHaiJ6JvNVgs1ej9uIAn4eGjkFt01G4NzUbS2s24IXuwFKCyoT1NUuwOTVikDudhPb6GlhxB6KNHgdLNWjtX9fios/j4RjErvsUleti5b9CfxhWCvcRdPp8PJyQjq1nalC253XEH3sDT6z4ENaBnBR1tOCiXb3fx2VcadPjyBY/N9htGjAF83I2ugJr+wEcLnjFGVRbcxIQPOg9TpJfTjP6fTl1zWgb5ifvDCuFZMeVOjsM983H47HB8psyAsFxz2D1ijig8hCOVXeqKw7WRIwP0uOxjPrnY5sqgZX9EqIL0vH89nDkZN9IUA2AcQKChvnezLDq1xhMuXuq/O8Az4omReHBGAO6LreiQ21SLi8bzlsBQxhCJ49Q20g3bmSbSg04mrsJpzO2oXB5FXJyD8N2Q2c+gZgUPQcx8rnV5VaPg+e1Jpw/0QzDzFBMZlj5gUAjZj8aqt1pargfs+8ZrT64Dq2OV6keJy21MKQ8glh3By3px2C3qRJUOeuxc5oJ7yxPRkLGW9g4zYwVOTcWWFqd6dL5k7A0hyJl/gx9j6EaAO5BThPx0JqXsQAf4c23Lc6CkmyfY/OWSiS++jTm9CrKa2hruSL/ewUtvc64RiEs8Sn5OQ5gZ8kZtCuDDffuwm5rMv645ufDvpB0TepAy6Uu4FKLV7/PYLZpJ6zFua6gWub+VHksIpZtcAbW6mLfnwZKbS24hC755XT0XjfgbiTK4YeS3Sg5J8dV+znsLdoD64KXseahno8Ahi11CEM3/x1n5RBtVWZhSgp3vgchIQuFqbhMNHqOCv2+XOTN8hxcN9VrzIrXcyRlCXOVP46wclHeg6Hu+/JNYpbH9gwJSRR55Z6j7H7MbaoxqHPWJlHea1CUe+CyslwZvGwWVW29hi4PC8rf7o0TRtAtw1qiwdKqHV4GEpEuMKyISBcYVkSkCwwrItIFhhUR6QLDioh0gWFFRLrAsCIiXWBYEZEuaI5gJyK63bxHsPPrNnTLsJZosPh1GyLSLYYVEekCw4qIdIFhRUS6wLAiIl1gWBGRLjCsiEgXGFZEfqMFFflLEJNfAT1OucuwIvILV2E7WozNn1arj/WHYUXkB6SGI/j4mzlYmnSH2qI/DKs+JLRbv0BJfhqijHfCaFyLUpvrpFmyfYmC9LnOrwIYjRFIztyFCttV5zK3XuskZ2JHhc3nPHF0O7XCerQURZmLYEwvhU1t1aRMXpotrxeTh4pbdR3VbsXR0g+QmTwX6aXfqI2e5DOk41uQHtV/DfYh1WGfGXjyuZ/pe/5K5buBnrTm6/If7jnZlPnYtom95Y2ie0a2tjKRl5ajzhnXJRrL/lGkRSrzyL0ryt3ztinrJM0USVmfiUaHvI7l9yIpMkOY67tcy/3M0K8lh/jWkiUinXPwybc0s2hUl/TVJerNGa51+8zld7M0CYsp3vVaQmaKNHOd2u7mEG3l78r1uVBkWeqFw1EvLFkLReRKs6h3lqDGvIPKbUmamNGrTeu5hxbldXpjWHVzF2OcSCs8JW92T0pR/6N4t/yK+ljxnagqfFZ+v9yTYqqPI7OE5Vs1vBxnReGicBFpssgx6H90U0vqdrpeWDnqzWLlrEdEUoK83i0LKxdHVaFYpBUoGvXkWjdemCxNasv1uMKs98S8Q5NW7fAyUCU17Efua3uABSbkprqn/XYLwNh5q5ARO059rAhE0LjxgOF+zL5ntPwEF3Cs5AQQH4tI93TzAXchPmUO7CVHUNHKi8EhK2A0xk0aqT7Q0H4axdmliNiej1Wh11nPqRPWolVILzqNdrXFpRXnitYipcj39PEBQeMwSb3vSar+CiWVIxEfF9Z9ORcQ9gBSYppQcuiU/BuGN4aVUyeqD36C/fYpiA87hexopT/gTkSlb8HxfvsDWnD2eCUMKY8gVgmn9kZYrXZMiDZiorpGd6DZq1F70Ue/Ag1RLah4fyP++uB6/CZW3pY+jcK9qdlYWrMBL3QHlhJUJqyvWYLNqRGD3OkktNfXwIo7EG30eB1q0Nq/rsVFn8fDMYhd9ykq18XKlak/DCuF+6xo+nw8nJCOrWdqULbndcQfewNPrPgQVo0ikBr+it377sMf1/zcdZTraMFFu3ORhsu40qbHkS3+Tg6Iip3YfOXXeKPP2fZ1BEzBvJyNrsDafgCHC15xBtXWnAQED3qPk+QSa0a/JVbXjLZhfvLOsFJIdlyps8Nw33w8HhssvykjEBz3DFaviAMqD+FYdae6okqqw95c5bLgH7B4ygi18XomYnyQHo9l/k2yWfDWZmD1K7/44SGjBFb2S4guSMfz28ORk30jQTUAxgkIGuZ7M8OqX2Mw5e6p8r/eZ0XyKX1xIU4+swG/8+zDmhSFB2MM6Lrcig61Sbm8bDhvBQxhCJ08kFCjoeMaLv2HGUUH38Ti6UZ1uMoDyDjYDDTnYfG0n/UztEClDHPI3YTTGdtQuLwKObmHYbuhM59ATIqegxj53Opyq8fB81oTzp9ohmFmKCYzrPxAoBGzHw3V7gh3d6A7KaOAt6EYT+LFeVN6v3lanelSPU5aanv6tUhHAhG8+D3nf63bc/sKBYkTgAkvorTmJLYtDlHX9aIEVc567JxmwjvLk5GQ8RY2TjNjRc6NBZZWZ7p0/iQszaFImT9D32OoBoB7kNNEPLTmZSzAR3jzbYuzoCTb59i8pRKJrz6NOc6gUYLqPfy5NgmvLevpv5Bsh5Gb/4VcPKMQlviU/BwHsLPkDNrlFuveXdhtTe7p16KhSepAy6Uu4FLLTej36YS1ONcVVN11MhYRyzY4A2t1se9PA6W2FlxCl/xyOnqvG3A3EuXwQ8lulJyT46r9HPYW7YF1wctY81DPxzrDljqEoZv/jrNyiLYqszAlhTvfg5CQhcJUXCYanUOm1AGeznbvm+cYF6/nSMpSB5H6J+U9GOq+L98kZvXanu5xc1rqhDlt5i0cZ6UxqLPP73IPXFaWK4OXzaLKPSh5GFH+dm+c3YZuGdYSDZZW7fAykIh0gWFFRLrAsCIiXWBYEZEuMKyISBcYVkSkCwwrItIFhhUR6QLDioh0QXMEOxHR7eY9gp1ft6FbhrVEg8Wv2xCRbjGsiEgXGFZEpAsMKyLSBYYVEekCw4qIdIFhReQ3WlCRvwQx+RXQ4yyWDCsiv6BMeFKMzZ9Wq4/1h2FF5AekhiP4+Js5WJp0h9qiPwyrPiS0W79ASX4aoozKxJZrUWpznTRLti9RkD7XObrWaJyL9PxDsLb3nlip1zrJmdhRYfM59RLdLvLZxvEtSI9StqdyW4TMHcf7zO33o27TdiuOln6AzGS5vjQnUfV8zRFIztyFCttVdVk/pDrsMwNPPvczfU8Jp3zdxpPWFDj+wz3NkTLF0Taxt7xRdE9y1FYm3jX9kyhr7JIfuKfmCheLCs/2WicvaaZIyvpMNDrUdSIzhLle+Rn/M7RrySHayv8s0txTWTkaRdm7y0Wksu3zykT3ZFw/6jZtEhZTvGsKrpCZIs1cp7a7Ka/5XbnuFoosS71wOOqFJWuhiFxpFvXOItSYyku5LUkTM3q1aT330KK8Tm8Mq25dot6cIRdrnEgrPNVTrP1yzSEXabLIEaf4TlQVPitCIrOE5Vs1vhxnReGicI91/MvQriU5GPK2iXLPOffU7dUzV9/t2aaOqkKxSCtQNH63a13PuSuvxxVms/LKxS2Z9vAm0qodXgaqpIb9yH1tD7DAhNzUnhmX+9VajeMnftEz27J0AcdKTgDxsYh0TxWvNaU8DRETMW9dGmLHeOwCAaMxbtJIGB6NxT2B8uNBbdNOWItWIb3oNNrVFpdWnCtai5Qi3zMyBwSNwyT1viep+iuUVI5EfFxY9+Wc1pTywxXDyqkT1Qc/wX77FMSHnUJ2tKsPIyp9C4736Q9Q+rQOIf/FTcA7WVg8ZYSrub0RVqsdE6KN8m7gFoigceMBezVqL/roV6DbTzkAHRuLlPkzXGEwqG06CvemZmNpzQa80B1YSlCZsL5mCTanRgxyp5Prrr4GVtyBaKP8+93UgLV/XYuLPo+HYxC77lNUrouV/wr9YVgp3EfQ6fPxcEI6tp6pQdme1xF/7A08seJDWLuLoB0V+X+H6QnPI//gf2JH6gKsKq1zHSk7WnDR7lxJw2VcadPjyBZ/chUNR8zY9/DLWPOQGk2D3aYBUzAvZ6MrsLYfwOGCV5xBtTUnAcGD3uMk+eU0o9+XU9eMNp9hpW8MK4Vkx5U6Owz3zcfjscHymzICwXHPYPWKOKDyEI5Vd6oruo5MdbUnUFrwIhINDdj/2jZ84fMSbyLGB+nxWOY/nN0A28Ow/e3HMGVAe4WPbaoEVvZLiC5Ix/Pbw5GTfSNBNQDGCQga5nszw6pfYzDl7qnyvxpH0IBgxC5ejVdffVi+HGhGS4ccVpOi8GCMAV2XW9GhrqZcXjactwKGMIROVi8XaehpP43iv9Tgma2revdh3cg2lRpwNHcTTmdsQ+HyKuTkHu4zJOKHCcSk6DmIkc+tLre6D56ya004f6IZhpmhmMyw8gOBRsx+NFS709RwP2bfM1p94GkEJoeGwTA9HEalwLU6XqV6nLTUwpDyCGLdHbQ0tCihkvcvQGo65gV7hc9gt6nynDnrsXOaCe8sT0ZCxlvYOM2MFTk3FlhanenS+ZOwNIf29LMNY9yDnCbioTUvYwE+wptvW5wFJdk+x+YtlUh89WnM0ShKyWbB229+jaf+/knEOI/GoxCW+JT8HAews+QM2uVysu7dhd3W5J5PDGloUULljfdR+/Q6LItwbyHlaykFyD96Wb4/mG3aCWtxriuolrk/VR6LiGUbnIG1utj3p4FSWwsuoQuXWjp6rxtwNxLl8EPJbpSck+Oq/Rz2Fu2BdYFHP9twpg5h6Oa/46wcoq3KLExJ4c73ICRkoTAVl4lG5/Aa92A896A6jUGjTl7PkZQlzFX+OMLKZUjXkjqg0rU9vW6e46p+1G2qMaize8yXm3vgsrJcqUN1UOswo/zt3jhhBN0yrCUaLK3a4WUgEekCw4qIdIFhRUS6wLAiIl1gWBGRLjCsiEgXGFZEpAsMKyLSBYYVEemC5gh2IqLbzXsEO79uQ7cMa4kGi1+3ISLdYlgRkS4wrIhIFxhWRKQLDCsi0gWGFRHpAsOKiHSBYUXkN1pQkb8EMfkV0OOUuwwrIr+gzNpTjM2fVquP9YdhReQHpIYj+PibOViadIfaoj8Mqz4ktFu/QEl+GqKMd8JoXItSW9+TZsl2GNnJEX1OqSXblyhIn+v8uoAxORM7Kmw+54mj20U+2zi+BelRynZWbouQueO410SkrbAeLuhZ51Zv03YrjpZ+gMzkuUgv/UZt9OT5miOQnLkLFbar6rJ+SHXYZwaefO5n+p6/UvluoCet+br8h3tOtv7mBVQ5LgjzyjjnezUrr1x0T+vWVibykmaKpKzPRKOjSzRafi+SIjOEub5LXcG/DO1aUuaC/LNIc8+752gUZe8uF5HKts8rE23OdbpEvTlDREauFIVnlbkCvxVnC1fK6ywWeeVXnGvcXE3CYop3vm8hITNFmrlObXdzz1+5UGRZ6oVDnfswcqVZ1DsLVWPeQeW2JE3M6NWm9dxDi/I6vTGsuqmFGRIn0gpPqcWqxbXerKREkSC/Vz1h9Z2oKny29wSZjrOicFG4iDRZ5DL3P0O7luRgyNsmyj0nCFW3V/fEourjXgck9zqLCkWV5pHsxjmqCsUirUDRqCfXuvHCZGlSW67HFWa9/p4hSqt2eBmokhr2I/e1PcACE3JT3dN+e5MvEc99hOztYdj+55UIVVudpAs4VnICiI9FpHu6+YC7EJ8yB/aSI6ho5cXg0DIR89alIdY59b8qYDTGTRoJw6OxuCdQfnzpDP5aaXctc1O3Kaw1qG/X2qadsBatQnrRabSrLS6tOFe0FilFvqePDwgah0nqfU9S9VcoqRyJ+Liw7su5gLAHkBLThJJDp+TfMLwxrJw6UX3wE+y3T0F82ClkR7v6J6LSt+C4Z39AezneX///8ODGZYgN8nrr2hthtdoxIdoo7wZugQgaNx6wV6P2oo9+Bbr9Wqtx/NhYpMyf4QqD0eMw2QB0XW5Fh3MFD/ZmtHRoxc4o3JuajaU1G/BCd2ApQWXC+pol2JwaMcidTj5Q1tfAijsQbZRryk0NWPvXtbjo83g4BrHrPkXluli5MvWHYaVwnxVNn4+HE9Kx9UwNyva8jvhjb+CJFR/C6iyCFlS8vxNXXjQhNUKjm7KjBRe9DsI9LuNKmx5HtviTq2g4Ysa+h1/GmofUw83YGZifEgr7jk34U+k5Z/BItgr8+1+tgGECxo3uZ/cJmIJ5ORtdgbX9AA4XvOIMqq05CQge9B4nySXWjH5LrK4ZbT7DSt8YVgrJjit1dhjum4/HY4PlN2UEguOeweoVcUDlIRyrboXt6BZsxlK8Mm/KIN60iRgfpMdjmf9wdgMol/dvP4Yp3RtYvlR87QMU/BrYkZGA6cZFyPrnf8X+gw0wpDyCWPflvhYlsLJfQnRBOp7fHo6c7BsJqgEwToD3yf5wM8z/vBsxBlPunir/q5wV1eM/dn6Ig/lL5IJVP8KeuxYH5aXN+Y9hmjK8wXEvHowxeF0ydKLhvHIUDkPo5BFqGw057adR/JcaPLN1Ve8+LMWYCCz+w785/9fKurp/xauzR+IC4rDiiZjrDwOQGnA0dxNOZ2xD4fIq5OQe9hoS8UMFYlL0HMTI51aXWzvVNtm1Jpw/0QzDzFBMZlj5gUAjZj8qn+5rdYQb7sfse+SC3XZSLVj1VvYeEuXFE9btRU3de1g8NaxvZ7pUj5OWWt9HYbp9lFDJ+xcgNR3zgq93QFE+XNmFF3/7Ee5aZ8JvYsep7RqU58xZj53TTHhneTISMt7CxmlmrMi5scDS6kyXzp+EpTm0p59tGOMe5DQRD615GQvwEd582+IsKMn2OTZvqUTiq09jzoCCZhTCEp+Sn+MAdpacQbsymHDvLuy2JuOPa34+7AtJl5RQeeN91D69Dsu6+yGVr6UUIP/oZfWxTBmoWZSFJ+f/GfhtIYp+F9fPp8WKTliLc11Btcz9qfJYRCzb4Ays1cW+Pw2U2lpwCV241NLRe92Au5Eohx9KdqPknBxX7eewt2gPrAs8+tmGM3UIQzf/HWflEG1VZmFKCne+ByEhC4WpuEw09jeWptEs0nqNs1J4PUdSljBX+eMIK5chXUvqgErXtva6ucfKfV8u8mYpbT4GCd80GoM63WO+urkHLrtflzqodZhR/nZvnN2GbhnWEg2WVu3wMpCIdIFhRUS6wLAiIl1gWBGRLjCsiEgXGFZEpAsMKyLSBYYVEekCw4qIdEFzBDsR0e3mPYKdX7ehW4a1RIPFr9sQkW4xrIhIFxhWRKQLDCsi0gWGFRHpAsOKiHSBYUXkN1pQkb8EMfkV0OMslgwrIr+gTIRRjM2fVquP9YdhReQHpIYj+PibOViadIfaoj8MK6LhTqrDPjPw5HM/0/WUcAyrPiS0W79ASX4aoozK7MtrUWrrucKXGkqxKkqdldl5ew5F1p4ZciXblyhIn+talpyJHRU2n/PE0e0iXxod34J09/aMSkP+YSva1aW9Xb8ubhpljsLSD5CZPBfppd+ojZ48X3MEkjN3ocJ2VV3Wjor8JNff4nl7MhemP6Xjb0MjsTj/NJrzl2GV5nMPccp3Az1pzdflP9xzsvU3T9wVUZ73K5GUVyba1JZe2spEXtJMkZT1mWh0dIlGy+9FUmSGMNd3qSv4l6FdSw7RVv5PwvTuMdfckN3zCD4rCqu+c63SzVdd3CxNwmKKd75vISEzRZq5Tm13U17zu/LrWCiyLPXCob7myJVmUT+gF+Sal7D3XJdDk1btMKy6dYl6c4aIDIkTaYWnNMPIUW8WK2f1Fz7fiarCZ3smyFQ4zorCReEi0mSRy93/6K6WnBPXxguTpUltUPiui5vNUVUoFmmFlUY9udb1fs390XdY8TJQJTXsR+5re4AFJuSmuqf99nQZX/zlbexv3oOMhN8iv+QLWNs9LvCkCzhWcgKIj0Wke7r5gLsQnzIH9pIjqGjlxeDQJqH1bAVOeE3F7rsuvHXCWrQK6UWnvS4nW3GuaC1SinxPHx8QNA6T1PuepOqvUFI5EvFxYd19TwFhDyAlpgklh07Jv8GXMYhd9ykq18UiUG3RE4aVUyeqD36C/fYpiA87hexo17V+VPoWHO/uD5iIeX/4K+pqT6D0rZ/j8tbfICFuNYrOqSXS3gir1Y4J0UZ5TbdABI0bD9irUXvR/Tw09LTCevg9vPgC8M5bj2FK914xkLrwNgr3pmZjac0GvNAdWEpQmbC+Zgk2p0YMcqeT0F5fAyvuQLRRrim3gNEYN2kk7F/X4uIwPx4yrBTus6Lp8/FwQjq2nqlB2Z7XEX/sDTyx4kNYPYsgIBixi9Lwh38pQW58BbJ/V4QK5QyrowUX7eo6fVzGlTY9DsPzA9cqkB8zHQnPv4WDzduRev86lDaoQfRD6sJTwBTMy9noCqztB3C44BVnUG3NSUDwoPc4SS6xZvRbYnXNaGNY+QHJjit1dhjum4/HY4PlN2UEguOeweoVcUDlIRyr7vm0r9uYaKS+uhox/7UPn1d1qI39mYjxQXo88fYDgbFYV/nfqC3bi4J1STDY9+C1v3zpuqQaTF24KYGV/RKiC9Lx/PZw5GTfSFANgHECgob53syw6tcYTLl7qvxv/2dFAZNDMdOgPpgUhQdjDOi63Iqe6OpEw3krYAhD6OQRahsNRQHBsVj8u1fwqrwN7RdbPLahN9914SQ14GjuJpzO2IbC5VXIyT0M2w2d+QRiUvQcxMjnVpdbPULyWhPOn2iGYWYoJjOs/ECgEbMfDdXuCDfcj9n3jFYf9CZdrMXXdz2Kh8Pl5Vqd6VI9TlpqYUh5BLHuTncaugJ+gtCZd2J69FRXR/og68IZVDnrsXOaCe8sT0ZCxlvYOM2MFTk3FlhanenS+ZOwNIciZf4MXQ/4HBD1U8Fu/jp0wTksITJcHSMlP278TGQlxXd/XO1oLBd7zf8uyhuVYQsO0VZlFqakX4m88ivOn1e4nsP9Ebc6NofjrHRCHReX9HthcW5jF1910ZcyhGWlxvJvxdnCNWJJ4VmfY7RcwxHCxaI+66rDKCJXisKz3wrRdlaYTT9knJV+aNUOw6qbO4DCne9BSMhCYSoucw0YVJY6i9S9TBkcWCgsVd6jp7yeIylLmPus4z+Gdi0pA3wXq9tT3d6F+9SDkafr18XN5RoH1fOa5NusTaK816Ao9wBVZblSh2ZR1TbMkkqm/O3eOLsN3TKsJRosrdphRwoR6QLDioh0gWFFRLrAsCIiXWBYEZEuMKyISBcYVkSkCwwrItIFhhUR6YLmCHYiotvNewR7n7AiIhqKeBlIRLrAsCIiXWBYEZEuMKyISBcYVkSkA8D/BxnO/4Q3RYsYAAAAAElFTkSuQmCC"></p>
</div>
<div class="specification">
<p>Find an estimate of</p>
</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>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac></math> is always positive.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></munder><mi>k</mi><mo>=</mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>k</mi><mo>=</mo><mn>0</mn></math>, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) the gradient of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>;</p>
<p>(ii) the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use chain rule, including the differentiation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>×</mo><mfrac><mi>c</mi><msup><mi>T</mi><mn>2</mn></msup></mfrac><mo>×</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>this is the product of positive quantities so must be positive <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong> </em>may be awarded for correct argument from <strong>their</strong> derivative. <em><strong>R1</strong> </em>is not possible if their derivative is not always positive.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for an increasing graph, entirely in first quadrant, becoming concave down for larger values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, <em><strong>A1</strong></em> for tending towards the origin and <em><strong>A1</strong> </em>for asymptote labelled at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi></math> of both sides <strong>OR</strong> substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>A</mi><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>(A1)</strong></em></p>
<p><br>(i) so gradient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><br>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The implied <em><strong>(M1)</strong></em> and <em><strong>(A1)</strong></em> can only be awarded if <strong>both</strong> correct answers are seen. Award zero if only one value is correct <strong>and</strong> no working is seen.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>an attempt to convert data to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p>e.g. at least one correct row in the following table</p>
<p><img src="data:image/png;base64,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"></p>
<p>line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mo>-</mo><mn>13400</mn><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>13400</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>13383</mn><mo>.</mo><mn>1</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">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rearrange or solve graphically <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi><mo>=</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mo> </mo><mn>300</mn><mo> </mo><mn>000</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo> </mo><mn>304</mn><mo> </mo><mn>258</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> <strong>Note</strong>: Accept an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3269017</mn></math>… from use of <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mn>3</mn><mi>sf</mi></math> value.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question caused significant difficulties for many candidates and many did not even attempt the question. Very few candidates were able to differentiate the expression in part (a) resulting in difficulties for part (b). Responses to parts (c) to (e) illustrated a lack of understanding of linearizing a set of data. Those candidates that were able to do part (d) frequently lost a mark as their answer was given in <em>x</em> and <em>y</em>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>3, −3 <em><strong>A1</strong></em><em><strong>A1</strong></em> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>stretch parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis (with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis invariant), scale factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>translation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 0.003} \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.003</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (shift to the left by 0.003) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Can be done in either order.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>correct shape over correct domain with correct endpoints <em><strong>A1</strong></em><br>first maximum at (0.0035, 4.76) <em><strong>A1</strong></em><br>first minimum at (0.0085, −1.24) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> ≥ 3 between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.0016762 and 0.0053238 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.011676 and 0.015324 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for either interval.</p>
<p>= 0.00730 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}} = \frac{1}{{0.007}}\int_0^{0.007} {6\,{\text{sin}}\left( {100\pi t} \right)} {\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right){\text{d}}t">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</mfrac>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</msubsup>
<mrow>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 2.87 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>in each cycle the area under the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis is smaller than area above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis <em><strong>R1</strong></em></p>
<p>the curve begins with the positive part of the cycle <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{{4.76 - \left( { - 1.24} \right)}}{2}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 3.00">
<mi>a</mi>
<mo>=</mo>
<mn>3.00</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = \frac{{4.76 + \left( { - 1.24} \right)}}{2}">
<mi>d</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1.76">
<mi>d</mi>
<mo>=</mo>
<mn>1.76</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{2\pi }}{{0.01}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>0.01</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 628\left( { = 200\pi } \right)">
<mi>b</mi>
<mo>=</mo>
<mn>628</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>200</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.0035 - \frac{{0.01}}{4}">
<mi>c</mi>
<mo>=</mo>
<mn>0.0035</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>0.01</mn>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.00100">
<mi>c</mi>
<mo>=</mo>
<mn>0.00100</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the system of paired differential equations</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot x = 3x + 2y">
<mrow>
<mover>
<mi>x</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot y = 2x + 3y">
<mrow>
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mi>y</mi>
</math></span>.</p>
<p>This represents the populations of two species of symbiotic toadstools in a large wood.</p>
<p>Time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in decades.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the eigenvalue method to find the general solution to this system of equations.</p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given the initial conditions that when <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 150">
<mi>x</mi>
<mo>=</mo>
<mn>150</mn>
</math></span>, <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>, find the particular solution.</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>Hence find the solution 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">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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>, find an asymptote to the trajectory of the particular solution found in (b)(i) and state if this trajectory will be moving towards or away from the origin.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>The characteristic equation is given by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} {3 - \lambda }&2 \\ 2&{3 - \lambda } \end{array}} \right| = 0 \Rightarrow {\lambda ^2} - 6\lambda + 5 = 0 \Rightarrow \lambda = 1{\text{ or 5}}">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>λ</mi>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mtext> or 5</mtext>
</mrow>
</math></span> <em><strong>M1A1A1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 1{\text{ }}\left( {\begin{array}{*{20}{c}} 2&2 \\ 2&2 \end{array}} \right)\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right){\text{ gives an eigenvector of form }}\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right)">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> gives an eigenvector of form </mtext>
</mrow>
<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>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 5{\text{ }}\left( {\begin{array}{*{20}{c}} { - 2}&2 \\ 2&{ - 2} \end{array}} \right)\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right){\text{ gives an eigenvector of form }}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right)">
<mi>λ</mi>
<mo>=</mo>
<mn>5</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> gives an eigenvector of form </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>General solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = A{e^t}\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right) + B{e^{5t}}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \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>
<mo>=</mo>
<mi>A</mi>
<mrow>
<msup>
<mi>e</mi>
<mi>t</mi>
</msup>
</mrow>
<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>
<mo>+</mo>
<mi>B</mi>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>5</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Require <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A + B = 150,{\text{ }} - A + B = 50 \Rightarrow A = 50,{\text{ B = 100}}">
<mi>A</mi>
<mo>+</mo>
<mi>B</mi>
<mo>=</mo>
<mn>150</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mi>A</mi>
<mo>+</mo>
<mi>B</mi>
<mo>=</mo>
<mn>50</mn>
<mo stretchy="false">⇒</mo>
<mi>A</mi>
<mo>=</mo>
<mn>50</mn>
<mo>,</mo>
<mrow>
<mtext> B = 100</mtext>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Particular solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = 50{e^t}\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right) + 100{e^{5t}}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \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>
<mo>=</mo>
<mn>50</mn>
<mrow>
<msup>
<mi>e</mi>
<mi>t</mi>
</msup>
</mrow>
<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>
<mo>+</mo>
<mn>100</mn>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>5</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</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="t = 1 \Rightarrow \left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {15000} \\ {14700} \end{array}} \right){\text{ }}\left( {3sf} \right)">
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<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>
<mrow>
<mn>15000</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>14700</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>s</mi>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
</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>The dominant term is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="100{e^{5t}}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right)">
<mn>100</mn>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>5</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> so 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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) \simeq 100{e^{5t}}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \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>
<mo>≃</mo>
<mn>100</mn>
<mrow>
<msup>
<mi>e</mi>
<mrow>
<mn>5</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>Giving the asymptote as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>The trajectory is moving away from the origin. <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math>.</p>
</div>
<div class="specification">
<p>The shape of a piece of metal can be modelled by the region bounded by the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math>, as shown in the following diagram. The units on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> axes are measured in metres.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><msqrt><mi>x</mi></msqrt><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>16</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>16</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></math></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is obtained from the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> by:</p>
<ul>
<li>a stretch scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction,</li>
<li>followed by a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> direction,</li>
<li>followed by a translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> units to the right.</li>
</ul>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>825</mn><mo>)</mo></math>. Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> under the given transformations and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><mi>h</mi><mfenced><mi>x</mi></mfenced><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mi>b</mi><mo> </mo><mo> </mo></mtd><mtd><mi>a</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>p</mi></mtd></mtr></mtable></math></p>
</div>
<div class="specification">
<p>The area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> correct to six significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</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>Hence show that the equation of the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</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>Find the area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region on the diagram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em> </p>
<p> </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>gradient at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>16</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><msqrt><mn>0</mn><mo>.</mo><mn>16</mn></msqrt></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>25</mn></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>16</mn></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"><mn>0</mn><mo>.</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></mfenced><mo>+</mo><mi>b</mi></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not allow working backwards from the given answer.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>45</mn><mo>,</mo><mo> </mo><mo> </mo><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4125</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>413</mn></math>) (accept " <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>45</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4125</mn><mo>)</mo></math> ") <em><strong>A1A1</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"><mfenced><mrow><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></msqrt></math> <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>if only two correct transformations are seen. </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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn><mo>.</mo><mn>28</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>Correct substitution of their part (b) (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>28</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math>) into the given expression <strong><em>(M1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>.</mo><mn>25</mn><mo>×</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for transforming the equivalent expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> correctly.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>b</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing need to add two integrals <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></msubsup><msqrt><mi>x</mi></msqrt><mo>d</mo><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>d</mo><mi>x</mi></math> <strong><em>(A1)</em></strong></p>
<p><br><strong>Note:</strong> The second integral could be replaced by the formula for the area of a trapezoid <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>34</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>825</mn></mrow></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>251</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>250916</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>area of a trapezoid <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>05</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4125</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>825</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)(A1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>45</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>8</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)(A1)</em></strong></p>
<p><strong><br>Note:</strong> If the rounded answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>413</mn></math> from part (b) is used, the integral is <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>45</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>8</mn><mo>.</mo><mn>24</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>.</mo><mn>295</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>03095</mn></math> which would be awarded <strong><em>(M1)(A1)</em></strong>.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>shaded area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>250916</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>0627292</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for the subtraction of both <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo>…</mo></math> and their area for the trapezoid from their answer to (a)(i).</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>157</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>15725</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The differentiation using the power rule was well done. In part (ii) some candidates felt it was sufficient to refer to the equation being the same as the one generated by their calculator. Generally, for ‘show that’ questions an algebraic derivation is expected.</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;">
<p>The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function <em>h</em>. In most cases it was due to not appreciating the effect the horizontal transformations have on <em>x</em>.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function <em>h</em>. In most cases it was due to not appreciating the effect the horizontal transformations have on <em>x</em>.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Part (i) was frequently done well using the inbuilt functionality of the GDC. Part (ii) was less structured, and candidates needed to create a clear diagram so they could easily see which areas needed to be subtracted. Most of those who were successful used the formula for the trapezoid for the area they needed to find, though others were also successful through finding the equation of the line AB.</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A sector of a circle, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math>, is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>A square field with side <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>m</mtext></math> has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> from the post.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAcwAAAEJCAYAAAAZ/heBAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAByVSURBVHhe7d0NeBTVof/xX/OnCoQioEIEjIEKkSAKAZE3RZS3IvJE4LZaJYBSKoJAEEWNIjcqFKUEI1IuLyWCKNWCexGoBCmBvyICBlCJ0ChiijFQjDQQRYzbO2d2AkkIyYDJspv9fp5nn8ycednZ2WR+OWfOzPzsPxYBAIByhTk/AQBAOQhMAABcIDABAHCBwAQQvHI9GhHZVJGRY+TJLbQK/inPiGus8T5Kzjjmmwfl8OpYVrpSk5Yow+y+M8pXVnqqklJ3qtzZAkqh9esxxvpdaKq2yRmVst0EJgCEqtyVGn/L3Zq84l9OQVlM8DymW+If14pvvE5ZaCIwAVQjlytuwYfKzl6rhNg6ThlQOQhMAEHihHIzliqxb7TdzNZqRIrW7/namVakrCbZksudXDYr35lu8eYqY0mi+trTb1OiZ7eynOa8yBEe5VqzFGbMVFt7fKHWp96nVmY9iemy11JiefPqqBHJ65R1zNTITjUNmmXT16doRCszT7T6Tl6vXK+1fdvmnipL9DjLlaV4M+N6fVJ8XYlLlZF7wpnPMJ/bo+QRHX3vXXoe05zdcYzSzHDeTMU1L6vp0rzfeHUc67HH8pIHqHnxfWs+t2emsw3mZe27Jdusz+SbXLZ8ZZ3c7jNtu+VYltJTi+3TvolKTc/SqYZ2F99rGby5GfIk32t/f773TlV6BcsUITABBAGvjmXM1bC4SVqSWWCXFKQ9q+HxT/oO+OXwZr2ikcWWM+xlB8xQer45sh9RxvOjFJf4kjLtqTu0ZOydemDuTnvsNGlPavjkVTJru/CSuqqt48pa/GCx5Y0cpSUP14Bpm3yBWsRaNn74s0qzN6VAman3a9jIkRo26OlTZUsm6cG/ZlmfuHx5yUPVp/i6rOXuemGz835F+2uMktNy7JKieeJ6PCRPTqlwOifOfhs709kGw9p3iberx/0e5ZT5AbzKT5+hASe323C2a9hcZRT9o+DNlmfiXYqfXGyfZr6kyfHDNS39sD1a8fdahmPb9PywOzU2ea31roZ578cVP+AJV/uEwAQQBPK0fflr9sEzvPc0rcvMVvb+rVo8rJ1v8hkd085Vr2qnGqtfymbtzz6g/Vtmq1+4NangM+0/aB0k83dq+bxtVkFj9U5aq0wzz9ZkddSZzuu107DFW6117dFbd7ZUjcJMrZq10dqwQUrZsk/Z2fu0JWWQ7Lf4aL8Oljh2Fy37sTwJ11nj1gE77Wt1LFW2851PdMievxzh/ZW0LrPk+63O0KemiujN0l+nvGjtL+szTXxNW/cfUHbmeqUMsfZXwXI9+qIVrBFxWrB1tnqbdTWYIM++A9qZEKsaZvykGoqIm6WtKXH2WIOEldrnNHd7szyakmztt/A+mrh8u7X92cpcP1tDYsJVsGaGXtzkC7aSvtWnO963PmG42iatt7+P7Mw3lGAto8zXtHx7njWPFaqbFujRNVbQn/YZ92vJzFXK8rr4Xk9zQjlvv6x51i9RzLCXfPskO1PrkvorvOAtzf3LrmK117IRmAACX2G2dqzebw201sj7b1d0HevQFdZYN91/r++Af0Z1FJuw1jowvq0H6+3SSs9CPXHPJK2xqxeH9c3RQnkP7tdHZrzBHbr/7tbWEtaqI7rr/vt6mZlO16CH4ro1tg6edRQRYc1dI1YJO62D77Yxqrd1jTyp/617xi731WCy83S0eGC2HaQhN5pl6+rKdm3skDtVVk/XdL9BDewZKxY+8A4NjK5rDV2gxp16qKuv2Ob97D2t2GltgflM93dRhDnS14lW3AO+/VWw4u/KOFMtzJXj+uzddVZgWW8xcrTuvy7C2n5rj7Torwfs/bZfK9Z9XLJ2bauhX9S/xPpp/VMweYD6JS6U5+0vdeXUjVbovaNnbjLTTujg/s/s/VfiM8Y9r09MwK4cphZhFX+vp8vXP7Z9YK3X+icldag6Rpkm2Rj1slsLrLINu/VVBbuEwAQQRFqoWeOazrDlkki1Ljdhis4PxuiW+FEaO/bJEk14hvdonrLNQIfmanyyelXDWnXzssMrsoF+UfzI6c3VtpR71Sqmh+LHjtHY4s2IpTWs5ywbptp16+nCEmVnx9ccXLayP5OlaH8V5OnItz8lMAt19BtTg2ygDs0uLVYrPbXfCg4eseqTpdVUi/g/avnEPtY/C6Y59EnrOxmj0XEdFNX3aaXb5zGL1l3eZ6z4ez3ddzpysJxzlaX/uSkDgQkg8IWFq36kqY9l6fOc474y43C2dptWvDPxfq63nkpWWkFj9U6YoRc927V/30ol2El4ier/oobCftFAkWZ0+z7lnKyYFFqr3qcyV10q4LyfvaWnZqxVQXgfJcxaIM/WfdrnmeAL29Lh6idlfyZL0f4Kb6B6tX/KhhXVFPO0/fN/FesodGq/hTeqV3bYhUXourELrdriPm31LFBKygwl9G4sZc7VKPscbNG6pe8P55cRuhYX3+vpaqleI1NbbaDeKe9ZtVPTJFvstXOCYstarBgCE0DgC7tYUW0utQZ2a96cN7TXdA7x5ih9zsLyO/0c+kTvmKZJNVHr7n11W2wDHXpnjdYWS8KwRlFqY7I4b5nmvLzbPo/lzd2oOXPX2dPLV6hDu7fbTZO64lp1791bsQ0P6x3PhrLD1k/CLmutHua8oPlMczb7eq0e2yvPC779FT7wZsXWtQ7/RTXOvH3KPlxWM6ZxqtaYtztbvrrfBbrsmusVY8rmvag523LlNR2NslbpBXu/RWlgr6tl4qkE716lDjC9WjtqRGqW6sT2VVxcX3Vv3cSe7KuVXqBGUb/0nZNdsUwr9ppaoenElOLrMdtqstKzPq7wez1dHTVpcYX1M09pc1/y1WZN56L7TC/iaA1I3Wu9S/kITABB4BLdOHqi3amjIO1R9YqJVGRUR8Wn7nCmn0HDVurW1hx6tyk57mrrwNhcHePnOk2mzrmuup00bFJ3azxHaZP7KMY6KEd1fFY5TaPsucpXQw1bd1BbM5j5rOJKb5eLZr4qUeda3fnIECvQrM8049e+83Uxt2jsEmu7wgdp2ugupcLMo7Edoyq+I07aGHW0Lyv5VnXaDtYjptNVwVrNGNRBUZGRirlljN00Gt5vokbf6KsllhDWQoOnjPZtl7OvIyOvVpzpPGSF7JC7u8mcDa174whN62fVOgtWaXKvGGsea91xz1rfW7hiRt6mDi2urvh7PU1NtRg83ulgNFfxHZtb31UXjbU7F/XVPb2bVRiIBCaAoBDWuJ+mLJ1u98I0wns/rEWL/7v8Tj9h0Yqft0gTTZOfETNUz3j+v9YlmYDcrw0ffmnVKnzn1TzPDLVrTL5enwuU1L+FGatQWIvfat7yx9Xb3izrgD5kujxb1ijJHNDzPtCHnxdrQvabCxRx08NK9cz2NXfanG3b8JziGl/gK6pxte6cf2rbI/SjytraGm1/q/n2eUfDNLRagWQ6XU2ZL0/KBGd5o52GPPOGNsyJU+My0yXMqlXeV2q7LNb3krR4kR61O/1YwiIVN2OpFic534lhmrxTXlXquOtUx9X3WoY612lc6qtKSSj6LNZqze/RyqdO7ZNy8DxMAKGtMEPJHQYoOc8KlISX9dcE64Bsmnun/N6uKZpLKbafdrkFQhGBCSDEHVZ6Ypzil5jLVkq7TgmeRUqIreeMI5TRJAsgxF2imx5dVLL5zxLee4JSPH/SOMISDmqYAAC4QA0TAAAXCEwAAFwgMAEAcIHADGHe3G1aknibip4lZ543tyTD3LEDAFAagRmqvNlaOWW0Ej9op5T15vE5mVp3x9eaGvegFmedjwutASCwEZihyntYn2/JUXj7m3VzC3OTrLqK7ttXXfWFsr6s6KlwABB6CMxQFXaJmnVqrIIVb+jvzpPGvUeP6FD4zeoVa9/yHwBQDIEZqux7NS5SUtc9WpWWpWPeHG1ask4NJ/1GHcxTDAAAJXDjgpBmHsKaquSn/qKVO4+qa9IizRrme+I8AKAkqhIhywrL9Gc17Inj+s0rq7Vh+T3S9D6KOfnUcwBAcQRmqMrfrBdGLdEFd/RR2zo1FXHdSM3yTFPvL5Zo5lufc2kJAJRCYIYo78H9+sg8sPykMNWJvkn9u16o7G8KCEwAKIXADFFhv+ysgW2lndOTtXhvvlVimmhf0ty0X2po9+Y8+w8ASqHTTwgzd/pZ+kKSEpfs8BWYp5ZPfUB3xUbwnxQAlEJgAgDgAhUJAABcIDABAHChUptkzRMvAPhXdvYBZwhAVar0wOSPF/Af/uYA/6FJFgAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcCMrAzMvL0+bNm7Vo0SKNGzdOmzZtcqYAAFA1guo6zKysLM2c+Udt2bJFTZv6bpKQm/uVunTppueff94eB0IJ12EC/hM0Ncy5c+dq0KCB+u6779SzZ08VFhbqq6++0uDBv9aQIUOcuQAAqBpBUcOcMWOGVq9+U7fc0tMOyTffXGnVNGfp1ltvVa1atZy5gNBDDRPwn4CvYZrzk6+//podlv/+97/tsHzzzdVWzXIwYQkA8JuAD8xp06aqc+fOqlGjht599x39z//M17XXXutMBQDAPwI6ML/88kt99VWOIiIilJ+fr4suqqdf/epXzlQAAPwnoAPz22+/1WWXNbaHDx8+rJtu6mEPAwDgbwHfJGt6wxonTpxQw4YN7WEAAPwtIHvJmhsTbN26VR7PG3av2IsuukjfWbXNC2vW1KWXXOrM5dOmTRvdc++9zhgQWuglC/hPwASmub7S9Ih97bXXlJm5W7169rI799SyQvJMcnJytDszU8mzZjklQGghMAH/Oa+BaUJyx44dWrdunRYunK//Gvxfim0Xq6uuukoXXnihM9eZLUpdpGuubauhQ4c6JUBoITAB/zkvgblr1y6t/N//1fwF89TvV/10Q7cbXIekYa7HXJuWpvV/X6+3317P9ZgIWQQm4D9+C0xziciKFcv16quvKuqKKN3co4datWpln59064XZL+hvb/3NHn7wwYkaMiReDRo0sMeBUERgAv7jt8C8Z/hwRbdsqW7dupUZkt9//70SH09U0yZNNX78eKe0JBOYEx96WC1atHBKgNBGYAL+45fLSsxTRn4o/MG+9+uZapTHjx/XF9lfKH1jut3kWpbmzZrro48+csYAAPAfv12HeVHdus5Q2UyQzvxjsl54YfYZQ9U80mv16tXOGAAA/uO3wFz55kr169/v5Kssl1uBaF5nYjoGZWZ+bF+nCQCAP/ktMAfcNkBrVq05+ToXphftnXf8VlOnPuOUAADgHwF/a7zSunbpopwvczR79mynBACAqueXXrLmBgVjxoxRyyuvVK3atZ3SkmrXqqVOnTo5Y+U7dOiQHns8Uf3799eHH36kLVs2W7XOP+juu+925gBCA71kAf/x22UlJjRXrXpTa99aq7D/F6aaxW559+mnn6pzp8768cdCDR402Cktn+lJ+/tR9yk//1SPWvNgaZ6ViVBCYAL+4/c7/YwbN04XXPBzXXzxxfZ4bm6u9TqoZcuW6amkJF0WEeG6plm681Dnzl10ReQVat+hvVPiXuEPhdr83nv6/vvjdhjXr19f4eHhzlSpcePG6t27D4GMgEJgAv5zXs9hmkd37dq18+S9YFtGt9S3Vk3UrdGjRjtDPm2ubqPrr79eB60APtvX119/bd9Y4Zo216jVVa20IyNDe/Z8oho1aqhRo0Z2M/CkSQ+rgxXGf/nLMucdAQCh4rzVMGvXrq1NmzaqdeurNXv2i/Y0E0QmvMxt89za+49/6Ju8PHm9Xj099WlNe2ZapdUCzT1vn095Xm3btVP79u3185//XEeOHFFaWppatozWvffeS40T5xU1TMB/zksN891337HPZ9511xA999wMp/TcmFqhacLt0qWLHZaPJj5qB11lMGH4pzl/0tH8o9b2rtIPP/ygevXq6fbbb7em/kcPPDDGPmCZV0JC2bfzAwBUD36vYZqbsH/77bdl3g/2XGqYpZmwNKFZmTVNw9zH1jQX9yhj2/bt2yezF595hutD4V/UMAH/8XsNs0mTJlV683QTkpVd0zRG/m6k9u7ZowMHSh6cTK3TPPja3CcXAFB9Bd2NC9yoitA0dxkaN3ac1q9/2+4AZF6mZrl06VLFxd1uNwkDAKqvahmYRlWEplnnhRfW1Mcf77bC8vOTzbATJkxw5gAAVFfVNjCNqgjNTtd30sCBAzVv3jw7LG+44UZnCgCgOqvWgWlUdmhGXn65MjN3O2MAgFBR7QPTqMzQNHf8yf93vjMGAAgVARWY5vxgVamK5lkAQOgIqMBs1qxZlTZ3EpoAgHMVUIHZsmVLrfnbGn3//fdOSeUjNAEA5yKgArNWrVp68MGJenfzZqekahCaAICzFXCdfoYMideMPz5n3xigKhGaAICzEXCB2aBBA82fv1CPJT5WpU2zBqEJAHAr4ALT6NOnj5o0barnZjxHaAIAAkJABqYRGRlpP3+S0AQABIKADUyj9dVX60evV6+8+opTUnUITQBAeQI6MI2ePXtqQ3q6srOznZJTTM3TlFfWq379+ho/drwdmmlpaVXe8QgAEDz8/gBptxISxqtRo0Zq2LChvvjiC/t2dPfec48z1feg6NlzZlvzRKhZVDOntHJ8+NEuZWVl2cNdOnfRQxMfsh/vZZhgTd+4Uc8+95w9DpxPPEAa8J+gCEzzkOYF8+dr0Z8XOVOl3/3+d2rbNlYzZ860r9+sTOZzxMcP1eLFL9njo0eNPvmAaAITgYTABPwn4JtkDdP55+Chg86Y7KZSU7OsirAs0rx5czs0jfe3vm//BACErqAIzNKOHz9uN8NWVVgWMaF5220DtP2D7XQEAoAQF5SB6U+XX365OrTvQO9ZAAhxBKYLl1566clLTvbs2eOUAgBCCYHpUtF1mrNSZumr3K+cUgBAqCAwz4IJTXOd5saN6dpcxU9UAQAEFgLzLF111VXq3v0m3XHHrwlNAAghARuYrVrFnLfzhY0bN9GRI0fs4e+++87+WdxlEZdp2bLXKjc0vTlKn3ybItvOVEahU3Ym3mx57utoX4Pne0VrQOpeeZ3JAIDKF7CBOXLkSF15ZQtNmfKk/fKn5ctXWIH4qv2+ixb92SktqUuXLpUYmieUs3K6RqXucMbL49WxnR7N3X+3PJnmln4HrNderRwWTXMBAFShgD7GPv74404gHFDPnr1PXtaRk5NjPzezqjRp0kRbt2633/e9997Xl9b7FT0xZcv776tVTIw9XFmh6c1Zo6QpH+uKluFOSTm8B/T2vNWKui9ObesQkQDgL0FzxH3xxRe1fsMG9evfT+9v3aqJDz3kTKlaJjwfeeRR3T7odvu9TVAPHz7cmVoJoXlstxZP9ij6z8m6L8p3v9oz8yp/0wI9uma31owdrJHJrys9K9+ZBgCoSgF7L9lgY8LShKYJTxOi7hxRRvJ4zan/kGYNq6u3R9yqsduHybN9gmJrOLOU6YRyM97WW8v/pMlLDqp30iJr+daq40xF6OBesoD/0KZXSc6+punVsYyXNeebIXo6/mzD7gJFxPbTsGeWal1SrN6d/LjmZ/g6KQEAqgaBWYnOJjS9uRv07Bzp/od7KOKcv4W6io5P0KS2mXpp4z5V1LkWAHDuCMxK5i40C3VoyxtKTfuD4mIinUtDOmtsWp6UN1Nxza/RCM8/nXkrEHaxotpc6owAAKoKgVkFKg7NGoqIm32yB7Dv9Z5SejeQGkyQZ9+HWhB3uTNvBbxfa/9HTTS0e3NrrQCAqkJgVpGzP6fpgjdXGW969GZGru8mBcf2yvPEI1rWY6J+F1vPngUAUDUIzCr000Pzn/KMuEaRkWPkyTVnKL3K37ZQo+M6KMo04w5erCO9ntVfE66jhywAVDEuK/EDE5YmNE14mhAFKgt/c4D/UMP0gyppngUA+BWB6SeEJgAENwLTjwhNAAheBKafEZoAEJwIzPOA0ASA4ENgnieEJgAEFwLzPCI0ASB4EJjnGaEJAMGBwAwAhCYABD4CM0AQmgAQ2AjMAEJoAkDgIjADDKEJAIGJwAxAhCYABB4CM0ARmgAQWAjMAEZoAkDgIDADHKEJAIGBwAwChCYAnH8EZpAgNAHg/CIwgwihCQDnD4EZZAhNADg/CMwgRGgCgP8RmEGK0AQA/yIwgxihCQD+Q2AGOUITAPyDwKwGCE0AqHoEZjVBaAJA1SIwqxFCEwCqDoFZzRCaAFA1CMxqiNAEgMpHYFZThCYAVC4CsxojNAGg8hCY1RyhCQCVg8AMAYQmAPx0BGaIIDQB4KdxEZgnlJuxVIl9oxUZ2dR6Ratv4lJl5J5wpiNYEJoAcO4qDExvzhpNuStJH7SfrvWZ2crOXKE7Dj6vuJGvKMvrzISgQWgCwLmpODBz92lLwaVq36ubWtSxZq9j1TD7d5Sy9unLYyRmMCI0AeDsVRiYYRHN1Sl8v1Yse0c5dj4W6uiRbxQ+8GbF1nXRoouAVDw0O3Robze3v/zyy85UAEBpFQdm4wGa4ZmmrlveUlpWvlXjfFdLVtTXpPhY1XXmQXAyodmwYSMdOnTQHn/ssUe0a9cuexgAUJKLKmKY6rTopd/fU0crxnVTVI/X1fy5qRoWTVxWB0VhWaSgoMAZAgAUV3FgenOUPmWUnvjxN3plTZqWj/pB03tdp76T1yuXU5hBb+rUPzhDUrNmzdWuXTtnDABQ3M/+Y3GGy+BVfvoUXR//mSatX6hhLWpaZfnam/qw4iYfLVbmY86DZWcfcMYQLEwz7L/+dUhdu3ZTrVq1nFIEA/7mAP+poIZ5Qgf3f6aSjXR1Fd23r7rqsL45WuiUIZhde+216tmzF2EJAOWoIDBr6pdde6mtNmr69Ne111xGYppo5yxUWsyt6t6ytjMfAADVW8W9ZFv8VvM80zXwwNPqFROpyKge+sOPd8iTep9izXWZAACEgArOYZ4dzqcA/sXfHOA/VBEBAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAFwhMAABcIDABAHCBwAQAwAUCEwAAF372H4sz/JNFRjZ1hgD4S3b2AWcIQFWq1MAEAKC6okkWAAAXCEwAAFwgMAEAcIHABADABQITAIAKSf8HHGDHW6lhIZoAAAAASUVORK5CYII="></p>
<p style="text-align: center;"><sup>[Source: mynamepong, n.d. Goat [image online] Available at: <a href="https://thenounproject.com/term/goat/1761571/">https://thenounproject.com/term/goat/1761571/</a></sup><br><sup>This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)</sup><br><sup><a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">https://creativecommons.org/licenses/by-sa/3.0/deed.en</a> [Accessed 22 April 2010] Source adapted.]</sup></p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> be the volume of grass eaten by the goat, in cubic metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> be the length of time, in hours, that the goat has been in the field.</p>
<p>The goat eats grass at the rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>t</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext></math>.</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 the area of the shaded segment.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the field that can be reached by the goat.</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at which the goat is eating grass at the greatest rate.</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>The goat is tied in the field for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> hours.</p>
<p>Find the total volume of grass eaten by the goat during this time.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mtext>AÔB</mtext><mo>=</mo></mrow></mfenced><mo> </mo><mtext>arccos</mtext><mfenced><mfrac><mn>4</mn><mrow><mn>4</mn><mo>.</mo><mn>5</mn></mrow></mfrac></mfenced><mo>=</mo><mn>27</mn><mo>.</mo><mn>266</mn><mo>…</mo></math> <em><strong>(M1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext><mo>=</mo><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo><mo>≈</mo><mn>54</mn><mo>.</mo><mn>5</mn><mo>°</mo></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>951764</mn><mo>…</mo><mo>≈</mo><mn>0</mn><mo>.</mo><mn>952</mn></math> radians) <em><strong>A1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Other methods may be seen; award <em><strong>(M1)(A1)</strong></em> for use of a correct trigonometric method to find an appropriate angle and then <em><strong>A1</strong> </em>for the correct answer.</p>
<p> </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>finding area of triangle</p>
<p><strong>EITHER</strong></p>
<p>area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>×</mo><mi>sin</mi><mfenced><mrow><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for correct substitution into formula.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>.</mo><mn>24621</mn><mo>…</mo><mo>≈</mo><mn>8</mn><mo>.</mo><mn>25</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext><mo>=</mo><mn>2</mn><mo>×</mo><msqrt><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>-</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><mo>=</mo><mn>4</mn><mo>.</mo><mn>1231</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p>area triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mo>.</mo><mn>1231</mn><mo>…</mo><mo>×</mo><mn>4</mn></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>.</mo><mn>24621</mn><mo>…</mo><mo>≈</mo><mn>8</mn><mo>.</mo><mn>25</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p>finding area of sector</p>
<p><strong>EITHER</strong></p>
<p>area of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo></mrow><mn>360</mn></mfrac><mo>×</mo><mi mathvariant="normal">π</mi><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup></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>63661</mn><mo>…</mo><mo>≈</mo><mn>9</mn><mo>.</mo><mn>64</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p>area of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>9517641</mn><mo>…</mo><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup></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>63661</mn><mo>…</mo><mo>≈</mo><mn>9</mn><mo>.</mo><mn>64</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>area of segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mo>.</mo><mn>63661</mn><mo>…</mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>24621</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>39</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>39040</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong> </em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>63</mn><mo>.</mo><mn>6172</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>39040</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo> </mo><mo> </mo><mo>(</mo><mn>5</mn><mo>.</mo><mn>56160</mn><mo>)</mo></math> <em><strong>(A1)</strong></em></p>
<p>subtraction of four segments from area of circle <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>58</mn><mo>.</mo><mn>1</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>58</mn><mo>.</mo><mn>055</mn><mo>…</mo><mo> </mo></mrow></mfenced></math> <em><strong>A1</strong> </em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>angle of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>90</mn><mo>-</mo><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mn>0</mn><mo>.</mo><mn>951764</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>area of sector <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>90</mn><mo>-</mo><mn>54</mn><mo>.</mo><mn>532</mn><mo>…</mo></mrow><mn>360</mn></mfrac><mo>×</mo><mi mathvariant="normal">π</mi><mo>×</mo><mn>4</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mo>.</mo><mn>26771</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>area is made up of four triangles and four sectors <em><strong>(M1)</strong></em></p>
<p>total area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mn>4</mn><mo>×</mo><mn>8</mn><mo>.</mo><mn>2462</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>4</mn><mo>×</mo><mn>6</mn><mo>.</mo><mn>26771</mn><mo>…</mo></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>58</mn><mo>.</mo><mn>1</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>58</mn><mo>.</mo><mn>055</mn><mo>…</mo><mo> </mo></mrow></mfenced></math> <em><strong>A1</strong> </em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sketch of <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></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>110363</mn><mo>…</mo></math> <strong>OR </strong>attempt to find where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>V</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math> hour <em><strong>A1</strong> </em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>d</mo><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>8</mn></msubsup><mn>0</mn><mo>.</mo><mn>3</mn><mi>t</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>d</mo><mi>t</mi></math> <em><strong>(A1)</strong></em></p>
<p>volume eaten is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>299</mn><mo>…</mo><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>299094</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong> </em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Generally, this question was answered well but provided a good example of final marks being lost due to premature rounding. Some candidates gave a correct three significant figure intermediate answer of 27.3˚ for the angle in the right-angles triangle and then doubled it to get 54.6˚ as a final answer. This did not receive the final answer mark as the correct answer is 54.5˚ to three significant figures. Premature rounding needs to be avoided in all questions.</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;">
<p>Unfortunately, many candidates failed to see the connection to part (a). Indeed, the most common answer was to assume the goat could eat all the grass in a circle of radius 4.5m.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates completed this question successfully by graphing the function. A few tried to differentiate the function again and, in some cases, also managed to obtain the correct answer.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was a question that was pleasingly answered correctly by many candidates who recognized that integration was needed to find the answer. As in part (c) a few tried to do the integration ‘by hand’, and were largely unsuccessful.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of the minimum point on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x">
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi ">
<mn>2</mn>
<mi>π</mi>
</math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule or product rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}{\text{ }}\left( { = \frac{1}{{2\sqrt x \sin x}} - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}} \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{2\sqrt x \sin x}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> or equivalent and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}">
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> or equivalent.</p>
<p> </p>
<p>setting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} - \sqrt x \cos x = 0">
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
<mo>−</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} = \sqrt x \cos x">
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x">
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.17">
<mi>x</mi>
<mo>=</mo>
<mn>1.17</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x \leqslant 1.17">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo>⩽</mo>
<mn>1.17</mn>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
</math></span> and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \leqslant 1.17">
<mi>x</mi>
<mo>⩽</mo>
<mn>1.17</mn>
</math></span>. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < 1.17">
<mi>x</mi>
<mo><</mo>
<mn>1.17</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.19.25.png" alt="N17/5/MATHL/HP2/ENG/TZ0/10.b/M"></p>
<p>concave up curve over correct domain with one minimum point above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis. <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> asymptotically <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi ">
<mi>x</mi>
<mo>=</mo>
<mi>π</mi>
</math></span> asymptotically <strong><em>A1</em></strong></p>
<p> </p>
<p>Note: For the final <strong><em>A1 </em></strong>an asymptote must be seen, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> must be seen on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis or in an equation.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x){\text{ }}\left( { = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}} \right) = 1">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.96">
<mi>x</mi>
<mo>=</mo>
<mn>1.96</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(1.96 \ldots )">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1.96</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1.51">
<mo>=</mo>
<mn>1.51</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi \int_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{{x{\text{d}}x}}{{{{\sin }^2}x}}} ">
<mi>V</mi>
<mo>=</mo>
<mi>π</mi>
<msubsup>
<mo>∫</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
</mrow>
<mrow>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
</msubsup>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>M1 </em></strong>is for an integral of the correct squared function (with or without limits and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span>).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2.68{\text{ }}( = 0.852\pi )">
<mo>=</mo>
<mn>2.68</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.852</mn>
<mi>π</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> measures the concentration of mercury in micrograms per litre.</p>
<p>The situation is modelled using the second order differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math> is the time measured in days since the leak started. It is known that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>If the mercury levels are greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> micrograms per litre, fishing in the river is considered unsafe and is stopped.</p>
</div>
<div class="specification">
<p>The river authority decides to stop people from fishing in the river for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> longer than the time found from the model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the system of coupled first order equations:</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi></math></p>
<p style="text-align:left;">can be written as the given second order differential equation.</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 eigenvalues of the system of coupled first order equations given in part (a).</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>Hence find the exact solution of the second order differential equation.</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>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, labelling the maximum point of the graph with its coordinates.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the model to calculate the total amount of time when fishing should be stopped.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down one reason, with reference to the context, to support this decision.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>differentiating first equation. <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>substituting in for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong><em>AG</em></strong></p>
<p><br><strong>Note:</strong> The <strong>AG</strong> line must be seen to award the final <em><strong>M1</strong></em> mark.</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>the relevant matrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> is also possible.</p>
<p><br>(this has characteristic equation) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>λ</mi><mfenced><mrow><mo>-</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER </strong></p>
<p>the general solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Must have constants, but condone sign error for the <em><strong>M1</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>M1A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to find eigenvectors <em><strong>(M1)</strong></em></p>
<p>respective eigenvectors are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> (or any multiple)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>the initial conditions become:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mo>-</mo><mi>A</mi><mo>-</mo><mn>2</mn><mi>B</mi></math> <em><strong>M1</strong></em></p>
<p>this is solved by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p>
<p>so the solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,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"> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct shape (needs to go through origin, have asymptote at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> and a single maximum; condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>). Award <em><strong>A1</strong></em> for correct coordinates of maximum.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>intersecting graph with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p>so the time fishing is stopped between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>1830</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>11957</mn><mo>…</mo></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>06</mn><mo> </mo><mfenced><mrow><mn>343</mn><mo>…</mo></mrow></mfenced></math> days <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>Any reasonable answer. For example:</em></p>
<p>There are greater downsides to allowing fishing when the levels may be dangerous than preventing fishing when the levels are safe.</p>
<p>The concentration of mercury may not be uniform across the river due to natural variation / randomness.</p>
<p>The situation at the power plant might get worse.</p>
<p>Mercury levels are low in water but still may be high in fish. <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for a reasonable answer that refers to this specific context (and not a generic response that could apply to <em>any</em> model).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates did not get this far, but the attempts at the question that were seen were generally good. The greater difficulties were seen in parts (e) and (f), but this could be a problem with time running out.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>area of segment <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times {0.5^2} \times (\theta - \sin \theta )">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = {\text{area of segment}} \times 10">
<mi>V</mi>
<mo>=</mo>
<mrow>
<mtext>area of segment</mtext>
</mrow>
<mo>×</mo>
<mn>10</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{5}{4}(\theta - \sin \theta )">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{5}{4}(1 - \cos \theta )\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008 = \frac{5}{4}\left( {1 - \cos \frac{\pi }{3}} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mn>0.0008</mn>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128{\text{ }}({\text{rad}}\,{s^{ - 1}})">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.00128</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}\theta }}{{{\text{d}}V}} \times \frac{{{\text{d}}V}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}\theta }} = \frac{5}{4}(1 - \cos \theta )">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{4 \times 0.0008}}{{5\left( {1 - \cos \frac{\pi }{3}} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>0.0008</mn>
</mrow>
<mrow>
<mn>5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128\left( {\frac{4}{{3125}}} \right)({\text{rad }}{s^{ - 1}})">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.00128</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>4</mn>
<mrow>
<mn>3125</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad </mtext>
</mrow>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Charlotte decides to model the shape of a cupcake to calculate its volume.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>From rotating a photograph of her cupcake she estimates that its cross-section passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, where all units are in centimetres. The cross-section is symmetrical in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, as shown below:</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>She models the section from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> as a straight line.</p>
</div>
<div class="specification">
<p>Charlotte models the section of the cupcake that passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> with a quadratic curve.</p>
</div>
<div class="specification">
<p>Charlotte thinks that a quadratic with a maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> and that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> would be a better fit.</p>
</div>
<div class="specification">
<p>Believing this to be a better model for her cupcake, Charlotte finds the volume of revolution about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to estimate the volume of the cupcake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line passing through these two points.</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 equation of the least squares regression quadratic curve for these four points.</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>By considering the gradient of this curve when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, explain why it may not be a good model.</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>Find the equation of the new model.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for her estimate of the volume as a sum of two integrals.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of Charlotte’s estimate.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>625</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>625</mn><mi>x</mi></math>, <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn></math>.<br>Award a maximum of <em><strong>A0A1</strong></em> if not part of an equation.</p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>975</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>.</mo><mn>56</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>.</mo><mn>7</mn></math> <em><strong>(M1)A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>974630</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>.</mo><mn>55919</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>.</mo><mn>6569</mn><mo>…</mo></mrow></mfenced></math></p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient of curve is positive at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> <em><strong>R1</strong></em></p>
<p><em><br></em><strong>Note:</strong> Accept a sensible rationale that refers to the gradient.</p>
<p><strong><br></strong><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p>differentiating or using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p>substituting in the coordinates<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>7</mn><mo>.</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>(A1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>4</mn><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>4</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>6</mn></math> <em><strong>(A1)</strong></em></p>
<p>solve to get<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>192</mn><mn>49</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>90</mn><mn>49</mn></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>490</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>.</mo><mn>92</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>84</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Use of quadratic regression with points using the symmetry of the graph is a valid method.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>7</mn><mo>.</mo><mn>5</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>490</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mfenced><mrow><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>4</mn><mrow><mn>7</mn><mo>.</mo><mn>5</mn></mrow></msubsup><msup><mfenced><mrow><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math> <em><strong>(M1)(M1) (M1)A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)(M1)(M1)A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> is omitted but response is otherwise correct. Award <em><strong>(M1)</strong></em> for an integral that indicates volume,<em><strong> (M1)</strong></em> for their part (a) within their volume integral, <em><strong>(M1)</strong></em> for their part (b)(i) within their volume integral, <em><strong>A1</strong></em> for their correct two integrals with all correct limits.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>501</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>501</mn><mo>.</mo><mn>189</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">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.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>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 1 > 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>></mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < - 1">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 1">
<mi>x</mi>
<mo>></mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_15.40.09.png" alt="M17/5/MATHL/HP2/ENG/TZ1/12.b/M"></p>
<p>shape <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-intercepts <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is symmetrical about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f( - x) = f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is not one-to-one function <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>horizontal line cuts twice <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any equivalent correct statement.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1 + \ln \left( {\sqrt {{y^2} - 1} } \right)">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = {y^2} - 1">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}(x) = \sqrt {{{\text{e}}^{2x + 2}} + 1} ,{\text{ }}x \in \mathbb{R}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{1}{{\sqrt {{x^2} - 1} }} \times \frac{{2x}}{{2\sqrt {{x^2} - 1} }}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}} = 0 \Rightarrow x = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p>which is not in the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> (hence no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = \frac{{{{\text{e}}^{2x + 2}}}}{{\sqrt {{{\text{e}}^{2x + 2}} + 1} }}">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} > 0 \Rightarrow ({g^{ - 1}})'(x) > 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
</math></span> so no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept: equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> has no solutions.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \tan x">
<mi>u</mi>
<mo>=</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate(s) of the point(s) of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, labelling these clearly on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x">
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></span>.</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>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x">
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0">
<mrow>
<msup>
<mi>u</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>u</mi>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>, giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arctan k">
<mi>arctan</mi>
<mo></mo>
<mi>k</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[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="f’(x) = 4\sin x\cos x + 14\cos 2x + {\sec ^2}x">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
<mn>14</mn>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>sec</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</math></span> (or equivalent) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.47.49.png" alt="N17/5/MATHL/HP2/ENG/TZ0/11.a.ii/M"> <strong><em>A1A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correct behaviour at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <strong><em>A1 </em></strong>for correct domain and correct behaviour for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \frac{\pi }{2}">
<mi>x</mi>
<mo stretchy="false">→</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>, <strong><em>A1 </em></strong>for two clear intersections with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and minimum point, <strong><em>A1 </em></strong>for clear maximum point.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.0736">
<mi>x</mi>
<mo>=</mo>
<mn>0.0736</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.13">
<mi>x</mi>
<mo>=</mo>
<mn>1.13</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to write <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x">
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span> only <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x = \frac{u}{{\sqrt {1 + {u^2}} }}">
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>u</mi>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></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="\cos x = \frac{1}{{\sqrt {1 + {u^2}} }}">
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = 2\sin x\cos x{\text{ }}\left( { = 2\frac{u}{{\sqrt {1 + {u^2}} }}\frac{1}{{\sqrt {1 + {u^2}} }}} \right)">
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>2</mn>
<mfrac>
<mi>u</mi>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = \frac{{2u}}{{1 + {u^2}}}">
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>u</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\sin ^2}x + 7\sin 2x + \tan x - 9 = 0">
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{u^2}}}{{1 + {u^2}}} + \frac{{14u}}{{1 + {u^2}}} + u - 9{\text{ }}( = 0)">
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<mn>14</mn>
<mi>u</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>+</mo>
<mi>u</mi>
<mo>−</mo>
<mn>9</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{u^2} + 14u + u(1 + {u^2}) - 9(1 + {u^2})}}{{1 + {u^2}}} = 0">
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>14</mn>
<mi>u</mi>
<mo>+</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>9</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> (or equivalent) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0">
<mrow>
<msup>
<mi>u</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>u</mi>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 1">
<mi>u</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 3">
<mi>u</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \arctan (1)">
<mi>x</mi>
<mo>=</mo>
<mi>arctan</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \arctan (3)">
<mi>x</mi>
<mo>=</mo>
<mi>arctan</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only accept answers given the required form.</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">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">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A point P moves in a straight line with velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> ms<sup>−1</sup> given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>8</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> at time <em>t</em> seconds, where <em>t</em> ≥ 0.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the first time <em>t</em><sub>1</sub> at which P has zero velocity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the acceleration of P at time <em>t</em>.</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 value of the acceleration of P at time <em>t</em><sub>1</sub>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 0">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> for <em>t</em> or equivalent <em><strong>(M1)</strong></em></p>
<p><em>t</em><sub>1</sub> = 0.441(s) <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( t \right) = \frac{{{\text{d}}v}}{{{\text{d}}t}} = - {{\text{e}}^{ - t}} - 16t{{\text{e}}^{ - 2t}} + 16{t^2}{{\text{e}}^{ - 2t}}">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>16</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>16</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to differentiate using the product rule.</p>
<p><em><strong>[2 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="a\left( {{t_1}} \right) = - 2.28">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2.28</mn>
</math></span> (ms<sup>−2</sup>) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A curve <em>C</em> is given by the implicit equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y - {\text{cos}}\left( {xy} \right) = 0">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy = - \frac{\pi }{2}">
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> intersects <em>C</em> at P and Q.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P and Q.</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>Given that the gradients of the tangents to <em>C</em> at P and Q are <em>m</em><sub>1</sub> and <em>m</em><sub>2</sub> respectively, show that <em>m</em><sub>1</sub> × <em>m</em><sub>2</sub> = 1.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the three points on <em>C</em>, nearest the origin, where the tangent is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt at implicit differentiation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \frac{{{\text{d}}y}}{{{\text{d}}x}} + \left( {y + x\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right){\text{sin}}\left( {xy} \right) = 0">
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for first two terms. Award <em><strong>M1</strong> </em>for an attempt at chain rule <em><strong>A1</strong> </em>for last term.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 + x\,{\text{sin}}\left( {xy} \right)} \right)\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 1 - y\,{\text{sin}}\left( {xy} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>−</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy = - \frac{\pi }{2},\,\,{\text{cos}}\,xy = 0">
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x + y = 0">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - \frac{\pi }{{2x}} - {\text{cos}}\left( {\frac{{ - \pi }}{2}} \right) = 0">
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mi>π</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> or equivalent <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - \frac{\pi }{{2x}} = 0">
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = \frac{\pi }{2}\left( {x = \pm \sqrt {\frac{\pi }{2}} } \right)\left( {x = \pm 1.25} \right)">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mo>±</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mo>±</mo>
<mn>1.25</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\sqrt {\frac{\pi }{2}} ,\, - \sqrt {\frac{\pi }{2}} } \right),\,\,{\text{Q}}\left( { - \sqrt {\frac{\pi }{2}} ,\,\sqrt {\frac{\pi }{2}} } \right)">
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>Q</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong>or</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( {1.25,\, - 1.25} \right),\,Q\left( { - 1.25,\,1.25} \right)">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1.25</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>1.25</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>Q</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.25</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mn>1.25</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>m</em><sub>1 </sub>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \left( {\frac{{1 - \sqrt {\frac{\pi }{2}} \times - 1}}{{1 + \sqrt {\frac{\pi }{2}} \times - 1}}} \right)">
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>×</mo>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>×</mo>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em>m</em><sub>2 </sub>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \left( {\frac{{1 + \sqrt {\frac{\pi }{2}} \times - 1}}{{1 - \sqrt {\frac{\pi }{2}} \times - 1}}} \right)">
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>×</mo>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<msqrt>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>×</mo>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em>m</em><sub>1 </sub><em>m</em><sub>2 </sub>= 1 <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0A0</strong> </em>if decimal approximations are used.<br><strong>Note:</strong> No <strong>FT</strong> applies.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>equate derivative to −1 <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {y - x} \right){\text{sin}}\left( {xy} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x,\,{\text{sin}}\left( {xy} \right) = 0">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p>in the first case, attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = {\text{cos}}\left( {{x^2}} \right)">
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>(0.486,0.486) <strong>A1</strong></p>
<p>in the second case, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\left( {xy} \right) = 0 \Rightarrow xy = 0">
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y = 1">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p>(0,1), (1,0) <em><strong> A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following graph shows the two parts of the curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}y = 5 - {y^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span>, and the normal to the curve at the point P(2 , 1).</p>
<p style="text-align: center;"><img 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"></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there are exactly two points on the curve where the gradient is zero.</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>Find the equation of the normal to the curve at the point P.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The normal at P cuts the curve again at the point Q. Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of Q.</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>The shaded region is rotated by 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis. Find the volume of the solid formed.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>differentiating implicitly: <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2xy + {x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 4{y^3}\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each side.</p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> then either <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> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0 \Rightarrow ">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
</math></span> two solutions for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( {y = \pm \sqrt[4]{5}} \right)">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mo>±</mo>
<mroot>
<mn>5</mn>
<mn>4</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> not possible (as 0 ≠ 5) <em><strong>R1</strong></em></p>
<p>hence exactly two points <strong><em>AG</em></strong></p>
<p><strong>Note:</strong> For a solution that only refers to the graph giving two solutions at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and no solutions for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> award <strong><em>R1</em></strong> only.</p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at (2, 1) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 + 4\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 4\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mn>4</mn>
<mo>+</mo>
<mn>4</mn>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{1}{2}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p>gradient of normal is 2 <em><strong>M1</strong></em></p>
<p>1 = 4 + <em>c</em> <em><strong> (M1)</strong></em></p>
<p>equation of normal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2x - 3">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}\left( {2x - 3} \right) = 5 - {\left( {2x - 3} \right)^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>5</mn>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>4</mn>
</msup>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{{y + 3}}{2}} \right)^2}\,y = 5 - {y^4}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.724">
<mi>x</mi>
<mo>=</mo>
<mn>0.724</mn>
</math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition of two volumes <em><strong>(M1)</strong></em></p>
<p>volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \pi \int_1^{\sqrt[4]{5}} {\frac{{5 - {y^4}}}{y}} {\text{d}}y\left( { = 101\pi = 3.178 \ldots } \right)">
<mn>1</mn>
<mo>=</mo>
<mi>π</mi>
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mrow>
<mroot>
<mn>5</mn>
<mn>4</mn>
</mroot>
</mrow>
</msubsup>
<mrow>
<mfrac>
<mrow>
<mn>5</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mi>y</mi>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>101</mn>
<mi>π</mi>
<mo>=</mo>
<mn>3.178</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \int {{x^2}} {\text{d}}y">
<mi>π</mi>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</math></span>, <em><strong>A1</strong></em> for limits, <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{{5 - {y^4}}}{y}}">
<mrow>
<mfrac>
<mrow>
<mn>5</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mi>y</mi>
</mfrac>
</mrow>
</math></span> Condone omission of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> at this stage.</p>
<p>volume 2</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}\pi \times {2^2} \times 4\left( { = 16.75 \ldots } \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>16.75</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong> <em>(M1)(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \pi \int_{ - 3}^1 {{{\left( {\frac{{y + 3}}{2}} \right)}^2}} {\text{d}}y\left( { = \frac{{16\pi }}{3} = 16.75 \ldots } \right)">
<mo>=</mo>
<mi>π</mi>
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
<mn>1</mn>
</msubsup>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>16.75</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>total volume = 19.9 <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> satisfies the conditions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = - 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and its second derivative is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>15</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \int {\left( {15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}} \right)} \,{\text{d}}x = 10{x^{\frac{3}{2}}} - \frac{1}{{x + 1}}\left( { + c} \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>15</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong></em> for first term, <em><strong>A1</strong></em> for second term. Withhold one <em><strong>A1</strong></em> if extra terms are seen.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \int {\left( {10{x^{\frac{3}{2}}} - \frac{1}{{x + 1}} + c} \right)} \,{\text{d}}x = 4{x^{\frac{5}{2}}} - {\text{ln}}\left( {x + 1} \right) + cx + d">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mi>x</mi>
<mo>+</mo>
<mi>d</mi>
</math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Allow FT from incorrect <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> if it is of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = A{x^{\frac{3}{2}}} + \frac{B}{{x + 1}} + c">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mi>B</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</math></span>.</p>
<p>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left| {x + 1} \right|">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>|</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p>attempt to use at least one boundary condition in their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 4">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 4">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 = 4 - {\text{ln}}\,2 + c - 4">
<mn>0</mn>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>+</mo>
<mi>c</mi>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = {\text{ln}}\,2\left( { = 0.693} \right)">
<mi>c</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0.693</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4{x^{\frac{5}{2}}} - {\text{ln}}\left( {x + 1} \right) + x\,{\text{ln}}\,2 - 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Xavier, the parachutist, jumps out of a plane at a height of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> metres above the ground. After free falling for 10 seconds his parachute opens. His velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\,{\text{m}}{{\text{s}}^{ - 1}}">
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds after jumping from the plane, can be modelled by the function</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(t) = \left\{ {\begin{array}{*{20}{l}} {9.8t{\text{,}}}&{0 \leqslant t \leqslant 10} \\ {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }},}&{t > 10} \end{array}} \right.">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>9.8</mn>
<mi>t</mi>
<mrow>
<mtext>,</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>t</mi>
<mo>></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>His velocity when he reaches the ground is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.8{\text{ m}}{{\text{s}}^{ - 1}}">
<mn>2.8</mn>
<mrow>
<mtext> m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find his velocity when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 15">
<mi>t</mi>
<mo>=</mo>
<mn>15</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the vertical distance Xavier travelled in the first 10 seconds.</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>Determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(15) = \frac{{98}}{{\sqrt {1 + {{(15 - 10)}^2}} }}">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(15) = 19.2{\text{ }}({\text{m}}{{\text{s}}^{ - 1}})">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>19.2</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{10} {9.8t\,{\text{d}}t} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>10</mn>
</mrow>
</munderover>
<mrow>
<mn>9.8</mn>
<mi>t</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 490{\text{ }}({\text{m}})">
<mo>=</mo>
<mn>490</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>m</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }} = 2.8">
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2.8</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 44.985 \ldots {\text{ }}({\text{s}})">
<mi>t</mi>
<mo>=</mo>
<mn>44.985</mn>
<mo>…</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>s</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 490 + \int\limits_{10}^{44.9...} {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }}{\text{d}}t} ">
<mi>h</mi>
<mo>=</mo>
<mn>490</mn>
<mo>+</mo>
<munderover>
<mo>∫</mo>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>44.9...</mn>
</mrow>
</munderover>
<mrow>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 906{\text{ (m}})">
<mi>h</mi>
<mo>=</mo>
<mn>906</mn>
<mrow>
<mtext> (m</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\arcsin (x - 1) - \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>, 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="y = \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a definite integral to represent the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\arcsin (x - 1) - \frac{\pi }{4} = \frac{\pi }{4}">
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1 + \frac{1}{{\sqrt 2 }}\,\,\,( = 1.707 \ldots )">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>1.707</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{1 + \frac{1}{{\sqrt 2 }}} {\frac{\pi }{4} - \left( {2\arcsin \left( {x - 1} \right) - \frac{\pi }{4}} \right)dx} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</munderover>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for an attempt to find the difference between two functions, <strong><em>A1 </em></strong>for all correct.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0,{\text{ }}y = \frac{{ - 5\pi }}{4}\,\,\,( = - 3.93)">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mo>−</mo>
<mn>3.93</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1 + \sin \left( {\frac{{4y + \pi }}{8}} \right)">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>y</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for an attempt to find the inverse function.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{\frac{{ - 5\pi }}{4}}^{\frac{\pi }{4}} {\left( {1 + \sin \left( {\frac{{4y + \pi }}{8}} \right)} \right){\text{d}}y} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>y</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{1.38...} {\left( {2\arcsin \left( {x - 1} \right) - \frac{\pi }{4}} \right){\text{d}}x} \left| + \right.\int\limits_0^{1.71...} {\frac{\pi }{4}{\text{d}}x - \int\limits_{1.38...}^{1.71...} {\left( {2\arcsin \left( {x - 1} \right) - \frac{\pi }{4}} \right)dx} } ">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>1.38...</mn>
</mrow>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mo>|</mo>
<mo>+</mo>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>1.71...</mn>
</mrow>
</munderover>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mrow>
<mn>1.38...</mn>
</mrow>
<mrow>
<mn>1.71...</mn>
</mrow>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mrow>
</math></span> <strong><em>M1A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for considering the area below the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and <strong><em>A1 </em></strong>for each correct integral.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area}} = 3.30{\text{ (square units)}}">
<mrow>
<mtext>area</mtext>
</mrow>
<mo>=</mo>
<mn>3.30</mn>
<mrow>
<mtext> (square units)</mtext>
</mrow>
</math></span> <strong><em>A2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\left( {x - 1} \right)^2}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 1 and the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^2} + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is bounded by the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 9">
<mi>y</mi>
<mo>=</mo>
<mn>9</mn>
</math></span> as shown on the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The shape of a clay vase can be modelled by rotating the region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> through 360˚ about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<p style="text-align: left;">Find the volume of clay used to make the vase.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \pi {\int_0^9 {\left( {{y^{\frac{1}{2}}} + 1} \right)} ^2}{\text{d}}y - \pi \int_1^9 {\left( {y - 1} \right)} {\text{d}}y">
<mo>=</mo>
<mi>π</mi>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>9</mn>
</msubsup>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
<mo>−</mo>
<mi>π</mi>
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mn>9</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</math></span> <em><strong>(M1)</strong></em><em><strong>(M1)</strong></em><em><strong>(M1)(A1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for use of formula for rotating about <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis, <em><strong>(M1)</strong></em> for finding at least one inverse, <em><strong>(M1)</strong></em> for subtracting volumes, <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em>for each correct expression, including limits.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 268.6 \ldots - 100.5 \ldots \left( {85.5\pi - 32\pi } \right)">
<mo>=</mo>
<mn>268.6</mn>
<mo>…</mo>
<mo>−</mo>
<mn>100.5</mn>
<mo>…</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>85.5</mn>
<mi>π</mi>
<mo>−</mo>
<mn>32</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 168\left( { = 53.5\pi } \right)">
<mo>=</mo>
<mn>168</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>53.5</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> is defined by equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy - \ln y = 1,{\text{ }}y > 0">
<mi>x</mi>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>></mo>
<mn>0</mn>
</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="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> in terms 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>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equation of the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> e</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + x\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{1}{y}\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>y</mi>
</mfrac>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for the first two terms, <strong><em>A1 </em></strong>for the third term and the 0.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{y^2}}}{{1 - xy}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mi>y</mi>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - {y^2}}}{{\ln y}}">
<mfrac>
<mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mi>y</mi>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - y}}{{x - \frac{1}{y}}}">
<mfrac>
<mrow>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>y</mi>
</mfrac>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_T} = \frac{{{{\text{e}}^2}}}{{1 - {\text{e}} \times \frac{2}{{\text{e}}}}}">
<mrow>
<msub>
<mi>m</mi>
<mi>T</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_T} = - {{\text{e}}^2}">
<mrow>
<msub>
<mi>m</mi>
<mi>T</mi>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - {\text{e}} = - {{\text{e}}^2}x + 2{\text{e}}">
<mi>y</mi>
<mo>−</mo>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {{\text{e}}^2}x - y + 3{\text{e}} = 0">
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 7.39x + 8.15">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7.39</mn>
<mi>x</mi>
<mo>+</mo>
<mn>8.15</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An object is placed into the top of a long vertical tube, filled with a thick viscous fluid, at time <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> seconds.</p>
<p>Initially it is thought that the resistance of the fluid would be proportional to the velocity of the object. The following model was proposed, where the object’s displacement, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, from the top of the tube, measured in metres, is given by the differential equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>9.81</mn>
<mo>−<!-- − --></mo>
<mn>0.9</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The maximum velocity approached by the object as it falls is known as the terminal velocity.</p>
</div>
<div class="specification">
<p>An experiment is performed in which the object is placed in the fluid on a number of occasions and its terminal velocity recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9{\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)^2}">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>9.81</mn>
<mo>−<!-- − --></mo>
<mn>0.9</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>At terminal velocity the acceleration of an object is equal to zero.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{{{\text{d}}x}}{{{\text{d}}t}}"> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span> into the equation, find an expression for the velocity of the particle at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = f(t)"> <mi>v</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></span>.</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>From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.</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 differential equation as a system of first order differential equations.</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>Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0.6"> <mi>t</mi> <mo>=</mo> <mn>0.6</mn> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the differential equation to find the terminal velocity for the object.</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>Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}t}} = 9.81 - 0.9v"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mi>v</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{1}{{9.81 - 0.9v}}} {\text{d}}v = \int 1 \,{\text{d}}t"> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mi>v</mi> </mrow> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> <mo>=</mo> <mo>∫</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{{0.9}}{\text{ln}}\left( {9.81 - 0.9v} \right) = t + c"> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mn>0.9</mn> </mrow> </mfrac> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>t</mi> <mo>+</mo> <mi>c</mi> </math></span> <strong><em> A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9.81 - 0.9v = A{{\text{e}}^{ - 0.9t}}"> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mi>v</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>0.9</mn> <mi>t</mi> </mrow> </msup> </mrow> </math></span> <strong><em> A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{{9.81 - A{{\text{e}}^{ - 0.9t}}}}{{0.9}}"> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mn>9.81</mn> <mo>−</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>0.9</mn> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mn>0.9</mn> </mrow> </mfrac> </math></span> <strong><em> A1</em></strong></p>
<p>when <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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 0"> <mi>v</mi> <mo>=</mo> <mn>0</mn> </math></span> hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 9.81"> <mi>A</mi> <mo>=</mo> <mn>9.81</mn> </math></span> <strong><em> A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{{9.81\left( {1 - {{\text{e}}^{ - 0.9t}}} \right)}}{{0.9}}"> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mn>9.81</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>0.9</mn> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>0.9</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 10.9\left( {1 - {{\text{e}}^{ - 0.9t}}} \right)"> <mi>v</mi> <mo>=</mo> <mn>10.9</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mn>0.9</mn> <mi>t</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em> A1</em></strong></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>either</strong> let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> tend to infinity, or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}t}} = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </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="v = 10.9"> <mi>v</mi> <mo>=</mo> <mn>10.9</mn> </math></span> <strong><em> A1</em></strong></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = y"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>y</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{dy}}}}{{{\text{d}}t}} = 9.81 - 0.9{y^2}"> <mfrac> <mrow> <mrow> <mtext>dy</mtext> </mrow> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </math></span> <strong><em> A1</em></strong></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x_{n + 1}} = {x_n} + 0.2{y_n}"> <mrow> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mi>x</mi> <mi>n</mi> </msub> </mrow> <mo>+</mo> <mn>0.2</mn> <mrow> <msub> <mi>y</mi> <mi>n</mi> </msub> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y_{n + 1}} = {y_n} + 0.2\left( {9.81 - 0.9{{\left( {{y_n}} \right)}^2}} \right)"> <mrow> <msub> <mi>y</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo>=</mo> <mrow> <msub> <mi>y</mi> <mi>n</mi> </msub> </mrow> <mo>+</mo> <mn>0.2</mn> <mrow> <mo>(</mo> <mrow> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>y</mi> <mi>n</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.04"> <mi>x</mi> <mo>=</mo> <mn>1.04</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 3.31"> <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>3.31</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>3.3015 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 = 9.81 - 0.9{\left( v \right)^2}"> <mn>0</mn> <mo>=</mo> <mn>9.81</mn> <mo>−</mo> <mn>0.9</mn> <mrow> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow v = \sqrt {\frac{{9.81}}{{0.9}}} = 3.301511 \ldots \,\,\left( { = 3.30} \right)"> <mo stretchy="false">⇒</mo> <mi>v</mi> <mo>=</mo> <msqrt> <mfrac> <mrow> <mn>9.81</mn> </mrow> <mrow> <mn>0.9</mn> </mrow> </mfrac> </msqrt> <mo>=</mo> <mn>3.301511</mn> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>3.30</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the model found the terminal velocity very accurately, so good approximation <em><strong>R1</strong></em></p>
<p>intermediate values had object exceeding terminal velocity so not good approximation <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="question">
<p>An earth satellite moves in a path that can be described by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="72.5{x^2} + 71.5{y^2} = 1"> <mn>72.5</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>71.5</mn> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>1</mn> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = x(t)"> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = y(t)"> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></span> are in thousands of kilometres and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> is time in seconds.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}"> <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>7.75</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3.2 \times {10^{ - 3}}"> <mi>x</mi> <mo>=</mo> <mn>3.2</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span>.</p>
<p>Give your answers in standard form.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>substituting for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> and attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> (or vice versa) <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = ( \pm )0.11821 \ldots "> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> <mn>0.11821</mn> <mo>…</mo> </math></span> <strong><em>(A1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="145x + 143y\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0{\text{ }}\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{{145x}}{{143y}}} \right)"> <mn>145</mn> <mi>x</mi> <mo>+</mo> <mn>143</mn> <mi>y</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>145</mn> <mi>x</mi> </mrow> <mrow> <mn>143</mn> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="145x\frac{{{\text{d}}x}}{{{\text{d}}t}} + 143y\frac{{{\text{d}}y}}{{{\text{d}}t}} = 0"> <mn>145</mn> <mi>x</mi> <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>143</mn> <mi>y</mi> <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> <strong><em>M1A1</em></strong></p>
<p><strong>THEN</strong></p>
<p>attempting to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}}{\text{ }}\left( {\frac{{{\text{d}}y}}{{{\text{d}}t}} = - \frac{{145(3.2 \times {{10}^{ - 3}})}}{{143\left( {( \pm )0.11821 \ldots } \right)}} \times (7.75 \times {{10}^{ - 5}})} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mn>145</mn> <mo stretchy="false">(</mo> <mn>3.2</mn> <mo>×</mo> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>143</mn> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> <mn>0.11821</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>×</mo> <mo stretchy="false">(</mo> <mn>7.75</mn> <mo>×</mo> <mrow> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = \pm 2.13 \times {10^{ - 6}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>±</mo> <mn>2.13</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award all marks except the final <strong><em>A1 </em></strong>to candidates who do not consider ±.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = ( \pm )\sqrt {\frac{{1 - 72.5{x^2}}}{{71.5}}} "> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> <msqrt> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>72.5</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>71.5</mn> </mrow> </mfrac> </msqrt> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = ( \pm )0.0274 \ldots "> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> <mn>0.0274</mn> <mo>…</mo> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = ( \pm )0.0274 \ldots \times 7.75 \times {10^{ - 5}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo stretchy="false">(</mo> <mo>±</mo> <mo stretchy="false">)</mo> <mn>0.0274</mn> <mo>…</mo> <mo>×</mo> <mn>7.75</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = \pm 2.13 \times {10^{ - 6}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mo>±</mo> <mn>2.13</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award all marks except the final <strong><em>A1 </em></strong>to candidates who do not consider ±.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2\,{\text{ln}}\,x + 1}}{{x - 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 3.</p>
</div>
<div class="specification">
<p>Draw a set of axes showing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values between −3 and 3. On these axes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the coordinates of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f^{ - 1}}\left( x \right)"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{\frac{{2\left( {x - 3} \right)}}{x} - \left( {2\,{\text{ln}}\,x + 1} \right)}}{{{{\left( {x - 3} \right)}^2}}}\left( { = \frac{{2\left( {x - 3} \right) - x\left( {2\,{\text{ln}}\,x + 1} \right)}}{{x{{\left( {x - 3} \right)}^2}}}} \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mi>x</mi> </mfrac> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>x</mi> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt at quotient rule, <em><strong>A1A1</strong></em> for numerator and <em><strong>A1</strong></em> for denominator.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \left( {2\,{\text{ln}}\,x + 1} \right){\left( {x - 3} \right)^{ - 1}}"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \left( {\frac{2}{x}} \right){\left( {x - 3} \right)^{ - 1}} - \left( {2\,{\text{ln}}\,x + 1} \right){\left( {x - 3} \right)^{ - 2}}\left( { = \frac{{2\left( {x - 3} \right) - x\left( {2\,{\text{ln}}\,x + 1} \right)}}{{x{{\left( {x - 3} \right)}^2}}}} \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>x</mi> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for attempt at product rule, <em><strong>A1</strong></em> for first term, <em><strong>A1</strong></em> for second term.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>finding turning point of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> or finding root of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f''\left( x \right)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.899"> <mi>x</mi> <mo>=</mo> <mn>0.899</mn> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {0.899048 \ldots } \right) = - 0.375"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mn>0.899048</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>0.375</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>(0.899, −0.375)</p>
<p><strong>Note:</strong> Do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.9"> <mi>x</mi> <mo>=</mo> <mn>0.9</mn> </math></span>. Accept <em>y</em>-coordinates rounding to −0.37 or −0.375 but not −0.38.<br> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>smooth curve over the correct domain which does not cross the <em>y</em>-axis</p>
<p>and is concave down for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> > 1 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercept at 0.607 <em><strong>A1</strong></em></p>
<p>equations of asymptotes given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 3 (the latter must be drawn) <em><strong>A1A1</strong></em><br> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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RDwZN+3oqJibNoaCW1hB26TptenvCFiGvtLLNLSUlH1FGD/nAOCx9S8bGdD4hsjN3Dl/If/EBY2HjY2tlyZRw4dQco/Kdy5ra0NJk+umaXG2m7FF3Uv4ieEhYkcTMTGxODEqVNcHktLC0yfHs6ds3Zga7Bqj7HBwSI+bLuo5KQkLsrQ0LDBGqtlEUtQiSo8ePAAQSOHw9evZrcMtlfjT/v2cXnYu/RVa9bUisbSiAjcvXuXux46fDjYHzvOXTiHLd//IEr3+erVoqnla9Z+heuZNfdP/wEDMD4sjEvHhpLWffmVKM/Hn3wqYhq5aRNOJZ9AeUUlvLy9ET59Kpcuv6AQc+fUrRNbtmwJbG1rmG6I3ISkhEQunVsvF8ydV5OOtd2779ZtpOvi7Ij5CxZw6R6WluJ/b9U8qGZmpmHVqtUifbZu34nPPl2MvNx8zg+ssKKSc6TOdoB4bcJEVFZW4PbN28jIuAAHBzsu39bt27Fvzy7W50TXrl2wadMmLpy9Dx01ajQEAi0wH7PhM2aKZqjv378fO3bs4MJZG+3du1+kQ9iEMJSXlUMorMLrr7+G0Ccu9RLiE/DFl19y664FAh1ER0eLZuVOmjwZmZk1noVeDpuAOeE19wn7Tr47611OtkALOBR1SPTDO2fOHJw4VfPdCx4+EvOe8MnKysbESTXfQ5Zn+9a9sLWz4WRERCxB3NFY7tzP1w9Ln9yDhfkFGB06RlTOt9/+gB49apYsLImIwE8H9gEVQN+X+mFT5GYuXUlxMQa+1A+o0AP0hPjuu+/h5VWz6fbyiC+w60BNOkd7R+yPrplUVVpRgQEe7pws6AFsDWFAQM39uHrtV9iyPhIs/OmuXREbV7MQn7WDq2sPrkymw8efLUZoaM1vU2TkRixaFMHFmRh1xJWMur1K+7i7Iy8/n4tbtHAhpj9heig6Cm+9XdfRuZ6bK/r9GTEsEGfPXeDyTAsPR8STdfTJyUkICalzjXj5wgVRO0yeNAmxcXFcngnjJmDV2lXc+Y2sbO7+6NLtGc4xxqXLV2BvV3PfT5s2HTv2bINWJTB8bDD27qx5lXevoAgvvPgcl5/dP8eOHYO7uzt33V7/5DKW0hpKVrmKhw9x++5dOJrUeOXZsWtfzY0HYMjgISJjyYzKO/V22d730z6Ym9d0xX86eJD7YjJ5/fr2FhnLykePuB+UDto1xvKH73+Apb8fx/TA4cPYEhnJnTs97yAylszUbNy4AULmUZqt57G0FBnLf86cwebtWyGsAqwsLBoYy81bt+JBRQXYl8/AQEdkLNPTL2H9xo0QVAmho2fYwFju2bOP8zLB8kydMkVkDLKyc0U6sHzjnvwQM31SU9OQmpbG2XJTY2P4+tTUp6ykDMeOn6y18WCGD6gxllnXMnEy+RjYZkwCIx0E4MmyHC0hjh+ryVMlFGLchFdgjppF6Nk3spF4PJmT9/jRI5GxrEIVjp88CbYNHXueCAkN4Tixfw/LH+L27bvAf1UwrLduStihA+7m3al5mOighcdVj+ryPKrE7bw7QActaNXb247NlGMPBdCuaYeK8pqHCpax8mGFqJzO5hUiWezkzr17qKgUolMXQ+gZGIji2Llp587cNTOW9Q8TMzPRg079PB119UV5WPr6U9xNjI1FcSb16mqgV1cOy6NTr6xOxsawtbMFK/+Zp7uKVDDU04OLS93DZX0dHJ62RsmTOJa39jDQ02mQhz2c1B5a+lqiOENDHQQGDsbdO3fR0ciIcz7wxhtvAjpaMDcywdtPfhRZ3QYM6A9t7Q7IzcnF/ftlteLg7OSE4U8eIIwNDUXhLM/w4XVLvJ7t9owoztHJCQEB/twEI8ak/sGm9bOHGXbY2dUYZHZubW3VQF593iFjRqO4uIRzXWJf70HcxtYWU6ZMBVuqxu5hA7M6JwpBI4PQq1evJ+XUPPCyiy6dO9et066qQqdOplwa9m94gD/snnC2tq5jamhsKCqH3fwWljU9IZYnKDgIzk/ayNzCUiRLz8gIixd/KrqufehmASNfDoSDSzcuzszMXJSmo44OFi9eLLp2cnIWnQ8dMgS2zzzNXXfUq/NixjjVlFPzuO7uXjdEOXJkEJwcnbgHI/YAUv/49ttvUVlZycXVPvCyeE9vH0RF7UWVEFwc8+Fae3z66QpUVD7g4qys67w7OTm7YP/evVx6lq9+O8ydPQ9Tpkzh4jpZdqoVhUsXznHna1atgZ+fL56x7SKKi1gSgVlPHoJYZ6b26GRpjuRTKTUubLQEePYJj9r49vhUyASf9lCcymyeAHsKrf8j1HxqilUHAj169MCPP26Bh0fdj2htvfT19XEt4xqsbOp++BbMmwc2QjN//vzaZPRJBBROIHBYIPr37YsPF0WIeq4KL6QNBNY9SrRBYVRE2xEgQ9l2rFWlJMuuXeD0xK9yY52EaOhIncVbWFmiIK+wcVK6JgIKJfDbr78pVF57CSNj2V7kqVwiICeB9EtpmPfBPAwNGo5HpQ/xbvgM0ZyAxqIF3NvqhqEGBoYoelj3bqthLF0RgdYTSD5xAm4uLiq1xVbra1OTUyFOCeRVgvITASIgOwHHHk5Y/fXX8PX1x/Tp0zF6TM3EFHGShGi46whLw96h3r97R1xyCiMCrSbAXJUGDhmCWDZ5EQCbxLd8+XLus9VCVSAjGUsVaARSgQi0loC9vR2cHB1afIJnPUs2FFv/YNtyZec03d6ofho6JwKyEGAbaowYNgKrV3+NMaNHcVnZJL4PP/wQbCImnw8ahuVz65HuREBKAuJ6lg5OjsgvyJVSAiUjAi0TYDOdv/v2O0yYWLd07t8ns99bdMnYsvh2TUHGsl3xU+FEoG0IiOtZdrW2hrC8EmydIFtWQwcRkJcAWzZTf+lMrTwdtm6O5wf/a8DzBiD1iUBbEBDXs2Tr6oxNjXH73r22UIHKUFMCCYkJWBSxSGLt2H32+N//GjhSl5hYhSPIWKpw45BqREBRBMT1LJlsIxMTlBTR8hFFcdY0OcxQjhoxCjbWdU4n1JUBGUt1bVmqFxGoR0Bcz5JFG5uaI5/2jK1Hik6lJcBcJ4aODuEm80ydOlnabLxNR+8sedt0pDgRkJ6ApJ6lvc2zuHWblo9IT5JS1hJgO95cuXYV5i2872YTe0xMjHDj2nWRH9laGXz6pJ4ln1qLdCUCrSTAepbiDht7axQV0DCsODYUJp5ASUndhs0tGcpaCcznN/O1zedD/DeIzzUi3YkAEWhCQCDBWFpZWCEzJ6dJegogAuIIHD0aj66du6C+wRSXrn5Y7dKR+mF8PCdjycdWI52JgIwEhKgUm8PU2BDlD+6LjaNAIlCfAPPMM2rUMHzz7TcyzWxls2HZDiv1d9mpL5cv5/TOki8tRXoSATkICKDD7SXYWATbfik3l7z4NOZC100JnEhOxmeffY4pU2v2Z22aQnJIRETNXpuSU6h+DBlL1W8j0pAIyE2A9Swbu7tjQi0sbVBIS0fk5qsJAubN0+yt3GgYVhPucqqjxhNgPUtxxzN2XVH24AEePXwoLprCNJzA0fh4nLuQKhcFNht27doveX+PkbGU6zagzESAHwQkvbM00jGCnp4ebt25y4+KkJZtRoBN5hkVOAzHjyXKXeasWXPBZsTy+SBjyefWI92JgJQEJL2zZNltrK2Rk5UupSRKpgkEUlJSMGxYIFZ+vYbb/k0T6txSHemdZUuEKJ4IqAEBSe8sWdXMLSyQX1SsBrWkKiiKQJ8+fbBnzy6EhITKLVJdlo6QsZT7ViABRED1CUh6Z8k0N7XpirL7tHxE9VuxbTVUhKFkGrOlI9XV1W2rvBJKo2FYJUAlkURA1QhIemfJ9LTrbI2iwiJVU5n0aWMCzCm6r58fHguFbVwyP4ojY8mPdiItiYBcBJrrWVpYmiMrJ1su+ZSZ3wTYrNdhgYEIGjmS6wnyuzbK0Z6MpXK4klQioGIEJPcWHOztkZku3/IAFassqSMjgVlvv4vlK1dhzpw5MuZsOTnrqVpYmKO4kN8+iOmdZcttTSmIAO8JSHKkzipm2qkL7t2hd5a8b2Q5KnDm/Bno6urKIUFyVvbO8u7de+RIXTIiiiECREBVCLAtuiQdlhaWKPm3VFI0haspgUupdaMJyjKUDJ26vAOlYVg1/SJQtYhAfQLN9Sy7WFtC+KCC9x5W6teXzpsnwJyi9+3jjvj4o80nVECsujhSJ2OpgJuBRBABVScgaYsupjfbxNfE2Ayp6ZmqXg3STwEEkpKSEDhkCOcU3c/PXwESWxbBHKkbGXVsOaEKpyBjqcKNQ6oRAcURkDzBh5Vh+6ItcjLJi4/ieKuuJObeUFmTeVS31vJrRsZSfoYkgQioPIHmhmGZ8t26Po+75JhA5dtREQoy7zwzZ8xQhCipZLB3lhs3buT9MD8ZS6mamxIRAX4TaG6CD6uZkbk+ysvL+F1J0l4iAebr9VB0tMR4ZUawd5ZvvfUWKsrJkboyOZNsIkAEFECgpZ5l506dUVxI/mEVgFrlRDBDyd5RZmW3j+OJ2tmwQskTslWOmTiFqGcpjgqFEQE1I9DcBB9WVVsbG/yTdl7Nak3VKS4p4QzlRx9/rBSHA9IQ1tZSDzOj0k4JsrLY7DwBnrbtAn0Bv2dSSXNTURoioDwCzU/wsbdzQPoFmuCjPP7tI9nMxAR//PUXXHv2bB8FAGhpaamFI3WVNpZRBw9i4zff4b/yCri49YR9d0c87+CAwQN90c3uWXTsSAa03b4BVDCvCLQ0DGtra4eHFSW8qhMpK5kAG/pk7wrZ0Z6GUrKG/It5qpoHe6fk5OQgMTEB9+/fR2JCIpJSkoDyKjg6OcPH2xvWtjZ4+eVxMDQxEt0g/GsK0pgIKI+Avr42MjKuw8bGWmwhpRUVeN7GFgWFBWLjKZA/BJJPnEBoSAhSzp6DlaUFfxRXcU15YSzFMWRj8UcOHkDunTs4djQBKZdS0FFLHyOCgvHCi/YYGhAAB0cncVkpjAhoHAFtfV1cy7gm0VgyIJbWFkhOPAV7ezuN46MuFWaTeQYPHoyly5djxvTpKlEt1st1euEFnDl5EmYW/DXevDWWje8C1iDply4h5vffkHbpMv6K/xtaWlXo79Ufvr4+8BngC8ceZDwbc6NrzSCgr6+PjIyrzRpLb29vzJ49GyEhIZoBRQ1rOXnyZLi6umDmTMXvHtJaXFVVVejQoQPu3L4DCx73dFX6naUsjcPG53u6unJ/LB9nPK9k4PjffyI+PhGfr/wCBro6GDBgIPwD/OHp7QMbKytZiqC0RIDHBJqf4MMq5uTkjOzsLB7XkVTfvHkzQVASAbUxlo35cMazR3f07NEd06aHc8bzdEoKkhMS8MPG7zHrnXfh5OwMTy9vBAcHwcXVld53NoZI12pDoKUJPqyiVlYWqBS2bFTVBoqaVIS9o3RxcoaRiZFK1ojNhl28eDH0DAxUUj9plVKPBTBS1JYZT29PT8ybPx+/Hf0DlzIyMD08HOUPyhA2fjxesLPD9OnTsXP3dpSU0KxAKZBSEh4RaMmDD6uKjo4BKsrIiw+PmhXMUA4LDMSBgwdUWm1ypK7SzdO8cmz9EXs3s/rr1cjMykJ09CF07/4i1n25Hi+88DxCxodg1covUVhY1LwgiiUCPCAgTc/S0soC6Zk0DMuD5uRUZJN5mKEMnz4NkyZN4ovavNVTY3qWLbWQex93zJgxEydSTuBKxlX4efvhn/Nn0bPni/Ds0wdr1qzBFdqVoSWMFK+iBFry4MPUdnNzw5kTp1W0BqRWYwJ2dvZYtmwZli1f2ThKpa7ZBJ+oqL28d6SuNrNhlXV3lJY+RMyRw/g97i/8EnsIttbWCB47FgH+AXD36KOsYkkuEVAogZbWWbLCSh8+xPPdnsX1gnzoP1nQrlAlSJjGEnjqqadw48bNZmdjqzoc6lm20EJsw9LxYWH4YfN3uJ53DYsiluD69RsY82oI1+NctCgCF86da0EKRROB9iUgzTCsQFsbzI2n8N+H7asslS6RADrKu1IAACAASURBVBt6nfa//0mMV+UIaUY3VFl/MpYytI6ulj6GjxyO777/DhkXM7AoYhEK7t7C2LFj4eXliaVLliLzCu02LwNSStpGBKSZ4KOvo8Npk5Od10ZaUTGyEGCTeQYNGgTLrp1lyaYyaYXg97YjNAyrgFuJjclHH4pGbGwMYo4c4dzwjQwYirBJk8ndlAL4kgj5CUjjwYeV4u83BNPD3yLHBPIjV6gEtm68m7UVpk2biiVLlilUNgmTjgAZS+k4SZ2KGc6d27fiYGwMkuIS4OvngwD/oXh1ymRaxyk1RUqoaALSePBhZS5auAgGxoaYP2+eolUgeXISyMnOga2drZxSKHtrCdAwbGvJScjHFuBOnDQZUbv34+q1a/Dx9cX2vbvxnI01pk+bhpjYWDCDSgcRUAQB1uOIjo7C3r27WxAnnbMBM3MTPLh7twVZFN1WBEpLSkVFkaEUoWiXEzKWSsTOPGqEh89AQnw8EhKSuN1R5s2bC1en7li0cCEuX7qgxNJJtCYQeGPCRMQcicVnn69Az56unKcqcfWWZoIPy8cm+BSWklMOcQzbOoxN5nn6madx6tSJti5aoeWxBzovby8U5vN7Rxsylgq9LSQLc3B0wMIFC3HpwiV8Hfkd7t67h8DAEfAN8MWmjZEoKiqWnJliiIAYApevXMGHixYiclMkNyO7vPwBjiXEi0nJtlCXbjSjc+encSP9ulgZFNh2BGp3D3lv9mx4eHi2XcFKKIl5Tzt+7DgqeO5KkYylEm6OlkT6+vji22+/xeXLlzFtynTsPrQfPbu/iKmTJuFoUoLE3kFLcileswh0f/FF9HB2FlW6l5sbbGzqttdibhuLCwu5P6FQuq+6p0cfXMq6JJJJJ+1DIDMzEzNnhiNiyZL2UUAJpfJ96QhN8FHCTdEakczl3v69+7Et8nsYGhkj6OWxmDptOs2mbQ1MDczDhrrGjBmDmJ9/FtU+MHAw4uLiAQFbPFkl9aJwaytLXL6SARMTE5EsOiEC8hBgvmFnzp4N5maUrwcZSxVsucNHDmNv1F4kHD4Kv6H+CBkfiuCgYBXUlFRSFQJLl0Rg5sz3JO48IY0Hn9q62DvaI/bIUTg41PVSa+PoU3kE2NArm/zn4eGhvEJIcqsJSDc202rxlLE1BIJGBmHn5p24fO0qXHo5Y9GHC/F89xewatly5Ofnt0Yk5VFjAgcOHsT48WESDWVN1aXfjc9M3wi5edlqTEz1qsYMJXM4EBt7RPWUI404AmQsVfhGMDExwrx5C3D6/Dl89/V3uJieij7urggNDkEcLUFR4ZZrO9XWb9iA0uJiFBYVIP5oHPbu3iuhcOmWjrDMDk4uyEpPlyCHghVNgG24PXjwYLDJPIsWqc87yvqckpMT8fjx4/pBvDuX/nGTd1VTH4XZbDJ/fz/uL7+wELs2bcHc+fMgmD8PL7/6f5g8YRKsrKzUp8JUE6kIsCUFzMk/m7wjELDZrgK8/957EvJK/1V36eWE3DxyeScBpMKD2e4hu3Ztw4gRoxUuWxUEsjtzwICXpH5nrgo6i9OB3lmKo8KTsJgjMdi8eROOJx3HsOGBmDptGjw8+T3NnCfoeaemLO8sN2zYgAvnLmHDxvW8qyefFGbvJ5kTE0042K4jt27cgpUNfx/qNaOl1PRuZE7d90dF4diJ47Czd0BIaAg8vbywceNGWn6ipm3eFtXqZGqM27dutkVRGlvGuQupCAwM5P0ej7I0IN8dqZOxlKW1VTQtc4O1YOEC5ObmITx8Ovbs3IVnHWyxaMFC5ObmqKjWpFbbEpB+GNbO3hFpeVltq54GlXb58iWMCPRH3359od+xo0bUvLq6mtd7WbJGomFYNb1Vz6acwg8//oiD+w9jgHc/TAsPh7+/v5rWlqrVEgFZhmEfPnyI57o9ixsF+eT8vyWwrYhn30MvLw/aPaQV7NozCxnL9qTfBmUXFhVjY+RG7NmyDcamxpg8ZTJeGfd/LSwzaAPFqIg2JSCLsWSKOTjY4/CRWDg5OrSpnppQGJsVqqurqwlVVas60jCsWjVn08pYmJth4fz5OHP5DGa/9z727P4J7u49sGDhQuTk5jbNQCFqSkD6YVgGwMzMHPm5tNZSUTfDpdRUkShNM5Scd6mQEHKkLroD6ESlCehq6SM0NAR/xP+OrTt3Izc3F54e7pg0aSJOnTql0rqTcoogIP06S1aanb0dcnLoYUoR5JmhDPT3x87dOxUhjncy2NK3gwcOkCN13rUcKQwvTy9s37oVp06dg6ODE8aMHoWAgADs3b+X9tqk+4Mj4OzoiLyCPLof5LwfLl26wBnKCZMnY0LYBDml8Ts73x2p0zAsv+8/ubS3sbGumUWbl49Ro0bh808/g2uPHli/fp1GTWmXCyJvMss2DGtjY4vMtGyNWQeorGbU1dbF7Pfew8ply5RVBC/kzn5vBnQM9XihqyQlaYKPJDIaGM7eLfweG4d1a9Yh/WoaXnvtdUyfPo28A6nBvSDrBJ9ffonBFytWIiExQQ1qT1UgAvIToJ6l/AzVRgJ7tzBy5HD8dvRXHNj3E3JysjhftFOn/g8Xzp5Vm3pSRVomYG3dDXfu3G45IaVoQoCto1y/fkOTcArgNwEylvxuP6Vp7+7RB1u3bkdS8imYmXVC0NixCA0JQXxCAr3HUhp1ZQqWbRjWzs4WRcVF5AlKxiY5d+Ect3tITk6mjDnVNzlz65eZmc37e4mMpfreowqpGfvRXLlyOc6c+QdeXl6YOnkS/Hz9sP/gAd7f/AoBxBshss2GZTvemBqb4uKFc7ypYXsrWlpSihGBwzD25ZexcuXq9lZHZcpn/m9feOE53MkvUBmdWqMIGcvWUNPAPObmZtxEhcysbEyaPAGfzl+IAX3cOT+0bFcBOtSPwPMvvICsLHJ7J23LGpkY4dChQ9hAQ7DSIuNVOjKWvGqu9leWPSVOnjwVF65cxnsffoyd27fD7gVbrFv5JdiTNR2qSkC2YVhWi6e7dcP9e/dVtUIqqVcfDw+V1IuUkp8AGUv5GWqshLDxoUhMSsLW77Yi/ngievTsjoWLFiE/P19jmahuxWUbhmX1sLWyRmEetWVzbXrh0iVYW1shMz2tuWQaH3f12nV0sbLkNQcylrxuPtVQ3tfPF9HRhxC9PxpZOdno794bM2a9i6xs2vFENVqodVrY2tngn9SLrcusAbnYrNchgwdhfGgY7B2dNKDGra+ivZ0t753yk7FsfftTzkYE2Aza3Vu340jCn6iurICvlwfemvImPXU34tQ+l7IPwzo6OiMtjXpMktpr308/Ydy4MKxesxr0QyqJkvqEUxurT1uqTE16ODrim2+/55adGJt1ho+vHyZMmIBz52itZvs1kuzDsJaWVnj4sKT9VFbxkiMWRWDduq9VXMv2V48tHXln6tvIzeO3r2Eylu1/L6mtBmxT6pWrluPchQtwdHDAkMBAhISGIik5WW3rrE4V62ptiYryShSXkMGsbVe2jpJtsk6H9ATYpMD1P2xAfh4tHZGeGqXUSAKWFhZYFBGBa1evw62XC8aHhWLM8FE4Gk+u1NruhpB9GJZ5dLKzt0dyErUTa6fLZ9MROGQINm2ObLtmU6uSZB/dUKXqU89SlVpDzXUxMuqIhQsX4eqFK+jt2x/hU6dgWGAgEuLJK5Dym751P1ROTi64eZPc3rHJPIOGeWNC2ESw4Vc6ZCMQPHYsLMwtZMukYqnJWKpYg2iCOvomRlg4bwFSzp7D8CFDMTl8MkYNG4aY2FhypadiN4C1pRnKyspUTKu2V8fG2hYffPAhN5mn7Uvnf4nRUVHcKAWfa0LGks+tx3PdmceTGXNn43LKJc737OyZ4ZwrvZ9jyGgqvmllH4ZlOlh1tcGly5cVrw7PJLJ7dc6cOTzTmtRVJAEyloqkSbJaRUC/Y0dMmzYNF1LTETZxPN4Pfxv+/n6IiYmhnmariIrL1LphWJc+bhr7zvLsuYsICxsvDiaFyUig9OFD3vuSJmMpY6NTcuURYBNKpk2djstZVxEyfjzCw2ciwNcf0YcPK69QktwsAScHR40chmXvKEcNC4CpsXGzfChSOgLGhoY4fzZFusQqmoqMpYo2jCarxaaah0+bjiuZaQiZMAbz358HH19v7p2mJnORr+6tG4bVMzGCUFiFR48eyVc8z3K/+n//h7EvB2PDRpr5qrima909qLjy5ZNExlI+fpRbiQRYT3P6tBk4d/YMwkLDMHfuLAzxG0KzZ1vFvHXDsEZ6eujc2QrHjx1vVal8zfTr739g3TrawFmx7de6e1CxOrReGhnL1rOjnG1EgL3TnB4ejjMnz2DsmCBMnToFo0YNQ1zc0TbSQLOL8fLsh/MX/1F7CLn1NgBga4PpUByBv/76A06OzooT2A6SyFi2A3QqsnUEOKM5YwbOnruAgIDheOft/2HM6FFITiKPQC0Tbf0QmLWdDUpKHrZcBI9TZGVnYdDAATh6NI7HtVBd1X18/MBmFPP5IGPJ59bTUN2Zc4OZM2fizJnz8Orng5AJoQgLDUXy2RMaSkSaard+CMxARw/37t2RphBepsnKyuQ88wwa+BL8/QN4WQdSWvkEyFgqnzGVoCQC7En1/QXv49rlDDg6OSMkcBQmTZgI5r+TDsURcHV1Rfzv8YoTqGKSLmdmY8jgwYjcvFnFNFMfdSIiIni/z+1T1dXV1erTJFQTTSZQWlKKT1d8hn3bt2Gw/1AsXPQR7OzsNRmJqO76+vrIyLgKGxtrUZi0J6UVFXjBxgYZublgE37oIAKyEnjqqadw8uRxeHh4yppVZdJTz1JlmoIUkZcA62muXLYcx06dgYlZJwz0HIg5M2chJ5ffWwPJy6Umf+uHYTkDqaWFimL12X0kMz0dR4+qb29ZMfeMoqW0/r25ojVpjTwylq2hRnlUmoC1lRVWr16Fv5OPobS8FN4enpi3YAGKiopVWm9VVs7S0hL/XDyvyipKrduVzHT4B/gjLi5W6jyUUD4Cfd3coGegI5+Qds5NxrKdG4CKVx4BO3s7REZuwq9//Ib8nCz07ueG5UuW4dFD9Z7ZKZ6ofE/1Pv7+OHue/8tH2APTyGHDEBQ0EitXrhSPikIVTuDU2bPo2aOnwuW2pUAylm1Jm8pqFwI9uvfA9p17sW/bLvx9/BhcXFywYcMG3vuqlA1m64dhWTn2tta4X1gkW5EqmNrc3AwrPvucHA6oYNuoukpkLFW9hUg/hRHw8PLCL7/+gm+++xbbt+9E/z7u2Lt3r8Lkq7MgAwNjpGdlqkUVmd9hOoiArATIWMpKjNLznsDQgKFITk7CBx99hC9WrIC3t6cGeAOSbxi2b9++uJjCz3eWbDKPj7cX75cu8PmLp62tjVOn+L0Omowln+9A0l0uAuNDx+PY8eOYOGEyZs16G8NGj8DZlFNyyVTdzPINw7K1lhX/VaCwoFB1qyhGM2Yoff394ejgCDZJiY72ISAUsvtPvge29tG8rlQylnUs6EwDCejq6mLa9Gk4dvw0Xuo3AGNHj8PkiRNpuYmYe8HY0Bg5eTliYlQ3aNnKVRju74+NmzeD7WZDBxFoLQG6e1pLjvKpFQEzExPMX7AAJ88eh5G5Obz69MGcObNQojZrC+V/qndwcER6Whqv2n3zpsgaQ8krrdVP2R1btsDWzpbXFSNjyevmI+UVTcDS0gpfr1mDhMREFBUV4cXuL2Dl8uWKLqYd5Mk3DMsU7tu3Ny5eVH1jmZub02CmM/3ItcPt1qjICZMmge87udB91KhR6ZIIMAIOjo7YunU7oqIP4Y/4P9Dd8QXs3L1do+FYW1sjL1+1vSGxd5Te3t5Yt26dRrcVVV7xBMhYKp4pSVQjAl6envgt7nes/HI1vvz8Kwzy9uE2n+ZfFeUfhrW3t8fF8xdVtuq1k3kC/Pwxd/ZsldVTExVbvnw572cjk7HUxDuX6iwzgZEjR+L4uWN4OSwMk96ZgrCQUDC3ae19MI806776Ct6eHi2oIv8wbK9ebsjLy1NZD0gCHQEmTpjAvaNsAQZFtzGBDz/8EGx4nM8HGUs+tx7p3qYEdLX0ER4+HWf/ToG9oz0Ge/tixqx329XnrKGJEV4aNAh37txVOgsTMxNYmJsjNV0131uyHWZYD4Z+1JR+K7SyAPlHN1pZsEKy0X2lEIwkRJMIMJdpS5ctR0JiAspKitG7txtWr/2ywaSStuKhKxDAzNxCiuIU80Nl/7wjstLbv0ddW2HWu49YElF7SZ8qSsDU1BBairkF262GPFe/3bhRwUSAmwS0efN2JCUmYsGiRdizeSc+XBKBMUFBKkFn+/atSE1Lh45AAKGwQiE6OfdwRnq6ari9y8zK4pyiu7m5KaRuJER5BIqLS5UnvI0kU8+yjUBTMepLwNvHB4kJCZj94Qf4+P35GDVsGFJSUtq9whUVFWCeU8q5T8Wo4+zsiPMX23+Sz2OhEKMCAjGg/wDs3x+lmMqRFCLQDIGnqqurq5uJpygiQARkIFBa+hDr1q7Fhg3rERQ8BosiFit9fVlubh4GvzQQGdeuSdS0o7Y2rly7Dhsba4lppIlgPnXffvsdnDt3TprkSk1zKTUV3Z2d6R2lUimT8FoC1LOsJUGfREABBIyMOmLBwgVIPnEK5eWl8PLog3Wr1+KRsH330KwUKOar7ujojPy8/HabEVtaUTec3IMMpQLu2LYRwb4XZ8+ebZvClFSKYr5BSlKOxBIBvhJgPbjNm7di+9adOHg4GoM8ByMmJkbh1WHDkSdSTiG/sBD5hfmS5QurJMfJEMMmN3WyscTpf9p+I+jMzCz0cHTAqeRkGTSmpKpAoKysHEIF3YPtVR8ylu1FnsrVCAJePt74IyEBM94Lx7Rp0xA6Zgyys7IVVnc2G5bJLC0thaW55F01BAqcyufq4NTmPmLZrNdhw4bAw8MDbF9SOvhFwMHODvk8c8LfmDAZy8ZE6JoIKIHAhLCJuJqdDTsHe/Tx6oP5Sxbh8ePHCi2puV01uB2SFFSak5Mj8vPyFCRNOjE/Rx8Bm/VKk3mk46VqqWbMnAkHxx6qppZM+tAEH5lwUWIiID8BtnPHgtnzcSHnElYuW4kxY0LkF9qCBH19bWRkyD/BhxWzd/9+bFi/HgkJCS2UStFEQH0IUM9SfdqSasITAo5OToiKPYQvVnyJjxYsxqjhI8BmdirzUGTPsmdPV+RkZSlTXU42c492+bLqOEBQeoWpAJUmQMZSpZuHlFNnAsFBwTh56iR69+mN4QFDMH/+fJSWKGfxtiLfWT5nb4fyygqluvnLz8/HoEFDsGPbZnW+BTSmbpsiI5GdrfwHLGUCJWOpTLokmwi0QIBNqY9YsgR/JP6N9Ow09HupP6L270dVlWJmr9YWr8ieJZtU5OLsjLhYxc/uZfoWFBZi0KBB3P6ZS9ViL9HaVtDcz88+/xwXzil39ETZdMlYKpswyScCUhBwsLdH9N5DWL7kU3w8/yOEjBmNzCzFuZVTZM+SVadnH3fczrstRc1kT8I2CX7zzTewe/ducjggOz7KoSQCZCyVBJbEEoHWEAgKGoN/0i/DwdEZbC/NpUuWov5C/NbIZHkU2bNk8mytbTm/s63Vp6V8c+fOaykJxfOIgAJXLrVbrclYtht6KpgIiCfAhjlXrlyO3//4A38l/gXnF57D3Dlz5Hqfqeie5cCXBuLXo4fEV6AVoWwyz4SwsHbzDNQKlSmLDASYK8bRY1RjgwEZ1G6QlIxlAxx0QQRUh4BrT1f8fvR37NmzB1k52ejfrx+ijxxplYKK7lk6ODhCWF6lkF4vM5TsHaWBoSG0dHVbVT/KRASUTYCMpbIJk3zeEGCu44qKilVOXy8vH0RHRePjTxdj/syZmDB+PPILCrhe2KOH0vmcVXTP0qhjR1hYWeB4UpLcvN55510MGvgSIiMjwXrVdBABVSRATglUsVVIpzYnwGZg7tm8DW/Nmcn9YO/cvhNZWVkoK7+PkUHB8PH2aVGnqP1ROHP+NKoqhRjg549RQ4dKzMO29Dp4+DAcHOzx6v+9BiMTI7Ae1ldfrYONjSVKCh8jYtlCUf5hIwKRl5uLoqJS3LpzG6aGhlizZg3Ky8sw8KWX0KN7895RtLW1cU0Bu46IFAIwd848GHc1xsK5C9Cc96D6ecSds2UillZWNJlHHBw1CXN3dcXy5csQMHQ4f2vEtuiigwhoMoGKf/+tDg4eKUJw8uTJ6pcGDeKuKyoqqrt07iyKk3SScfVqda9ezqLorl27Vt+9d090Xf9k14491QHDA+oHced9+/atTktL5c5nz5xdveOnHdz5F199Uf3N199U37pxi/s7eex09dChAdXOzi7VwcFjq916OVffvnOnibz6AXp6guobN27WD5L7/IsvVlTPeHN6q+SUPXhQ/V+rclImPhKws7OrPnjgEB9VF+lMw7D8fc4hzRVEYO2qtfDz9RdJ27RpE7yeOOvW1dWFr68vt0elKIGYk41r18Hfv+6peUxICDZu3NgkJdsU+u133sS2LTsaxCXEJyA3OwcvvtidC3fr3QtrV6zlzqdNmYbwGeGwsrHi/uLiY7FixRc4c+4snF2ckZWVi9dfe7XZCUCKfmfJFOvj4YG4+KMN6iHNBetB93LrjU0bN0iTnNIQAZUgQMZSJZqBlGhPAuvXr0Hf/v1EKhw+cBB29TZJtrK0RE5+M9tfAUg8kYiuXTuLZHTp3Bnnz18UXdeefPHFCnh7+2Ljxg2YM2sOLjzZRPnchXOwt7evTQY3t744ffo02HtUIyMjUTg7+evPo+jZsyc3XLx0yRL8fSoZsbFxeLH7i1i5bHkDo8mGOOe+O4stHsHSpREoKGi+Hg0KauHC18cXD8rKUFhQ2ELKuujayTxubr0wZeq0ugg6U2sCszlH6o68riMZS143HykvL4Hc/HzcuHkTTo5OIlEPysvQqXMn0bWWjg6qKitF1+JO7t8vg5GRiShKh1vbWLdRcW3EgQOH0bevGz6YvwC9e/fFgIEDOeNWVlGOjiZ1RlFXV5/LUtFoAk9efj569/aoFcd99nRy5j6nT52C0+dPo1cPFyxYuBA5OTkYPHgQvly3lltn+f33P2Dw4CEN8sp7YWTVCRdTz0stJjc3F/0HvIQdzOGAFv38SA2O5wnZriPdu5Ox5HkzkvqaTCA7M42rvolJnaEzMNBDRUWdcaysqIRWC7M0DY0NIRTWywPAQM+wCVqWJihoHNcrnDAxDKysLVt+hJ5AgKp/haL0RcUF3Lmenp4ojJ2wod2gsWMbhLGLrl27Qggt7N8bhZ+ifkJhfi6Yw/O0tIaOyFNTUxEdHdUkf2sDQgKCkJSYKHV2Ly9vbN+6mWa9Sk2MEqoKAXq0U5WWID3ahYCBoTFXbv0lGGz2a3pmndNnNivWxsqqWf36enghN6duj8fM9Ew4OtQNq9ZmNjQ0QNmDu7WX8PL2gUBHD67u7sjIuiIK/+fMGbj1dgN7Z1r/OHn8OOfZp34YO2dG2NzMjAt29+iDyE1bsX27eCfkQgW+wLS1t23AqrFe7DonNxeHDh8WF0VhGkJg986dvHekDtFUHzohAhpIgM2E1dPTqc5IyxDV/tjx49WDBw/mrsvLyqq7dO1SzdKx4+PFi6tv3bolSlt7cvHSpepevdxqL6u7dOlcffduzWzYbTu2VaemXuTi3n///eoZM2Zz5//991+1vZ1tNftkB8t/9dpV7vy92e9Vb9u1izuv/Xfj1q3qj99/v/ZS9Ml0A1B948Z1URg74WbydunKxbH42r+xr7xc/cP33ytkNur5f/6ptn3WVqIsppOTk2N1YGBgA93oQrMI0GxYDXkqomqqLwG2CD44OBgnTp8SVZL5ZB01ahQWLFiAxZ9+ikPR0dywIZuM88WKz/Huu++K0tae9HB2xuyZszFnwTwsmD8PO3fvhbl5TU/vyy++wB9//sUlXblyJYpKCzBv4XwsjViCLdt2iN7d7dq1DcuWLMOy5ctgaGiIiWFhteK5z62RkfAf2dRl2O28PNjbO8DGxrZBetYr/fPP3zE2qCYP+zx5/DR8PLy5DZwtrS0wITQMmzdHorikpEFeaS96urpCUAWkpzd1+s62G2PbbDk5OSE2NlZakZSOCKgmAc16vqHaEoGmBNIyrlS/Mm5c0wgxIQ8elFUvXvyRmJiaINZLrO0p1iZqfM3CxYXVppf0KSnPJ4s/qf7l4EFJ2bhwHUHTdZasp/rN12uqR44cWd21S5fqwYMGVX/7zTfV58+fb1ZW48jAIUOqt23b0jiYu/7l119FvXKxCShQIwioQ8+SPPio5jMMadXGBPbv3w99gTZGjgmWWDLz8hMTcwQvB7/MedyRmLANIzLT03Ho50NoaZcObX1d3Mi4zq3TFKdefkEhjh6N5XqAyYlJYBOevH18MGTwIHh4+cDK0kJcNi5s1Vdf4XpWFr755huJaSiCCPCdABlLvrcg6a8wAsxhgLWVDaysLRUmU5mC2NZd51NS4O3t3WIxzFhey7gGm3rrRyVlYms7jyUm4cSJE4g9chiZ2Zno6dwTLu69EeDnh959+4mGmJmM2F9isOTzpUhOSkZeXh7CwsZj89atsLdrOsFJUpkUTgRUnQAZS1VvIdKPCCiAgLa2Lm5ck9yzbK4I9j4zLiYGqWmpOHjwIAoKCtDDsTt6e/aHm5szvLz80MfDFX/8/ifGjXsZzz77LGJiYkTvYpuTTXFEgC8EyFjypaVITyIgBwF9fX1kZFyVqmfZUjGlpQ9x8PAB3L1zB1F7DyEzNw26Vdp4VPkYlQIhfon6Dd7eni2JoXgNIuDBNjKPWMRrR+q0H44G3bBUVc0lIESdwwR5KRgZdcTECRM5MbNnvwdhVRXmzp2Li2fPw8XBEZPefB2VJcUwsrREsP9Q2NjbwsWlF/p6epIzAnnh8zT//Tt38OhRndMNPlaDjCUfW410JgKyEhAKIFDCpawzgAAAIABJREFUJljM16uBgTECAvxx6tQJrIn8FmsAbl/QM2dO4+zZFJxITsGaNatR9qAM1lbWcHJzgYdLHzzr0A2uPfvAWor3qLJWl9KrFgF+m8kaljQMq1r3FGlDBJRCQJHDsLUKMiftAwe+hHHjxmHme7Ph+mJ3ZFy9KnGmcEFREVL/OY+09HRcPHceZ9MvID8zBx2N9PHcc8/DrrsDXO27w9nNBbY2dmREa0Grwee6tWsxeMhQXvuHJWOpBjciVYEItERAW78DrmXcUMg7S1YWM5SDBg2Ck5MDoqIOcZN5fH39MGNGOEJCQlpSRxTPZt6mXbrEuczLvXkdJ0+cQGZaOgpLiiDQFcD1BVc8Y/cMrK2t4dnHk3O+0MnaFEY6dU7nRcLohAgokQANwyoRLokmAipDQMHDsDo6enh57FgsXbZMVEVPD0+kXrwok7FkHpRcXV25P5GgJydnU87h4sV/UFZehszMTMyZOQcF5QVAmRBWVtbo1b83rMwtYWpsgj59+qKXmxsMjDpy70VpR5PGNOlaXgLUs5SXIOUnAjwgoIxh2MbVTkg4infmzMKls5caRyn0ms3GPX36OC5ePI+qx1W4nHkVx5L+RPH9YlRWVcFUxxDunh5wc3ODQLcDLDp1Qi+XvnB27UETjBTaEtILOxQdBfc+nnKNbDB3k9k3bmD0E/eN0peumJRkLBXDkaQQAZUmoIhhWLZx9MbNm7Bo/kKxdWU7t3Tr9iyuZGQ0cFogNrESAtmQbuXjx0hNvYiLFy+iqLAA9+88QGZ2Ns5eOIWK8nLoCrRhaNgZji/awbGXC6zMLWBsaAwbO1vY29jB1t5OCZqRyOeeew5fffkFRo+Rfoi+MbVHQiE+mjcPq1evbhzVJtc0DNsmmKkQItC+BHSE8n3Vmau/QYOHoJeLC5hRYsOnjQ/9jh3h4++HPXt2IDx8RuNopV8zndgfGw5mf42Ph6WluHv3PnJys5GZlYWi/HxkXs5G4d08ZGZmI68gl8tibmKOLl06w9zWCs42jrCwtkDnTp1gYWWFLp27wt7OFqyudMhKQFtiBnZP7dm+FWmpaQgYORJ+vr5N0uqLueeaJFJiQNM7XomFkWgiQATah0ClnN/0kNGjOUO5Zft2sYaytlb9PT1x9sRZILw2RHU+OxoZgf3Z2tnAx8eniWJVbMlLQSFyC3ORn3Mbd27nI//uXaRcuIB7ubeRd6cQxaWFKLv7AB0NO6KrRRdYWluiU5euMDE3Qbeuz3A+dG1sbWFs3AnmZuYwtTSEvoAMaxPYjQK+WL4cfj6+cHRyQkWFkFuGdPr0SVGqgIAgODi0b6+fhmFFzUEnREB9CcjiG1YchcuXL+HFF7u36MKOrbt07eOOW3kFzRpVcWXwJYwNN1+/kYPc3NsoLsrHg7JyFBfdw42becjMSEfRnQIUP3yIyopyVJZXwsTUBHa2tjC17ARj084w79gRBqamsDA3g72dHZ5+5lnOcb2hng4ERkYw0tPjCwqF6MnumdDQ8Zwv4pYEzpkzh4ZhW4JE8USACLSegE4rVoXXH27t3r2HVIWzPTWtLK0Q9+uv3J6gUmXiWSI2BOvk1J37a0n1ktKHyM7OQnp6Oh48eIAqoRAVFWUoKSpFYmISNkVuRkFhPsDet+pUQascgJYWZ1w7d+0Cm+ftYKSnDz2BARduaKgHOzt7WFtbwsbaDsYW5hBoaYkeYsQNj7ekY3vGr1uzDtGHo2FiZMRxYnWTdDAfxT79+0t8DSApn6LCqWepKJIkhwioMAFZe5bsHeWQQYOwavUazjuPLFX7YvkK5N8pkKkHwGY6WtvYiiYG7d69E+bmllKVnZuXi4MHDuLNN94QvUs8d+EcTp88idcmTxH1cNmPbVrq+QZVsXdwgqWF5O3HahPv3L4ddrZ28PKp2eEl+UQyevfxEMmuTSfvZ1VVFTdB6uaNG8jJzUF58QMIq4DHlZUoEz5EQW4+crKzcDP3NkrKiqBVpQUhqiCo0kKVADDq2BFPP9MNNjY23PKaTp1MIRBooQM6QEub2VsBuj3zDCwsrWBuaYmnO3eR6ERC3rpIkz87Nxeho0bj408+gZ+fH5grRVU9yFiqasuQXkRAgQRkMZZs1iubzOPi7IxtO3fKbBDOppzldh9JzciQKm98fALKS+5ze4my3uy4MSH4ePHHuHfvLg4fPoxvvvlWLAk2HPq/t9+Cp7sHXn/zDXR8Mulm66bNyMnNxbTw6Zg4IYxzmsB+hOcvmI/4uDiYmdcYx8wrmdi1exc8vZpOBqotkNNn9GisWLMaZ86lICs1E4siIrhoNiQ4e/ZMsN50ex2PKish/PdfCIXluHe3DAUFecgvLMK927dQUlqK8op/ISxnn1Uor6xASXERbhfcRfGtPBSUFKFCWMm5QdTW0oZAF9DpYAiBthZ0dHRg2qkTLC0tYWBqCDMTE3Tp1IUzvLq6umDrbHUEAu5PYKAHEwMj6OgLYGFlA/Z63NS4E3T0dEQPP1OmvYmh/kO44db6rNjDgaWlBW7mq/6wvZyv/etXm86JABFoDwJFRcXg3hW6ukosXpZh2KTkE1JN5pFUmHsfd+jq6+J4chJ8fZrOaqyfj3kCWhe5FtG7o7ngA7t3w8DYAH369OGuZ82ahbOnUuDuUXNdm5cZseDgILzyyquYPGVybTDnk3bu/LmiH18fH1989umnWL5yOXy8fLF82XJR2hEjhjVrKFnCHzZuhIOjE150cOT+nv/gWYybEMadM7mBw0bg9/ijUj0UiApW4Im+jg7A/tARZiYWsJdx6QvjKHz0COXlFahgvdfycjx8WMINGVc8KENhURHKK8pRUVKBUmEFHty/j7KyClQ+qkBJeQmElUKUlz5AScVDVFVW4V7hXTyu+heVleVghlBYCc7A3n/wAO49ezWpeVTUXgT4D203fk0UaiaAjGUzcCiKCKg6gVOnTmHr1k0YHxqGsJAQ7IyKEusuXZbZsCFjxoD9yXNMmDAJP+3+qUVjuZANv3nWGdR9Bw6iX7/eoqIH9B+IPT/ta2IsmRErKCziDCX7wa99Vxdz5CCsLCxF12wCzZyF8zljOXzkUJHc3Nw8zgiKAiScREdHw6ve5tpOLm44HP0zXpznyA35Ojk54sD+vQgLmyBBgmoHc8ttnswSVrSmrF0q/2UTnYDRw4ZBIGBGveFx5vR5+Pr7NQxU0SstFdWL1CICREAKAv/735tYuDACPr6+cHDrhcgNG6TI1TQJe0eZkpLSNKKVISOHBuDw70fAhkolHezHdNumTQgeN06UJD09DUZGdX5f2Y4kbE1k44MtcO/UqRNWrlwJJ/sXsGzZUi5JTm4eTLt0FSVny0RuX7/JTQoRBQLYuXMrXvu/V+sHiT0/duxvWNV7p2lqaorbN2+K0jo42GPfvp9E13RSR4AZYiN9E5ibmWBa+DTOo1JdbM3Z5q2b8MrLde3fOF6VrslYqlJrkC5EQAYCOTk5yMnJhZWVFZfLxsICsXFxYiU0NwzLTeYZPASfffaZ2LytCezp7s6tO2TvHCUd9/ILIBQKYfNEf5ZOWFHRpAfCZpA2PphRXLhoIebNm4c/k/7ERx99jMz0NE6evm79xe81g2fMs0/9IyYmBs6uPesHiT2vqKiEsaFhg7jyR3V7g7K1lKdOJDeIp4umBCZOnAQPD48GEey+8/P3b9cJRg0UauGCjGULgCiaCKgqgfS0VFjU6/V06twJWVnpYtWVNAzLfrDY7iE2Xa3AhhwVeUyZMgk7tu+SKLKSTfNsdDzTrRu3tKI2OCc7F7a2TRejMyPLZnSyw9bGFs899yziExK5ma2lJcW12ZGVlYlOpqbcLNHaQFZnDw8v0VBtbbi4z169XJCTlyeKun/vNp61f0Z0zSa5sHd6dMhOYP369Zg9c7bsGdspBxnLdgJPxRIBeQkI2OL1er2usrJyGOg17AW1VAZbNjFj+nQciPm5paQyxwePHcf5aL1w4YLYvKadDbjw0ooKUby/vz9yc/JF1/+cP42hw+veNdZGODraI+NyncN2MyMzDBzQH/4BQ5GWnlmbDDk5eQgaO1Z0zU42rt+AscFBDcIkXfj5BaCinn6ZWTnw8qpZPsLylAvZDig2krJT+BMCbIifvSdmh5eHB06dOAH2vrdxb1OVgZGxVOXWId2IQDMEnJxcUFhc14sqyC+Ek7Oz2BzNDcNODw+XqpclVnAzgWy5QXBoMLZs+UFsKhMjczh2s8OZUydE8TNmzERcfBz3rjMzPR1WXSwx2N+fi2fr8Hbv3s2dr1jxFdatXQ/WN2WL/i2sLNG9R0+wd4gvvxyMuLij3HvKpMQELIlYJJLPTo4fPwbPepN2LqWmomfP7oiNi22Qjl189NFHOHL4ICeL/eC7ODs1cJXHnLX3F+M6r4kgDQ/4cN4HOHz4AEch6pdfYGllhbDxYc1SYe+0VekgY6lKrUG6EAEZCFhZWsDH2wcXzp7lct28dRvs3ZC4o/4wbFFhIUInhqK0pFRcUoWGzZgxAz/tPshtFi1O8FtzZ3KGrTbOyMQIv/z8M37Ysg1JSUk4+PMRkSEfOXwoHBwcuKSjR49CREQEvv7qK84TzsGf63rG32yIRG5eNrZs3Ih169Y1WAfJfoDfeP2NBjOGTYwNERz8Mk4k1xntWn3Mzc0QFR2NHzds4PbW3LGj4bDy338fQ/ibb9Qmp89mCNTOhmUTpmxtW16b+smiRZhSb1lQM6LbJIqcErQJZiqECCiHAHv/tuDThbDrbAUTYzPMmDVTbEG1TgkM9HQ4hwNPP90Vh47UGSKxmRQUOG3aW9DvqIM3Xn0dx06exIABA9CzZ93kGuakfceuXSLvOwoqViYxcVyvUgsBAQFS52M92rWr12HNujVS59HUhEP8BiM4JFim3WiYP+JBg4bA08sD0VHRIpd+khiWPnwoejddUlwMEzMzSUlbF15NBxEgAmpPwECgU33jxs3qV197vTowMLBN63v12vVqHT2DagCivw8+eF+kw92796o/+Oij6op//xWFteXJrVu3qk+ePClTkSzPis9XyJSHEstO4MaN69X29nbVLw0Y2Gzmbdu2VHfr9kz1yePHq7f8uK0aAq3qO7duN5tH1kjqWbbuGYNyEQFeEajtWZoaG7f5VH22pOOFF7s34ZWWcYXzhFMbwYaF2TAsHw7mZo7znsMHZXmuIxs9GTJ4CNioyJ9//ilxBGLd2rVISj6F8BnT0M9TutnOMqGR1bq2Nv29O3dET5X1nzDpvO5pm1gQC7oH6B6ge0DyPfDmm29KNEHnL56vdnFykhgvb0SbTfAxs7BAdXW1RvzNDA/H4sWL1bau7Gns3p07alk/1m7h08PVpm6379yBi0uNT85bN261S72u37jR5AFeIBDgxo3rcumzZs0qtWor9vt4/doNMDbq9lvpP8gP3333XavqlXHlCrp1ewbBwcH4phkPVWynloLi4ma9RjW5EWUIaDNjKYNOlJQIEAEFEGAO1tnwla2ttQKktV6ErY0NvvzyS84IMCnMGKxavbrBLNXWS1evnGy7LTrqCJw7lQpvH294+/hwTjNqfQDXpQDycvO45UN6Bobw9vbC4ZgjSEpMrJ9EIedkLBWCsaGQ/gMGoo9bw10SGqbg99W0N6ZBx6BmQTm/a9JUe9Zu/fsPaBrBw5C83Bw4OTrg559/gUCvqRPrtqwS287q5q1b+P7bb9Dlma54ZdzLchfv7NxLbdqqFoapqTGmvDal9lKjPxMSE/DSEC+MDwnFzu07JbJ4Z9Y7SIiPh5+vL6ZOnoKYw0caOL+XmFHGCJrgIyMwSk4E+Eigo7Y2rly7Dhub9u1lMnZvT3sL/1VVYWPk93xESTq3gkDk5k1wc+kl2npNGhErVy4H8827aFFDpxLS5FVGGjKWyqBKMolAOxFgMwcTEhIwPjS0gQZsNuyNjOuwsqnxp9ogso0v2B6WAz0HYNvuXfBqZuPlNlaLiiMCzRJQuWHY4pIShM8Ix/r167BgwYJmlVflSPa+KHT8eDg6OcHfz19pL53bi8HadetgbmkOExMTfPnVl+2lhtLKTTwRj0C/IWCb0/LlYIaSOUXftu1HsSo3fh/Gts+aO3cO912bP2+O2DzKCGS7pMyYMxPz589rtXjmom7ixIlYvXp1q2WoUkY25Ghvb8t9n0LHN3zQUSU9ZdWFLQeaOGki+vTsyfmBZb+LfD1Uzlh+PO8DjBo1mvP0INDSQuSmSF6yTU0/j/179yI9rWbboBoPIbysShOls7KzYGBgiKKCIrD9/ubPm4/MK3XOq5tk4GFAP7cBqBZUQ8zGGCpbG2Yo7WxscOCng0101BFTkcWffAK33r257xpzBr5u/fom+ZQV8FZ4OISVlYiMbN3328HREdZWVqisrNsuS1m6toVcNiHl8uUMXLt2HelpmVi+fFlbFKv0Mo6fPInvv/0OUT//DLad2eGDNf5hlV6wEgqQ21gyX4v1fyjZuTwOcCO3bEbvJ1Pd2W4DkZGblFBt5Yv08arb/d3dvQc8vX2UX2gbldCpU2dMfeKzsUePnhg4sD8eCR+1UeltU4yuri709Tu2TWEKKuXr1atx4OefwXRvfFQKtCBo4BEV2LBhPfx8a3apHzwkEKuWL2+cTWnXbFbjsmXLEbE0Aq3pbbD8emo0yey92e9x7cZ80QYFjcSDMvXY9isgwJ9zIvDGxDdwKeMy/ANqnOIr7cZSouCanVHlKGDfzu347PMV+PWXX2Bnb493Z72DT5d+1uBF7okTyTh6NAECgRaET55w2TlzphsWVud5nn1p2JOisYU5p5GdjT1Onzwph3btm5U9NKxfsxqP/v2P22evfbVRXOlsN4n6B3u4dxTjoaV+Gr6eC3jUcfFvwa9p42FYbkuvJwaHPfXfun27TZvJ188XAb6BmL9gLiI38vOhWFHA9DvWPZjl5+Vg6bKVihLd7nLY72BpeTFc/7+9c4Grqkr7/+9l0FQiMzWHHF4ickzzlhH5OuYtwSsimikSERHDOGVm3vASOWpEhIqamRmZKZoZ3lIiYnjNYdSUMUQR74Y0OU1/M9OUHOL9f9apQ1zOgcM5Z5+z1t6//fnwOfvsvS6/5/tseFh7P3utezrjdh/3J5jZC8ThYBkVHYPDJSdMSQUiWD4R8wS69qy5TFCvXr0hfhrabpT//Jep6l0aD4cHvg116bTzYpHayl9o3nH77ab/EoUdfgEdsGjpcgTcdTemTZ3qtP5c2ZB4tvXVv/+fqcubb22Gdq3aVXWfviYdqanJVStDVJ1QaEe8p1VeUQEPTw/89rbbakynVeHeNy7qpSieBwUHB+O111+r8c+ppUqWbsOKch4WRqGW6mt1LDn1FXTr3BkxUfmapPtrpVurdguLCtG7b3+I57p62cTfwaaezbAzOxsvzJ6NlBQ1/xFwOFgKh4pni6VlZaZleFp4eeMmj+Y1/Cxm5y85UlzjmPhyW5vb0ataNpxP+3amF5YvX75setB96lQJOnb8eUmeOpUlOxATE4HvvruGisoKbM3MRMAvSwmNCQ/Hte+/w87sXMkU2y7n4MGDeO65Z00VxGK4qYtTTfv79+/HHe19bfpHyPbeXF8yYXYCio8cFlcy3nzrzQYDj+sV1u1RJMKJQNmuXWt07dGjboFaR8Rt2NqbmByg/NJlePs0w9ffXMDtt7epXUTz72Lx6XkL52NGQoJpSS7NO5S4A5Gg9dlnnyE+Ll5ilfZJ2/XxR8j56yd4a9VK+xqQoJZTgmWzpk1xtbwcOTk5iI6uu56ev38AxI8t24iRI1BSXGIKovn79+PJp/5oSzW3l8nbnW9Vg3i28j+9g6yel/1E3/79caiw5mr3BwoKcO2b7zA09OdV7EuOFqNTl5p3FGS3y6xv3bp15l1lPtetWWUKlCKZp+pOTCPVT5gwHoePfI5BPkOwb99niI11zx/pifETsTZ9DVatXIX4ie7R0Eh0Ti8uAmXWth1VgVIsNB0YqJ+JTUTWfMsWN+OBB9Wd8MMpwVL8d5i7cSMWvPCCwxfR4tRUzEuci++vfo/zJaewNH2Vw226o4HoqCi0bN0a93XtitKyc0hMnO8OGZr0KZ5Bh4WG4SZvb/zXn/9jenE4JTVZ2WBpCVJpaSlKS8+h+PARXAkZJt1qGM8+NwPPPDutwTX+zLZZug2bnJyC56Y9b2qj4NABpC1eZi7u8s+UlFQ8EfsERj0y2qbn+2IB65KSE2jWopkpQUgkxqi6iUAppiW8ePEbvLjgL/jpp58QFh6ui2A5bVoCrl27hAfuD8KJf5bixZkzVXUTnDIpQUZGBvr27eO0uR7FA+Evy/6JAP+GV9OWlbx4znf6/HncecfvpPtDKysz6qqfgHhGee1GuU3BpHZL1iYlqKysxLlzpQgI8K9dxeXfY+Ji4flf/4fVb1p+T9TlgtihwwR+/PFHnDx7FhszMjAsJMQ0x6vDjbqpgboPMhohREwc8OGHH6Jjx45OC5Sie3FbSeVAKWwQ2W1dO3VioGzE9cSi1gmIQCneo3zl5VesF2rgTO1sWFHcw8NDikAptCTNX4hPcvJwYO/eBizhaVUIiNeYxN/Bg3v34Uhx3bwVVewQOh0KluJVj3t+/3td3C5QyWnUaiwC58rKTIHy1va3YfEvyVWNJWDpNmxj29C6vMgAnTVrFp557jmtu2L7biAg/jFTeXNIvZjgVsykwY0ESEA7Am18bsawIUOQsz3b7k4sZcPa3ZiGFePi49G8WXMsXaS/KRQ1xCZ90w8PehgB/u6/1e8IKKc8s3REAOuSAAloT0A8szx78qwUq440ZK24DTv60Ufwj88P2/V8tqH2eZ4E7CHg0MjSng5ZhwRIoGECG9dtxEe5zns3V4XbsGYqQb17Y3BYKKY984z5ED9JwO0EGCzd7gIKIIGaBNaty8CExyegiRMnCbc0N2zNXuX6lpaUij379yI7O0cuYVRjFwEx0cyVKz/YVVeWSgyWsniCOkgAwIH9+/H444/hfz/5BIOGDXMqE0vZsE7twImNebf0xsKkJMxNSNDd8nZOxKRMU888+xwWLXpVGb2WhDJYWqLCYyTgJgJBvXqh5ORx9B/k3NUZVLoNa0YfFRmFW9q3wYrlrls6zNw3P51LQMy7rPqmvgWqe4D6SQCoMXq6p4PzM8xVyYatfTEsfnkJXlueBrGGKjcScCcBBkt30mffJABAJPM88OCDDq0Dq1eQPbrdi/GRjyEhYYZeTTSEXfMTExEfr/a8vwyWhrhUaaSsBNauWYcJTz6Gl15+ye4J0W2xTcXbsGa7Zs6Zg5LDJabZwszH+KkWAbF2serLjjFYqnXNUa2OCIh5WddmrMXHuz4yTUyvpWmqZcNWZyEWG39hwYuY/vxUCGbcSMAdBDgpgTuos08ScDEBlSYlsIZmaFgoHri3O+YnLbRWhMclJZCxcRPatm2NECcnrrnSXI4sXUmbfZEAgKysLJc/n1T5Nqz5onlt8WKsXL0SRxWfkNtsj5E+c3JzsDff+pq/KrBgsFTBS9SoGwJr12UgLCwMB128soaq2bDVHR8Q0AET4ydh5lQm+1TnosJ+ZYX6t88ZLFW40qhRFwQyM7fhqSefwJYN7ym9rp87nTFn3lyc+9cZbNq4yZ0y2HcjCXTqGAA/P99G1pKrOJ9ZyuUPqtExgVMnTuDcuXMIGTLE5VZ6NWmC42e/UGIi9YbgiDV0Z06fjr/9fR9at27VUHGeJwGnEGCwdApGNkICchMQCT7nT34BH18fuYXaqC5i7Bi09w9AakqKjTVYjAQcI8DbsI7xY20SqJeAeI9yxvPP11vGVSdVmhu2ISZJKSnYlJGBwqLChoryPAk4hQCDpVMwshESqEtAJPM89ccn8YcB/eqedPERPWTDVkfm7x+AiU9PwcxZs6of5r6kBCIjozFv3jxJ1dkmi8HSNk4sRQKNInD50iU88+ensOX99zWfcMAWYXrIhq1t55TJE3HxnxeQkbG29il+l4yAZzNPyRQ1Xo76FjTeZtYgAc0JtGzVCl+e/wrik5s2BJp7eeHFpGRMmfhnjBwxGmJZL24koBUBJvhoRZbtGpKAGFHKGCD1lA1b+8KKiIhEu/ZtkZaaVvsUv0tC4FDBATRr0QKdO3eRRFHjZTBYNp4Za5CARQJbN2/GoxMm4Or165pOim6x8wYO6i0btrq5ZWWlCAwMxIcf7kJQUFD1U9wnAacR4DNLp6FkQ0YmsH3HDlOg3PLhLukCpdkvesqGNdskPn19/TB18lQkJCZUP8x9EnAqAQZLp+JkY0Yl8HlBgSmZJ3RIiJQI9JYNWxvyjNkJ+P5f32LtmvTap/hdAgKZmVuxe/duCZTYL4G3Ye1nx5okoAwBPaw60hDsrOxsPD/pWewrOAixrBc3eQjERMfCz99X6ddHOLKU53qiEsUIiGeUZ86cUky1fuUOGzIE9/e8DwsWLNCvkYpapodHAAyWil58lO1eAps3bcboCeNx6sw59wqxsXe934Y1Y1iYnIIPNm5AYeEh8yF+SkCg9a2tcPPNN0ugxH4JvA1rPzvWNCiB3JwcDB0+HB9//AkGDuyvBAU9Z8PWdsDChfOx7++fYddHu2qf4ncSsJsAR5Z2o2NFoxJ4aOBAfPyJOoHS7Cc93Aoz21Lf5+TJU/H11xewefPG+orxHAk0igBHlo3CxcIkoCYBPU9KYMkjW7ZtQ8L0GThSckzaV3ks6eYxeQlwZCmvb6hMIgLbt27H0MEPS6SocVL0ODdsfQRGjxqFe+/phBcT59ZXjOdcRCAuNl7pTFiBicHSRRcLu1GXQGbmNowaPQoTHntSXSMMqPylV1KwJj0dx0+dMKD1cplcXnFDLkF2qGGwtAMaqxiHQGVlJebMScD7772PqKhIZQ03SjZsdQd17twRsbGxmD59RvXD3HcDAQ9P9UON+ha4wfHs0jgEPDw8cOz4cYzG90iSAAAaqElEQVQdN1Zaoy9fvoKEGfUHBHEb1tOAN5JemPMCThSXIGvnDmn9ZwRhEaPHYtCgQUqbygQfpd1H8VoROHa0CJ27dNOqeae1e/2HH/Du+vX405/+hPL//MdqMosRZvCxBlWsd7nolaXYd+iAVT7W6vI4CZgJcGRpJsFPEviFwNatmeh+3/04cOCA9EzEmo6Px8SYdN7kaX15WiPehjU7LzIyGre0uRVLUxebD/GTBBpNwPpvV6ObYgUSUJ+AafWQR8fj7bffUW65J/F8Vdw2Nm/i5fyS4hOAhyeuodJ82JCfKcnJGDduDKKio+Dj42NIBu40evv2D9GuXWv06tXbnTIc6vvX3yyHmmFlEtAHgda3tcG776xXMpmneqAU3ujU8V7cd393dO3ayZDPK6tfkWKdy6EhoZg7e071w9x3EYF//OMgPtim9nNjPrN00cXCbkjAUQKXL1/GQw/9T1UzRUXHTPvXb9xAi5tuqveZZfPmTXDy5Bfw9W1fVd9oO19/8w3uv6873n//A/Ture4IR0W/zZs3D1fLy5GanKyifJNm3oZV1nUU7iwCYvUQr5YtERIi51qUZju9W7bErl2fmL/W+azvmWWFATNhawNq17YtJk2firmJc5GXm1f7NL+TQL0EOLKsFw9P6p2AeEY5akw4Nry9HhFREUqae/36dbRo0QI//fRTjWeW1Y3xanITjp89a+iRpeDxY0UF/tA9CJMTpiAqKqo6Iu6TQL0EGCzrxcOTeiZQWlqKu+++G+++/a6ygVL4Z/vWTHz51Vf43R13YEhomMXXI4z86kjtazgrKwtTpkxCQUERvL29ap/mdxKwSIDB0iIWHjQKATHhQOd77tG9uQyWNV0cHhaK7t0fwLz5iTVP8BsJWCHAYGkFDA/rl4B4kV+8n2ikzWirjjTk26NHizA4OBh79u5FgH9AQ8V53kECyclJuHq1HOJ1JlU3vjqiqueo2y4CIpnnzrvuND27sqsBRSsZdbo7a+7q0qUbIiKiMC9xnrUiPO5EAuXlN6D6ZOoMlk68INiU3AQ+ysnB6EcfxRuvvW7xuZ7c6h1XZ5TFn20lNX3WTOTv3oO8/XtsrcJyBibAYGlg5xvN9DdWrsCG9e8hfKy8k6Jr5RMjT3dnjal4leT5ydMwby7XvLTGyFnH+/fvjxBOpO4snGyHBEhAKwJM8LFM1vQqSWAgps+ciXERar46ZNkyHnU2AY4snU2U7UlFIC8vD1cuX5FKE8XIQ0BM5JC4cCFenPui4Z5jy+MFNZQwWKrhJ6q0g8CurCwMHjwYeZ/+rx219VWFt2Gt+3PkiBH47053YWlyivVCPOMQgf3792JPvtrPhhksHboEWFlWAh/l5mJ0eDg+yMxE2MiRssp0mS5mw9aPOmneQixaugRlFy7UX5Bn7SKQnZ2DHTuz7KorSyUGS1k8QR1OJdCze3dsZaCswZTZsDVw1PgSGBiIkSNHIemlv9Q4zi8kYCbAYGkmwU9dERCZjsNGjNCVTY4Yw9uwDdNLnDcXOzK3oqioqOHCLGE4ApzBx3Au16/BuXl52LltG9KWLdOvkXZaxmxY28DNnZuI4uJibN2aaVsFljIMAY4sDeNqfRsqJhwYOngwHnzg1/Ue9W0xrdOCwMzp03HsWCFyc3O1aJ5tKkyAwVJh51H6zwTE4sePjx2LDes3KL16iJb+5G1Y2+h6t/TGjGkzkMiJCmwDZqBSDJYGcrZeTW3etCnOf/UVxo4z3sw8tvqU2bC2kgJi4uJxo/wG0teusb0SS9ZLIDU1FYnz1Z1EXRjHYFmvi3lSZgIXL16qkme0VUSqDG/EDrNhbYMl/ii+sOAvWJT0MicqsA1Zg6WuXr2Ka9euNVhO5gIMljJ7h9qsEhDJPL/73W9x6sQpq2V44lcCvA37Kwtb9sJCQ3GH7514c/lSW4qzjAEIMFgawMl6M1EEytDhQ5GyOA0dOnbQm3ma2CNuw3JrHIH5iYlITktD9TsYjWuBpc0EOJG6mQQ/ScCFBFauWgVUVmLixIku7FXtrvjqiH3+ixoXiXYB7ZGaxKnw7COon1p8z1I/vqQlJGCVgFeTJjh+9gv4+ra3WoYn6hI4UVKC/n37Yv+hQ/Dz9a1bgEcMQ4D3ZgzjarUNzcvJRWFRodpGuFE9s2Htg9+xUyeMGT8OyQtfsq8B1jIRKCgowN79+5WmwWCptPuMIT5v924MHj4U+/YdNIbBGlnJbFj7wM6cMQvbsrZzGjz78Jlq7dy5Ezt27HCgBfdXZbB0vw+ooB4CRYWFGB48GIuXLcPE+Lh6SvJUfQSYDVsfnfrPiVvXsTFxWPDygvoL8my9BCoqK1FZWVlvGZlPMljK7B1qQ7cePbB114eYxGQeh64GZsM6hA9TJk/BofwDyM/Pd6whg9f28FA35DDBx+AXr6zm/1hRAbGKPTfnEGA2rOMcF6elIicrF9k52Y43xhaUI6BumFcONQXbSmBXVhb69e5ja3GWs4EAb8PaAKmBInGxE3G+9CyyshksG0Cly9MMlrp0q7pG7c7Nw6iwMMTGxatrhITKmQ3ruFO8vb0wddpUJCclOd4YW1COAIOlci7Tt+Bnpz2PtDfSEBcXo29D3WAds2Edh/5YTCyufvc917tsJMrlS5ci6ZVXGllLruJ8KCSXPwyv5mBBAZ9VanAV8Dasc6CK5+iTpk7GSwteQnj4GOc0aoBWLl66hKvl5UpbypGl0u7Th/gDe/fCnFDOpB5tfMpsWOdxjY6OgcjqXJexznmNGqClSgZLA3iZJmpGQCTzPNRvALJ27tSsDzZMAs4kIEYYk6dMxZJXF3EJLxvBmiZSDwmxsbScxTiylNMvhlCVv3cvxo8bh9defwMjRowwhM3uMpK3YZ1LPjIyAs2aNcP6dWud27BOWxPBcsiwYUpbx2CptPvUFn9LixZ4bdnrTOZxgRuZDet8yNNnTsfS1CX48ccfnd84W5SOAIOldC4xjiAxO090TJRxDHazpcyGda4DRILPzd43Y8OG9c5tWIetnTpxAkeLi5W2jMFSafepJ37//r1Yu4aJEa72HG/DakN81pw5WLSEo8uG6GZs3Ig3lq1oqJjU5xkspXaPvsTtyd+DAQMG4Pvyq/oyTAFrmA2rjZNCQ0Nx+61tsP69Ddp0oKNWPZuoHW7UVq+jC0nvplz4+hsMHzocy5a9wUnR9e5sg9k3bXYClr7yqtIrahjMZXaZy4nU7cLGSvYQOHOuFAH+fvZUZR0HCXg1aYLjZ7+AWG6Km/MJ9O8/ENHREYiJ4TJyzqcrR4scWcrhB0OoYKB0n5uZDast+8mTn8arSYu07YStu5UAg6Vb8eu7c5HM07p1K1y8eEnfhipiHbNhtXOUyIy9rV0bpKev1q4TtuxWAgyWbsWv38737s1HcPBgzFuYZAqY+rVUDcuYDau9n6ZMnYJFr3J0aYn0unVrkbZ0iaVTyhxjsFTGVWoJ/eD995GUksJkHkncxmxY7R0xJnwMbrnlVojAwK0mgTNnzuH0iTM1Dyr2jauOKOYwVeQuTlumilTqJAGnEZg8eTJeXfQKHo2I5Oo5tah6Kj40U1x+LW/wq1sJ7N6zm88n3eoB653zNqx1Ns48ExEZgRs3KrFr+3ZnNqt8W5wbVnkX0gBnERDJPKHDQ7Hlg/ec1STbsZHAjxUVEO+x1rcxG7Y+Os49N3P6FKQuetW5jSreGoOl4g6kfOcQKCgoMCXzzJo1C3HxE53TKFuxiUD6mrV4MLAHegX1RM+egbhy+YrVesyGtYrGqSeiomPw3cVvkZWd7dR22Zh7CfA2rHv566L3jh06YlnaYsyePVsX9qhihBhRVlaUo7DwKEpLy3DLzc2wZLnljEPehnWtVydPm2JakcS1vcrb24ULF1BaViavQBuUMVjaAIlF6ifg3dIbMbGcuaR+Ss4/e5OnJx6Pia1qOCQkBF5e3lXfRQbikeIi0881VFYd5472BCaMfxwnz51Gfn6+9p0p0MOqVauwZOHLCii1LpHB0jobnqmHQN6ePYiIHFdPCZ5yBQERMM3bkeIS/PHJp8xfMWPGc+jb5w+mH1RUwhP8da+Co/GOt7cXJsbHYekSyyN9jbuXsvmK36h9/amtXspLQv+ixIQDYcOH496O9+rfWIksvP7DDxD/oZt/qkvLzsnBhOhoiFG+ecvM3I5Ll66Yfpo1awY+szSTcc1nbGw89u7PR2FRoWs6lLwX1V8d+fXfUslBU548BMaOGweRzMNnlK71ScVvfoOSI0fqdHrs2FH858f/IDR0eJ1z5gMMlGYSrvsUUz1GRsTgjRVv4o1Vr7uuYwl7mjV3roSqGieJq440jhdLA6Z3KcUfAm7uJyBWoC85cQojR44wLRG1a9cuiDUWa2/NmzfByZNcdaQ2F62/l/2zDEE9AnGoqBA+Pj5ad8f2NSTA27AawtVT02UXLlSZw0BZhcKtOyLDMDIyCpOengg/X1/4+/lh377PLGqq4PNKi1y0Pujb3hdh4aFYuWKl1l2xfY0JcGSpMWA9NL8nPx9DBw/Gp3/7FIE9A/VgkuFs4MjSfS4vPFqIoQ8PxunTp+Hl/eszZfcpcn3PmZs349vvv0dc7K/Z265X4ViPHFk6xk/3tUXW6/ChQzFr5kwGSoW9LUaWzIZ1jwN7dOmBoKAgvPnOO+4RIEGvR4qLUfz55xIosV8Cg6X97AxR8/LXX2PWrOmYm5hoCHv1aqQnKpkN60bnxsbHYfWKFRATSRh1q1D8VV8GS6NeuTbaHT52LGbPZqC0EZe0xfjM0r2uGTliJJo2bYYdWze7V4ibeg8MDEK/fv3c1LtzuuUzS+dw1FUrYlL0q+XlGNR/oK7sMrIxzZs3x8mTp+Hr297IGNxq++r01diy8QN8lPuxW3Wwc/sIcGRpHzfd1hKToj/8cDAOf173fT7dGm0AwypwwwBWym3i49ExOH7qOPbv3S+3UKqzSIDB0iIWYx4sLT2H4cOHYs6cmZg6ZbIxIejUak80ZYKPm30rpiaMjonF6lWr3KzE9d1funwZFy9dcn3HTuyRt2GdCFMPTWVlZ2HYkGF6MIU2VCPQpPlNOHvyLG/DVmPijt2yslIE9gpC4aEi+LRr5w4Jbulz3rx5+O7Cv5Gm8ExGHFm65dKRt1MGSnl9Q2XqE/D19cPAgQOxdlW6+sY00oJKL7VnV2WwbKTD9VZcPKPs2aMbxG0SbvolIF4d4SYHgfi4eLy1/i05xLhQRWW52tcgg6ULLxbZuio8dAiDg4MxfsIEtGrZUjZ51ONEAnx1xIkwHWyqf9/+aPPbO7AxI8PBltSpPmnSZMx8YY46gi0o5TNLC1CMcmj48OHo1+8hzJiRYBSTDWsnXx2Ry/Xpq1Zj3caN2L07Ty5hVGOVAIOlVTT6PyFmE6m+eLD+LTauhU2a/wZnT55ngo8kl4BYm/Tu39+FXR/loUc3rgsriVvqlcHbsPXi0d/Jo8XFVUYxUFah0P2OeHWEmzwEmnt5YVxENN7bsE4eURoqyc7KQsY6tW87M1hqeIHI1nRRYSH69emDjRs3ySaNejQmwEkJNAZsR/NPxkQjPX01rt/Q/4QRBYcOIW/3bjsoyVOFwVIeX2iqRGS9DhgwADNnzURExDhN+2Lj8hHgyFI+n3S5915079od76Yb7zUS+bzRsCIGy4YZ6aJEixbNMDtxLpN5dOHNxhvBkWXjmbmiRmRUFDI26f9OT8eOHdGpW2dXINWsDyb4aIaWDZOAPASYDSuPL2orEZPb7/pwF7r16FH7FL9LRIAjS4mc4WwpBw4VICUl2dnNsj0FCXBkKa/TIiIj8fY778orkMpMBBgsdXohHCooQOiAUJSVfaVTC2lWYwjwmWVjaLm27PhHxmP79s0Qr5PodROvqVVWcgYfvfpXWbvEhRkcHIzo+CgsX75MWTso3HkEOLJ0Hktnt9QzsCf82vthx84dzm5amvYWJSchNiZOGj32COHI0h5qktcR70+KBZxTUlIkV0p5riLAkaWrSNvXT1RMLNas3WBfZSVqecDDU+1wo7Z6JS4S14m8Ul5e1VmHjp2q9rlDAhxZyn0NjBo1CsVHDuDMuTNyC7VTndo3YH82msHSTufLVu1Q4QHc076Dbn/ZZOOtmh6OLOX2WOvWrTCo/xBszdwqt1A71UVGRmDy5El21pajGoOlHH5wSMWBvYcxdPBwjIsKR4B/gENtsbI+CXBkKb9fIyIjkL5Bn1mx/v4B6Natm/xOqEchg2U9cFQ5lbt7OyIjIrA4jck8qvjM1To5snQ18cb312/QIFRcuoHdeWpPC9d4y9WowUkJ1PATVZKAQwS46ohD+FxWWSyX19TTEwuTFrqsT1d0tHv3bly9fBUjwka4ojtN+uDIUhOs2jcq5no9p9NkAO3pGa8HjizV8Plj48djdcYqNcQ2QmV+fj627tjeiBryFWWwlM8nDSoSgXJwcDAydZoM0CAAFmg0AT6zbDQyt1To1rMHfFv5YuvWTLf0z06tE2CwtM5GyjNiPcrhw4ciNiYG06ZNk1IjRclHgCNL+XxiTdGocWOR9+mn1k4rebxdu7Zo376dktrNovnM0kxCkU8xO8+7a9YiLi5WEcWUKQMBPrOUwQu2aSgtPYfAnkE4/+V5iAnwuclBgCNLOfxgswoxOw8Dpc24WPAXAhxZqnMp+Pn5w9fXBzu2bVNHtAGUMlgq4OSiwkKEh4freqJlBdygtEQ+s1TLfVGPRSM7N1ct0TpXy2ApuYOPHTuK4MHB6Nq1E5p7eUmulvJkJcCRpayesaxr5JhHkJ21E1eu6GMlkuSUJMREq/3oiMHS8rUqxdGKyko89eSTiI2Lw/z5SVJoogg1CXBkqZbfAvz94Ofrh9zcbLWEW1FbcaNS+YnUPa3YxsMSEPD08EDm1u3w8fGRQA0lqEyAI0v1vPfooxHIys5GePgY9cTrUDGDpYRO/friRbRr3dqkjIFSQgcpKIkjS/Wc1n/gACxZsRgiA14k9qm8jRwxAt9++53KJoC3YSVzn3hG2bNLF2Tt0O9CsJIhN4QcjizVc3PPnj3QrmVbfJqXo574Woq79eiB/gP71zqq1lcGS4n8JQLlgAEDEB0bg2EjR0qkjFJUJ8CRpZoeHDZqBLJz8tQUrzPVDJYSOfSrL7/C009PQtJCJvNI5BZdSOHIUk03hgwKQe5H6if5iLlhs3epbQeDpUS/Q4NCQpCYmCiRIkrRCwGOLNX05AO9euFq+TUcOHBATQN+US1WHdm8Re35bhks3XwJirleN2/e7GYV7F7vBDiyVNPDIrEneODDOHjwMzUN0JFqBks3OrNETIo+bAj2/f1vblTBro1AgCNLdb3cd+BAbN64SV0DAHh6epp+VDaCE6m7yXsXLlxAjx49ERkxFovTlrlJBbs1CgExIffJk6fh69veKCbrxk7xKlmPLp1x8uRZeHtzFi93OZYjSzeRF+9PrluXzkDpJv5G65YjS3U9Lt657tq1K7ZxYnW3OpHB0o34Q0KGubF3dq0HAl9/8w3i4uMxdOgg7Nq+06pJfGZpFY0SJ/r07o3i4mIltOpVJIOlCz1bVFyMoJ49cfHiJRf2yq70TGDHli2YO3s2Zk6fixGPhOHy5csWzeXI0iIWZQ4OHDQM72/ZoIze2kLT0tIQrfhE6mrPoVTbIxJ/LzpyBKEjhmFg/4Fo1bqVxEopTSUCYlQpNj8/P3Tv1AlNrUyLxpGlSl6tq/WBXoG4fukHlF24AF8F54ouLy9H+Y3yuoYpdIQjSxc5a3naMoQMHIQ1a9dyjkEXMTdSNzt27ERSckqNZdyeefZZBAUFoXevIIg/VtzUJSBeIRk4aBB2bPpAWSNUH5mprl+ZC2d1+mpltFKoWgQ2bd6M997bgOysLJw9+0XVKjXDhg3D/d27m4w5+I/P1TKKausQ6Nq9O0pOn6hzXIUDffr0gX9AgApSrWrkqyNW0Th+Qsz16tveD94tvR1vjC0YnsDlS5dw513/beJQUVFZZ2HgsWPH4L777sfs2bPrsGrevAlOnvyCr47UIaPOgUOHCjFmbBjOnSlVR7SOlPI2rEbOFDPziEnRV69eqVEPbNZoBFq2aoVjR0/j2LGzOHn8bB3zu3e/D506daxzXByo4M1/i1xUOihWISm/Vo7Sc2UqydaNVgZLDVx56sQJDB40CKNHjcJz02Zo0AObNCoBn/bt4OPTFuJTrHM4OyEBRYWFpp8L//4XwsLCLaLxZLC0yEW1g/ff/wBy89SbkLygoABZWVmq4a6hl8GyBg7nfLmtze2YPGkSVq5azT9RzkHKViwQEEkf4mX1T/76V5SWlWHFstfg4WH5V5qvjlgAqOCh4OABKC4uUU55bm4uMtZlKKe7umAm+FSn4aT91q1bYYaF50ZOap7NkEAVgYjIyKr9+nb46kh9dNQ55x/QAZs3qrnwQoU6mC0qtfxvqMWiPEgCJKAqgQpUqiqduqsRCB7wME6dOafkxCaqj8xU11/tMuIuCZCANQKeDJbW0Ch1vLmXF3za+6C4pBh9+/RRRntCQoIyWq0J5cjSGhkeJwEdEWA2rH6c+Ye+fbE3/+/6MUgRSxgsFXEUZZKAIwSYDesIPbnqduvaFYcPF8olygBqGCwN4GSaSALMhtXPNdCpQwecOnVKKYOWr1iO2Ng4pTTXFstgWZsIv5OADgkwG1Y/ThWzNJVdKFUqyeeHKz+g/No1pZ3AYKm0+yieBGwjwJGlbZxUKCWmz/Rt296U5KOCXrNGvjpiJsFPEiABaQlwZCmta+wS5t+pI86cUGdSdTGR+qOjR9llqyyV+OqILJ6gDhLQkED62rdwa9vWGvbApl1JIC4uFv6+HVzZpUN9iWCp+sZVR1T3IPWTAAmQAAloToDPLDVHzA5IgARIgARUJ8BgqboHqZ8ESIAESEBzAgyWmiNmByRAAiRAAqoTYLBU3YPUTwIkQAIkoDkBBkvNEbMDEiABEiAB1QkwWKruQeonARIgARLQnACDpeaI2QEJkAAJkIDqBBgsVfcg9ZMACZAACWhOgMFSc8TsgARIgARIQHUCDJaqe5D6SYAESIAENCfAYKk5YnZAAiRAAiSgOgEGS9U9SP0kQAIkQAKaE/j/9LTmuNlBTasAAAAASUVORK5CYII="></p>
<p>attempt to reflect graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p>smooth curve over the correct domain which does not cross the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and is concave down for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> > 1 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept at 0.607 <em><strong>A1</strong></em></p>
<p>equations of asymptotes given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 3 (the latter must be drawn) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> For <em><strong>FT</strong></em> from (i) to (ii) award max <em><strong>M1A0A1A0</strong></em>.</p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = x"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </math></span> to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0.372 <em><strong>(M1)</strong></em><em><strong>A1</strong></em></p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 0.372 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>FT</strong> </em>marks.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>A particle moves along a horizontal line such that at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≥ 0, its acceleration <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> − 1. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 6 , its displacement <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> from a fixed origin O is 18.25 m. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 15, its displacement from O is 922.75 m. Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to integrate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \int {a\,{\text{d}}t = \int {\left( {2t - 1} \right)} } \,{\text{d}}t">
<mi>v</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>t</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {t^2} - t + c">
<mo>=</mo>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \int {v\,{\text{d}}t = \int {\left( {{t^2} - t + c} \right)} } \,{\text{d}}t">
<mi>s</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{t^3}}}{3} - \frac{{{t^2}}}{2} + ct + d">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mi>t</mi>
<mo>+</mo>
<mi>d</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt at substitution of given values <em><strong>(M1)</strong></em></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 6{\text{,}}\,\,\,18.25 = 72 - 18 + 6c + d">
<mi>t</mi>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>18.25</mn>
<mo>=</mo>
<mn>72</mn>
<mo>−</mo>
<mn>18</mn>
<mo>+</mo>
<mn>6</mn>
<mi>c</mi>
<mo>+</mo>
<mi>d</mi>
</math></span></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 15{\text{,}}\,\,\,922.75 = 1125 - 112.5 + 15c + d">
<mi>t</mi>
<mo>=</mo>
<mn>15</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>922.75</mn>
<mo>=</mo>
<mn>1125</mn>
<mo>−</mo>
<mn>112.5</mn>
<mo>+</mo>
<mn>15</mn>
<mi>c</mi>
<mo>+</mo>
<mi>d</mi>
</math></span></p>
<p>solve simultaneously: <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = - 6{\text{,}}\,\,d = 0.25">
<mi>c</mi>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mo>=</mo>
<mn>0.25</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow s = \frac{{{t^3}}}{3} - \frac{{{t^2}}}{2} + - 6t + \frac{1}{4}">
<mo stretchy="false">⇒</mo>
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>−</mo>
<mn>6</mn>
<mi>t</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l"> <mi>l</mi> </math></span> be the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x{{\text{e}}^{2x}}"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> at the point (1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^2}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </math></span>).</p>
<p>Find the coordinates of the point where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l"> <mi>l</mi> </math></span> meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>equation of tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 22.167 \ldots x - 14.778 \ldots "> <mi>y</mi> <mo>=</mo> <mn>22.167</mn> <mo>…</mo> <mi>x</mi> <mo>−</mo> <mn>14.778</mn> <mo>…</mo> </math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 7.389 \ldots = 22.167 \ldots \left( {x - 1} \right)"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>7.389</mn> <mo>…</mo> <mo>=</mo> <mn>22.167</mn> <mo>…</mo> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.667"> <mi>x</mi> <mo>=</mo> <mn>0.667</mn> </math></span></p>
<p>meets <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis at (0.667, 0)<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \left( {\frac{2}{3},\,\,0} \right)} \right)"> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{2}{3}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.667"> <mi>x</mi> <mo>=</mo> <mn>0.667</mn> </math></span> seen and <em><strong>A1</strong></em> for coordinates (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>, 0) given.</p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p>Attempt to differentiate <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = {{\text{e}}^{2x}} + 2x{{\text{e}}^{2x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1"> <mi>x</mi> <mo>=</mo> <mn>1</mn> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 3{{\text{e}}^2}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>equation of the tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - {{\text{e}}^2} = 3{{\text{e}}^2}\left( {x - 1} \right)"> <mi>y</mi> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 3{{\text{e}}^2}x - 2{{\text{e}}^2}"> <mi>y</mi> <mo>=</mo> <mn>3</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </math></span></p>
<p>meets <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{2}{3}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{2}{3},\,\,0} \right)}"> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{2}{3}"> <mi>x</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.667"> <mi>x</mi> <mo>=</mo> <mn>0.667</mn> </math></span> seen and <em><strong>A1</strong></em> for coordinates (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>, 0) given.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>